Incidence Graphs and Unneighborly Polytopes

vorgelegt von Diplom-Mathematiker Ronald Frank Wotzlaw aus G¨ottingen

Von der Fakult¨at II – Mathematik und Naturwissenschaften der Technischen Universit¨at Berlin zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften – Dr. rer. nat. –

genehmigte Dissertation

Promotionsausschuss: Vorsitzender: Prof. Fredi Tr¨oltzsch Berichter: Prof. G¨unter M. Ziegler Prof. Gil Kalai

Tag der wissenschaftlichen Aussprache: 6. Februar 2009

Berlin 2009

D 83

. . . for though it cannot hope to be useful or informative on all matters, it does at least make the reassuring claim, that where it is inaccurate it is at least deﬁnitively inaccurate. In cases of major discrepancy it’s always reality that’s got it wrong. The Restaurant at the End of the Universe Douglas Adams

For Finja and Nora

Preface

This thesis is about polytopes and “their life in high dimensions.” Nearly all questions considered here are trivial or simple exercises when restricted to dimensions that we can “see.” More often than not, the interesting examples start to appear in dimensions d = 6, 7, 8 or even much higher. Nevertheless, I have tried to illustrate ideas with pictures throughout. The reader should be warned however that they rarely show the real nature of the mathematical problem at hand. Some of the results are generalized to objects that are more abstract than polytopes. I have taken a pragmatical point of view with respect to the level of generality that theorems are stated and proved: If a generalization was asked for in the existing literature, it is provided (if possible). The following is a list of the main results in this thesis in order of appearance: The connectivity of (k,ℓ)-incidence graphs is determined for the case • ℓ 2k + 1 in Chapter 3. Athanasiadis’≥ conjecture on incidence graphs of Cohen-Macaulay cell • complexes is proved in Chapter 4. Linkages in polytopes on at most f0 (6d+7)/5 vertices are analyzed • completely in Chapter 5; this is based≤ on joint work with Axel Werner. Perles’ Skeleton Theorem is proved and generalized in a number of • ways in Chapters 6, 7, and 8. In particular, this theorem is re- established for graded relatively complemented lattices and for pyramidally perfect lattices. A complete solution to Mani’s problem on nonsimplicial illuminated • polytopes is obtained in Chapter 9. Counterexamples to Marcus’ conjecture on positive k-spanning sets • are constructed for all k 2 in Chapter 10. ≥ Bienia & Las Vergnas’ problem on bounding the size of positive k- • spanning sets in oriented matroids is solved in Chapter 11.

vii Apart from these results, many small improvements are made to existing results, and some unpublished results are re-established. The following is a list of minor results (some of them only partially new): Counterexamples to simplex refinements of polytopes with certain • properties are given in Chapter 2. These examples are originally due to Lockeberg, but our analysis of them is simpler. Counterexamples to a strong version of a conjecture about centrally • symmetric polytopes due to Grünbaum are given in Chapter 2. Two generalizations of Balinski’s theorem are combined in Chapter 3 • to yield the so far strongest version of this theorem. A partial answer to a question by Athanasiadis on polytopality of • incidence graphs is given in Chapter 3. Larman & Mani’s lower bound on linkedness of general polytopes is • improved marginally in Chapter 5. This improvement implies exact values for linkedness of d-polytopes for d 5 and for d = 7, 8, 10, 13. ≤ This thesis was written in the time period between April 2006 and De- cember 2008. I had the enormous luck to be financially supported by the Deutsche Forschungsgemeinschaft (DFG) via the Research Training Group Meth- ods for Discrete Structures (MDS) and by the Berlin Mathematical School (BMS). In particular, the additional support for students with children has made my life a lot easier. A number of people had their influence on this thesis, and I want to mention them here. First and foremost, I thank my advisor Günter M. Ziegler. Without him, this thesis simply would not exist. He offered me a scholarship, first on his Leibniz grant and later in the MDS and in the BMS. These scholar- ships made it possible to attend conferences around the world and to visit Julian Pfeifle in Barcelona in the spring of 2008. Günter also helped me get invited to the MSRI Program on Computational Applications of Algebraic Topology in 2006. Besides this, he was responsible for keeping me busy with mathematical problems, helping me out with clever ideas when I got stuck, and for providing a relaxed work environment on floor MA 6-2. I would also like to thank Gil Kalai, who agreed to be co-referee for this thesis on rather short notice. Peter McMullen shared his ideas on unneighborly polytopes and carefully read and responded to mine. He also copied large parts of Lockeberg’s thesis for me and sent them to Berlin. Thanks! Without Christos A. Athanasiadis, Chapters 3 and 4 would not exist. Apart from this, I thank him for coming to Berlin to present his results on

viii connectivity of incidence graphs of polytopes in the MDS lecture on June 16, 2008. Micha A. Perles shared his knowledge on Perles’ Skeleton Theorem during a brief conversation at the “11th Midrasha Mathematicae” in Jerusalem. Many of the figures in this thesis were programmed in PostScript. I learned to program PostScript from Bill Casselmann in a course offered by the BMS. Besides being the math graphics guru, he is also a really nice guy. I received a lot of help from everyone at the BMS office: Nadja Wis- niewski, Anja Bewersdorff, Tanja Fagel, and Mariusz Szmerlo. Julian Pfeifle and his wonderful family, Lourdes, Nina, and Theo, made sure that my family and I had a great time in Barcelona. My friends in the Discrete Geometry Group made my time as a PhD student all the more enjoyable. In a very particular order (beauty? height? in- telligence? age?), they are Axel Werner, Thilo Rörig, Raman Sanyal, Niko- laus Witte, Anna Gundert, Anton Dochtermann, Bruno Benedetti, Carsten Schultz, and Benjamin Matschke. Out of these, Anna, Raman, Anton and Axel deserve a special mentioning: They read parts of this thesis and helped to improve the exposition significantly. Elke Pose was of great help with administrative things. She also made sure that the supply of mineral water did not run out. I had a lot of fun presenting a program on Mathematik im Film together with Thomas Vogt on two occasions during the Jahr der Mathematik (2008). I have written this thesis in LATEX using the excellent memoir class by Peter Wilson. Figures were either programed in PostScript, as mentioned, or drawn in xfig. Life is not all mathematics (or is it?), and some of the things that helped me stay alive during the last years were sports (running in Kleistpark, swimming in Stadtbad Schöneberg, and playing tennis with Till Plumbaum, Axel Werner, Andreas Profous, and Sebastian Stiller), learning Spanish (at the Sprach- und Kulturbörse der TU Berlin in the highly enjoyable courses by Florencia, Carmen, Liliana, and Miguel), the brilliant pieces by Les Luthiers, un grupo de música y humor de Argentina, que he conocido gracias a Florencia, and some of the most interesting cities in the world: San Francisco, Jerusalem, Barcelona, Paris, Amsterdam, Berlin. I also wish to thank my parents for their continuing support. The two most important people in my life deserve the final spot: Nora and my wonderful epsilon Finja.

Berlin, December 2008 Ronald Frank Wotzlaw

Contents

Preface vii

Contents xi

0 Introduction 1

1 The Basics 7 1.1 Graphs...... 7 1.2 PosetsandLattices ...... 10 1.3 Polytopes ...... 13

I Connectivity of Polytope Skeleta 21

2 Refinement Homeomorphisms of Polytopes 23 2.1 Simplex Refinements of Polytopes ...... 27 2.2 Lockeberg’s Counterexamples ...... 30 2.3 Refinements of Centrally Symmetric Polytopes ...... 33

3 Incidence Graphs of Polytopes 37 3.1 Balinski’s Theorem for Polytopes ...... 38 3.2 HigherIncidenceGraphs ...... 41 3.3 Connectivity of Incidence Graphs ...... 43 3.4 Polytopality of Incidence Graphs ...... 48

4 Athanasiadis’ Conjecture on Incidence Graphs 53 4.1 Regular Cell Complexes ...... 54

xi Contents

4.2 Athanasiadis’Conjecture ...... 54 4.3 GraphManifolds ...... 56 4.4 Basic Properties of Graph Manifolds ...... 60 4.5 Connectivity of Graph Manifold Skeleta ...... 64 4.6 Proof of Athanasiadis’ Conjecture ...... 72

5 Linkages in Polytope Graphs 75 5.1 Linkages...... 77 5.2 Simplicial Polytopes and 3-Polytopes ...... 78 5.3 Minimal Linkedness of Polytopes ...... 81 5.4 Linkages in Polytopes with Few Vertices ...... 84 5.5 Minimal Linkedness in Small Dimensions ...... 96

II Perles’ Skeleton Theorem 99

6 Skeleta of Polytopes with Few Vertices 101 6.1 Reconstruction of Skeleta ...... 103 6.2 Perles’ Skeleton Theorem ...... 105

7 Perles’ Skeleton Theorem for Polytopes 109 7.1 Empty Faces in Polytopes ...... 111 7.2 A Bound on the Number of Flat Empty Faces ...... 112 7.3 Disjoint Empty Faces ...... 115 7.4 Pyramidally Inequivalent Complexes ...... 116 7.5 Empty Simplices in Simplicial Polytopes ...... 119

8 Generalizations of Perles’ Skeleton Theorem 121 8.1 Strong Spheres ...... 122 PL 8.2 Empty Faces in Strong Spheres ...... 125 PL 8.3 Simplex Reﬁnements of Spheres...... 129 PL 8.4 Perles’ Skeleton Theorem for Spheres...... 133 PL 8.5 Pyramidally Perfect Lattices ...... 136 8.6 Boolean Intervals ...... 137 8.7 Reconstruction of Skeleta ...... 139 8.8 Relatively Complemented Lattices ...... 141 8.9 EmptyPyramidsinLattices ...... 142 8.10 Empty Pyramids in Upper Intervals ...... 145 8.11 Proof for Relatively Complemented Lattices ...... 148 8.12 Proof for Pyramidally Perfect Lattices ...... 149

xii Contents

III Unneighborly Polytopes 153

9 Nonsimplicial Mani Polytopes 155 9.1 Illuminated Polytopes ...... 157 9.2 Mani’s Simplicial Illuminated Polytopes ...... 160 9.3 Nonsimplicial Mani Polytopes ...... 163

10 Counterexamples to Marcus’ Conjecture 169 10.1 Unneighborly Polytopes ...... 171 10.2 The Sizes of Minimal Positive Spanning Conﬁgurations . . . 172 10.3 Counterexamples to Marcus’ Conjecture ...... 177 10.4 UpperBoundontheSize...... 179

11 Positive Spanning Sets in Oriented Matroids 181 11.1 OrientedMatroids ...... 182 11.2 Oriented Matroids, Polytopes, and Duality ...... 183 11.3 Positive Spanning Sets in Oriented Matroids ...... 185 11.4 The Topological Representation Theorem ...... 186 11.5 Upper Bound on Minimal Positive k-SpanningSets . . . . . 186

Appendix 191

A Poset of Structures 191

Bibliography 193

List of Symbols 203

Index 209

xiii

Chapter 0

Introduction

This thesis revolves around two topics in the theory of polytopes: incidence graphs and missing faces of polytopes. In this introduction we will take a “brief tour” through this thesis and highlight some of the main results. Figure 0.1 shows the relations between the diﬀerent chapters, and we will explain the dependences in the following. A good start into the topic are two articles by Kalai [62] [63]. The article Polytope skeletons and paths [63] from 1997 (updated 2004) is a survey article on topics such as Balinski’s theorem on the connectivity of polytope graphs and the Hirsch conjecture. Part I of this thesis is concerned with connectivity questions of polytope skeleta and contains to a large part results that are related to Kalai’s survey [63]. Balinski’s theorem states that graphs of d-polytopes are d-connected [5]. It has spawned many generalizations, questions, and conjectures. In the work related to Balinski’s theorem we can identify three distinct directions for generalizations. Authors have strived for (a) graph properties that generalize d-connectedness, for example, d-rigidity of polytope graphs [59] [63], subdivisions of complete graphs with two prescribed “principal vertices” [46], or monotone versions of Balinski’s theorem [54]; (b) structures that generalize polytopes, most notably the graph manifolds by Barnette [8]; (c) larger classes of graphs associated to polytopes including the incidence graphs introduced by Sallee [98] and Athanasiadis [3].

1 0. Introduction

Chapter 11

Chapter 8 Chapter 10

Chapter 7 Chapter 9

Chapter 4 Chapter 5 Chapter 6

Chapter 3

Chapter 2

Figure 0.1: Interdependence of chapters. A dashed line means a thematical continuation. A solid line means a strong dependence on results, that is, a main result of the earlier chapter is needed in the later chapter. A curved line means a mild dependence on results. For example, some lemma might be reused in the later chapter.

We will go in all three directions, further in some than in others. Chapter 2 lays the foundation for later chapters. The most important result that we prove in this chapter is a variant of Grünbaum’s theorem that every polytope is a refinement of the simplex of the same dimension [50]. We apply this theorem in Chapters 3, 5, and 7. In Chapter 2 we also introduce a number of “connectivity type” problems and conjectures that would follow from a strengthening of Grünbaum’s theorem. The aforementioned problem on complete graph subdivisions by Gallivan, Lockeberg & McMullen [46] is one of them. We present counterexamples, originally due to Lockeberg [69], to this strengthening. A short side note on refinement homeomorphisms of centrally symmetric polytopes concludes this chapter. Incidence graphs of polytopes are considered under two different aspects in Chapter 3. We define the (k,ℓ)-incidence graph as the graph on the set of k-faces in which two faces are connected by an edge if they lie on a common ℓ-face. These graphs were considered previously by Sallee [98] and Athanasiadis [3]. We determine the connectivity of (k,ℓ)-incidence graphs for ℓ 2k + 1. ≥ 2 Balinski’s theorem serves as an important ingredient in the proof of this result, and we establish a strong geometric version of d-connectedness of polytopes. Polytopality of incidence graphs is also discussed briefly in this chapter. Athanasiadis [4] had asked for (k,ℓ)-incidence graphs that are also graphs (that is, (0, 1)-incidence graphs) of other polytopes and we provide some interesting examples. A different question by Athanasiadis will keep us occupied throughout Chapter 4. He determined the connectivity of (k, k + 1)-incidence graphs of polytopes [3] and, motivated by a result by Fløystad [41], conjectured that his result extended to Cohen-Macaulay regular cell complexes with intersection property [3, Conjecture 6.2]. This conjecture is established in this chapter by considering the question stripped off its algebraic ballast in the context of Barnette’s graph manifolds. The last chapter of Part I deals with linkages in polytope graphs. Link- ages are an important concept in graph theory related to the work of Robert- son & Seymour [95] on graph minor theory. In polytope graphs they were first considered by Larman & Mani [66]. We prove some new results on linkages in Chapter 5, which is based on joint work with Axel Werner. The main results of this chapter were published in [116]. I believe that Chapter 5 is a complete account of everything that is known to date about linkages in polytopes. In particular, linkages in polytopes with “few vertices” are discussed in great detail. Some aspects of the analysis in Chapter 5 hint towards a theorem by Perles on skeleta of polytopes with few vertices [89], and they lead us gently into Part II of this thesis. This theorem by Perles, which we call from now on Perles’ Skeleton Theorem (following Kalai [62]), appears in the other article by Kalai that I had mentioned above, Some aspects of the combinatorial theory of convex polytopes [62] from 1994. This article served as an important inspiration for Part II of this thesis. Perles’ Skeleton Theorem states that the number of combinatorial types of k-skeleta of polytopes on d + γ + 1 vertices is bounded by a function of k and γ that is independent of d. With little knowledge of polytopes with few vertices one can easily prove Perles’ Skeleton Theorem for γ = 1, that is, for d-polytopes on d+2 vertices. The characterization of these polytopes [51, Theorem 6.1.4], which is for example implied by a theorem we prove in Chapter 5, says that every such polytope has the combinatorial type of an iterated pyramid over a direct sum of two simplices. The k-skeleta of these polytopes are easily listed. For example, there are only three types of graphs (1-skeleta) that can appear, the complete graph on d + 2 vertices, the complete graph minus one edge,

3 0. Introduction and the complete graph minus two disjoint edges. In contrast, there are d2/4 combinatorial types of d-polytopes on d + 2 vertices. ⌊ ⌋ Chapter 6 is a brief introduction to topics related to Perles’ Skeleton Theorem. In particular, the problem of bounding the number of combinatorial types is reduced to the problem of bounding the number of empty pyramids of bounded dimension. In Chapters 7 and 8 we give in total four proofs of Perles’ Skeleton Theorem. The mostly geometric proof given in Chapter 7 is generalized to strong spheres (that is, cellular spheres with intersection property) in ChapterPL 8. PL Chapter 8 also contains proofs of this result for two types of lattices that both generalize face lattices of polytopes: the class of graded relatively complemented lattices, and even more general, the class of pyramidally perfect lattices. Although the proofs given in Chapters 7 and 8 are distinct enough to really consider them as different, they also share some characteristics. Three out of the four proofs can be roughly divided into a local and a global part. The local part consists mainly in understanding the behaviour of empty pyramids in quotients. The global part, which in all three cases is an induction by taking vertex figures or upper intervals, puts together this local information in a careful way. We have to ensure that we count all the missing faces, but that we do not count too many too often (remember that we want to bound the number of empty pyramids by a function that does not depend on d). There are different tools that may be used for the global part, among them: (a) Grünbaum’s theorem on simplex refinements [50], which we also generalize to strong spheres, (b) the Erd˝os-RadoPL Sunflower Lemma [40], as was done by Kalai [62], and (c) an innocent looking lemma on “large simplex faces” of polytopes with few vertices, see Lemma 5.4.2, that generalizes to “large boolean intervals” in pyramidally perfect lattices. These tools yield the different bounds on the size of the set of vertices in empty pyramids, which we call the kernel, as summarized in Figure 0.2. The proof for pyramidally perfect lattices is slightly different, as the technique of induction by taking quotients does not work here. Nevertheless, the “large boolean intervals” play an important role here as well. In Chapter 7 we also consider the related problem of bounding the number of disjoint empty faces. In this case we even get a characterization of the extremal examples by applying a theorem that we already proved and

4 structure technique bound on kernel size k 1 polytopes refinement (k + γ 1)γ(γ + 1) − − k+γ 1 sunflower (k + γ 1)! (2γ) − − k 1 strong sphere refinement (k + γ 1)γ(γ + 1) − PL − relatively complemented boolean interval (k + γ 1)(2γ)k lattices − .2 pyramidally perfect lat- .. boolean interval 22 tices 2k+γ 1 − | {z } Figure 0.2: Different proof methods for Perles’ Skeleton Theorem and the resulting bounds on the kernel size. Here, k denotes rank. used in a different context in Chapter 5. We also give a complete list of graphs of d-polytopes on d + 3 vertices. We study unneighborly polytopes in Chapters 9, 10, and 11. These chapters make up Part III of this thesis. A problem by Mani on illuminated polytopes [70], which are a special case of unneighborly polytopes, is solved in Chapter 9. Hadwiger had conjectured that an illuminated d-polytope, that is, a d- polytope in which every vertex lies on an inner diagonal, has at least 2d vertices. This was disproved by Mani [70]. Mani [70] also gave a construction for simplicial illuminated polytopes on the minimum number of vertices possible and asked for nonsimplicial examples. We provide these examples and prove for which dimensions only simplicial ones exist. Chapter 10 deals with a conjecture by Marcus on the size of minimal positive k-spanning vector configurations. This conjecture can be interpreted via Gale duality in terms of unneighborly polytopes. It is disproved in the special case k = 2 by Mani’s illuminated polytopes on the minimum number of vertices. We construct from these examples counterexamples to Marcus’ conjecture for all k 2. We also identify special classes of unneighborly polytopes ≥ that satisfy the conjectured bound, and we prove a general bound on the size of minimal positive k-spanning configurations by using Perles’ Skeleton Theorem. The problem that we consider in Chapter 11 is a similar question on the size of minimal positive k-spanning sets in oriented matroids. It was posed by Bienia & Las Vergnas [23, Exercise 9.35(iv)*].

5 0. Introduction

Once again an application of Perles’ Skeleton Theorem yields a solution to this problem. We achieve the best bound on the size with the version for strong spheres via the Topological Representation Theorem for oriented matroidsPL by Folkman & Lawrence [42], Edmonds & Mandel [39], and Lawrence [68]. The appendix contains a poset that relates (by inclusion) the diﬀer- ent structures and combinatorial abstractions of geometric objects that are relevant to this thesis. Questions, open problems, and conjectures are scattered throughout the text.

6 Chapter 1

The Basics

This chapter collects general notations and notions used throughout this thesis. Special terminology, or terminology that pertains to only a single chapter, is defined when it first appears. The material on graph theory is based on the third edition of Diestel’s graph theory book [38]. The part on posets and lattices is taken from Stan- ley’s book [107, Chapter 3] and partly from Ziegler’s book on polytopes [118, Definition 2.5]. The basics in polytope theory are drawn from the books by Ziegler [118] and Grünbaum [51]. The reader acquainted with these sources should have no difficulty following later chapters without having read this one. We write R for the real numbers, N for the natural numbers, and we use the notation [n] := 1,...,n . { }

1.1 Graphs

A graph G is a pair (V,E) consisting of a ﬁnite set V , the vertices of G, and a set E V , the edges of G. If G is a graph with vertices V and edges E, ⊆ 2 we write V (G) := V and E(G) := E. We write uv := u, v for the edge ¡ ¢ u, v . The vertices u and v are called the endpoints of{ the edge} uv. { } Two graphs G1 and G2 are called isomorphic if there is a bijection φ : V (G1) V (G2) such that uv E(G2) if and only if φ(u)φ(v) E(G2). In this thesis,→ we will usually not∈ distinguish between isomorphic∈ graphs. In particular, we will write Kn for any graph that is isomorphic to the

7 1. The Basics

(a) A graph. (b) A subgraph that is a cy- (c) The subgraph induced cle, but not induced. by the vertices of the cycle.

Figure 1.1: Subgraphs and induced subgraphs.

[n] complete graph G = ([n], 2 ) on n vertices. A graph is called bipartite if it admits a partition of its vertex set into ¡ ¢ two classes such that every edge has its endpoints in diﬀerent classes. We write Km,n for the complete bipartite graph. This is the bipartite graph on vertex set V = V V with V = m, V = n in which all edges 1∪· 2 | 1| | 2| between V1 and V2 exist.

1.1.1 Subgraphs

If G1 and G2 are graphs with V (G1) V (G2) and E(G1) E(G2), then G is a subgraph of G . If E(G )= uv⊆ E(G ) : u, v V (⊆G ) , then the 1 2 1 { ∈ 2 ∈ 1 } graph G1 is an induced subgraph of G2. Let G =(V,E) be a graph and v V . We write N(v) for the neighbors of v, that is, N(v) := u V : uv ∈E . A path of length k is{ a∈ nonempty∈ graph} P =(V,E) of the form

V = v0,v1,...,vk , E = v0v1,v1v2,v2v3,...,vk 1vk , { } { − } where the vi are all distinct. We say that the path P connects or joins v0 and vk. The vertices v1,...,vk 1 are the inner vertices of P . We − will often notationally orient the path P by writing P as v0v1 ...vk 1vk − or (v0,e1,v1,...,vk 1,ek,vk), where ei = vi 1vi for every i = 1,...,k. If P1 − − and P2 are paths such that their union is a path as well, then we call this union the concatenation of P1 and P2 and denote it by P1P2.A subpath of a path P is a subgraph of P that is a path. Distinct paths P1, P2,...,Pm are independent if none of them contains an inner vertex of one of the others. A walk of length k is a sequence (v0,e1,v1,...,vk 1,ek,vk of vertices − such that e = v v is an edge for every i = 0,...,k 1. Clearly, a walk k i i+1 − contains a path that connects v0 to vk.

8 1.1. Graphs

(a) The join operation applied to a cycle (b) The join operation in the comple- of length 5 and a path of length 3. ment.

Figure 1.2: The join of two graphs.

A cycle of length k is a graph of the form

V = v0,v1,...,vk , E = v0v1,v1v2,v2v3,...,vk 1vk,vkv0 , { } { − } where the vi are all distinct. Figure 1.1 shows a graph, a cycle of length 4 in that graph, and the subgraph induced by the vertices of the cycle.

1.1.2 Connectivity A graph G is connected if for every two vertices u, v V (G) there is a path in G that connects u and v. A graph G is k-connected∈ if V (G) k + 1 and the graph G U is connected for every set U V (|G) of| size≥ less than k. The greatest\ integer k such that the graph G ⊆is k-connected is the connectivity κ(G). We need the following important result by Menger [80]; see also [38, Theorem 3.3.6]. Theorem 1.1.1 (Menger’s theorem [80]). A graph G is k-connected if and only if for every two vertices u and v there are k distinct independent paths connecting u and v.

1.1.3 Graph constructions If G and G are graphs with disjoint vertex sets we write G G for the 1 2 1 ∗ 2 join of G1 and G2 deﬁned by the vertex set V (G H) := V (G) V (H) and the set of edges ∗ ∪

E(G H) := E(G) E(H) vv′ : v V (G),v′ V (H) . ∗ ∪ ∪ { ∈ ∈ } 9 1. The Basics

u (a) A 3-connected graph with three independent paths drawn between the vertices (b) The complete graph on 4 vertices is a u and v. topological minor of the graph shown. A subdivision of K4 is drawn bold.

Figure 1.3: Connectivity and topological minors.

The complement graph G of a graph G is the graph on vertex set V (G) in which uv is an edge if and only uv is not an edge of G. The join operation is exempliﬁed in Figure 1.2(a). The displayed graph is the join of a cycle of length 5 and a path of length 3. In most cases it is more transparent to display the result in the complement graph. The complement of the join is the disjoint union of the complements; see Fig- ure 1.2(b).

1.1.4 Minors A graph G is a minor of a graph H if G is obtained from H by deleting vertices and contracting edges. A graph G′ is a subdivision of G (or a reﬁnement) if G′ is obtained from G by replacing edges by pairwise independent paths (possibly of length one). If G′ is a subgraph of another graph H then G is said to be a topological minor of H. A topological minor is always also a minor, but not conversely.

1.2 Posets and Lattices

A finite partially ordered set (or short poset) is a pair (L, ) consisting of a finite set L and a binary relation , the partial order on≤L, that satisfies the following three axioms: ≤ (i) For all x L we have x x (reflexivity). (ii) If x y and∈ y x, then≤x = y (antisymmetry). (iii) If x ≤ y and y ≤ z, then x z (transitivity). ≤ ≤ ≤ 10 1.2. Posets and Lattices

We simply write L for the poset (L, ). ≤ Let L1 and L2 be partially ordered sets. Then L1 and L2 are isomorphic, in notation L1 ∼= L2, if there is an order-preserving bijective map π : L1 L whose inverse is again order-preserving, that is, x y in L if and only→ 2 ≤ 1 if π(x) π(y) in L2. If L≤is a poset, we denote by Lop the opposite poset. This is the poset on the set L such that x y in Lop if y x in L for all x, y L. ≤ ≤ ∈

1.2.1 Intervals, Chains, and Rank For x, y L with x y we write ∈ ≤ [x, y] := z L : x z y { ∈ ≤ ≤ } and call [x, y] the (closed) interval between x and y. An element y is said to cover an element x, denoted by x ⋖ y, if x < y and [x, y]= x, y . The poset{ L has} a 0ˆ if there is an element 0ˆ L with 0ˆ x for all x L. It has a 1ˆ if there is an element 1ˆ L with∈ x 1ˆ for≤ all x L. Intervals∈ of type [0ˆ, x] with x L are called∈ lower intervals≤ , intervals∈ of type [x, 1]ˆ are called upper intervals∈ . A chain in a poset L is a totally ordered subset X L. The length of a chain X is X 1. The poset L is graded if every maximal⊆ chain has the same length.| If|L − is graded, the rank rk(x) of x L is deﬁned to be the maximum length of a chain in the interval [0ˆ, x].∈ The rank of L, denoted by rk(L), is the maximum length of a chain in L.

1.2.2 Lattices If every two elements x and y of a poset L have a unique upper bound x y, the join of x and y, we say that L is a join-semilattice. If every two elements∨ x and y have a unique lower bound x y, the meet of x and y, we say that L is a meet-semilattice. ∧ A lattice is a ﬁnite partially ordered set that is both a join-semilattice as well as a meet-semilattice. The following is a useful criterion for telling when a poset is a lattice.

Proposition 1.2.1 (see [107, Proposition 3.3.1]). Let L be a ﬁnite meet-semilattice with 1ˆ. Then L is a lattice. Dually, a join-semilattice with 0ˆ is a lattice.

11 1. The Basics

1.2.3 Atoms and Coatoms

If L is a meet-semilattice, which has a 0,ˆ then the atoms (L) of L are the elements that cover 0.ˆ If L is a join-semilattice, whichA has a 1,ˆ then the coatoms of L are the elements that are covered by 1.ˆ A lattice L is atomic if every element x = 0ˆ is the join of k 1 6 ≥ atoms, that is, there are a1,...,ak (L) with x = a1 a2 ak. It is coatomic if every element x = 1ˆ is∈ the A meet of k 1 coatoms.∨ ∨···∨ We write (x) for the atoms6 below element x. ≥ A

1.2.4 Boolean lattices

Let A = a ,...,a be a set of n elements. We denote by B(A) or { 1 n} B(a1,...,an) the boolean lattice of rank n on atoms a1,...,an, that is,

B(a ,...,a ) := B(A) := (2A, ). 1 n ⊆

We say that L is boolean if it is isomorphic to a boolean lattice.

1.2.5 Relatively complemented lattices

A lattice satisﬁes the diamond property if every interval of length 2 is boolean, that is, if it has exactly 4 elements. A lattice L is complemented if for each element x there is an element y such that x y = 1ˆ and x y = 0.ˆ It is relatively complemented if every interval [x, y∨] L is complemented.∧ The following⊆ theorem gives a useful criterion for deciding when a given lattice is relatively complemented. It implies that every lattices that satis- ﬁes the diamond property is also relatively complemented.

Theorem 1.2.2 (Bj¨orner [20]). A ﬁnite lattice is relatively complemented if and only if every interval of length 2 has cardinality at least 4. Every relatively complemented lattice is both atomic and coatomic.

There are indeed lattices that do not satisfy the diamond property, but neverthelesss are relatively complemented. One such lattice is shown in Figure 1.4.

12 1.3. Polytopes

(a) A boolean interval. (b) An interval of length 2 that is not boolean. Figure 1.4: A lattice that does not satisfy the diamond property, but that is relatively complemented.

1.3 Polytopes

Let V be a subset of Rd. We denote by k k aff(V ) := λ v : v ,...,v V, λ = 1 , { i i { 1 k} ⊆ i } i=1 i=1 Xk X cone(V ) := λ v : v ,...,v V, λ 0 , { i i { 1 k} ⊆ i ≥ } i=1 Xk k conv(V ) := λ v : v ,...,v V, λ = 1, λ 0 { i i { 1 k} ⊆ i i ≥ } i=1 i=1 X X the affine hull, conical hull or cone, and convex hull, respectively, generated by the set V . An affine k-dimensional subspace of Rd, which can always be written as the affine hull of k + 1 affinely independent points, will be called a k-flat. A k-flat A and an ℓ-flat B are skew if dim(aff(A B)) = k + ℓ + 1. A polytope P is the convex hull of a finite set∪ of points of Rd. Thus if V = v ,...,v is a set of n points, we can write a polytope as { 1 n} n n P = conv(V )= λ v : λ = 1, λ 0 . { i i i i ≥ } i=1 i=1 X X Equivalently, we can write every polytope as the bounded intersection of a finite number of halfspaces—this is the Main theorem for polytopes; see [118, Theorem 1.1]. Using this equivalence, one easily shows that the intersection of a polytope with an affine space is a polytope and that the projection of a polytope is again a polytope.

13 1. The Basics

The dimension dim(P ) of a polytope P is the dimension of its aﬃne hull, and we say that P is a dim(P )-polytope. If d = dim(P ), the polytope P is full-dimensional. A face of a polytope is a set of the form

F = P x Rd : aT x = b , ∩ { ∈ } where aT x b is a valid inequality for P . With this definition, the polytope P itself≤ and the empty set are faces—they are given by the valid inequalities 0T x 0 and 0T x 1,∅ respectively (the symbol 0 denotes the column vector in≤ which every entry≤ is zero). A proper face is a face that is not P , and is the only trivial face. Clearly, a face of a polytope is a ∅ polytope. The faces of dimension 0, 1, dim(P ) 2, and dim(P ) 1 are called vertices, edges, ridges, and facets, respectively.− We write −(P ) for the set of vertices of P . V If P is a d-polytope in Rd, then int P is the set of interior points of P , that is, the set of all points that do not lie on a face of dimension smaller than d. If P is a k-polytope that is embedded in a k-flat H of Rd, then the relative interior points of P are the interior points of P considered as a Rk full-dimensional polytope in H ∼= . A flag in a polytope is a sequence of faces such that every face is properly contained in the next face of the sequence. The set of faces of P ordered by inclusion forms a poset (P ), called L the face poset of P . The standard d-simplex is the polytope

∆ := conv e ,e ,...,e Rd+1 . d { 1 2 d+1} ⊂ This is a d-dimensional polytope. The standard d-cube is the polytope

C := conv +1, 1 d Rd . d {{ − } } ⊂ We also denote the 2-cube by ¤. This polytope is also called a quadrilateral. The standard d-crosspolytope is the polytope

C∆ := conv e , e ,..., e Rd . d {± 1 ± 2 ± d} ⊂ ∆ In dimension d = 3, we will also call C3 the octahedron.

1.3.1 Vertex Figures and Face Figures Let P be a d-polytope in Rd, where d is at least one. Let v be a vertex of P , and take an inequality aT x b that deﬁnes this vertex, so that we ≤ 0 14 1.3. Polytopes

d T have v = P x R : a x = b0 . If b1 < b0 is a real number such that aT p <{ b} for all∩p { ∈ (P ) v , we} call the polytope 1 ∈ V \ { } P/v := P x Rd : aT x = b ∩ { ∈ 1} a vertex figure of P at v. This is a full-dimensional polytope in the (d 1)- Rd T Rd 1 − dimensional space x : a x = b1 ∼= − . This construction{ generalizes∈ as follows:} Let F be a face of P of dimension k 0. If dim F = 0, then P/F is the vertex figure at F . Otherwise, choose≥ a vertex v of F and inductively define P/F :=(P/(F/v))/v. We call P/F a face figure of P at F . The polytope P/F is a polytope of dimension d k 1. It can be obtained from P by a cut with a suitable (d k 1)-flat. − − − −

1.3.2 Polarity Every polytope P has a polytope that has an anti-isomorphic poset. If we assume that P contains the origin in its interior and is given by

P = x Rd : Ax b , { ∈ ≤ } where A is a real m d matrix and b Rm, then the polar of P is the polytope × ∈ P ∆ = y Rd : y = cT A, c 0,cT 1 = 1 ; { ∈ ≥ } see [118, Section 2.3]. The face poset of P ∆ is anti-isomorphic to the poset of P . This gives a new interpretation for face ﬁgures: Taking a face ﬁgure of P corresponds to taking a face of P ∆. We write F 3 for the face of P ∆ that corresponds to the face F of P .

1.3.3 Properties of the Face Lattice The face poset of a polytope has a number of speciﬁc properties that we list in the following theorem.

Theorem 1.3.1 (see [118, Theorem 2.7]). Let P be a d-polytope.

(i) The face poset (P ) is a lattice. We will therefore also speak of it as the face latticeL of P .

(ii) The lattice (P ) is graded with rank function rk(F ) := dim(F ) + 1. L (iii) The lattice (P ) is atomic and coatomic. L 15 1. The Basics

(iv) The lattice (P ) has the diamond property, that is, every interval of length 2 hasL exactly 4 elements.

(v) The opposite lattice (P )op is the face lattice of the polar polytope P ∆. L (vi) Every interval [F,G] is the face lattice of a (dim(G) dim(F ) 1)- − − polytope, namely the face lattice of G/F .

1.3.4 Combinatorial Types

Two polytopes P1 and P2 are combinatorially isomorphic if their face lattices (P1) and (P2) are isomorphic. AllL the followingL definitions are clearly invariant under combinatorial isomorphisms. ∆ Every polytope that is combinatorially isomorphic to ∆d, Cd, or Cd is called a d-simplex, d-cube, or d-crosspolytope, respectively. A polytope P is simplicial if all its facets are simplices, that is, every facet contains exactly dim(P ) many vertices. It is simple, if every vertex is contained in exactly d facets. A polytope is simplicial if and only if its polar polytope is simple. A polytope is ℓ-simplicial, if all faces up to dimension ℓ are simplices. (The polar notion of ℓ-simplicity does not appear in this thesis.) We define a vector f(P )=(f 1, f0,...,fd 1) with integral entries by − − letting fk be the number of k-faces of P . Thus f 1 = 1, f0 is the number of − vertices, f1 is the number of edges, fd 2 is the number of ridges, and fd 1 − − is the number of facets. The vector f(P ) is the f-vector of the polytope P . We write fk(P ) for the kth entry of f(P ). A related notion is the h-vector. Given the f-vector of a simplicial polytope, the h-vector is defined componentwise by

k k i d i hk := ( 1) − − fi 1, − d k − i=0 X µ − ¶ for every k 0,...,d . ∈ { } Finally, we define the g-vector for simplicial polytopes by g0 := h0 = 1, and gk := hk hk 1 for every k 1,..., d/2 . − The definition− of the h- and∈g {-vectors⌊ for⌋} general polytopes is much more delicate; see for example [62, Section 4] and [13]. As we will we not be dealing with higher entries of the g-vector, we define γ(P ) := f d 1= 0 − − g1(P ).

16 1.3. Polytopes

1.3.5 Gale Duality A Gale diagram of a polytope on d+γ +1 vertices is a vector conﬁguration, that is, a ﬁnite multiset of vectors, in Rγ that encodes the combinatorial data of the polytope. For detailed information on Gale duality, in particular, how they are constructed, we refer the reader to the extensive survey by McMullen [78], the exposition in [118] together with a crash course in oriented matroid theory, or the elementary account in [74]. Gale duality of polytopes is intimately related to the notion of positively spanning.

Deﬁnition 1.3.2 (Positively spanning). Let V =(v1,...,vn) be a vector conﬁguration in Rr. We call V positively spanning for a vector space W Rr if ⊆ cone(V )= t v + + t v :(t ,...,t ) 0 = W, { 1 1 ··· n n 1 n ≥ } that is, nonnegative combinations of the vectors in V span the space W .

Here we state the fundamental principle for reading oﬀ the combinatorics of a polytope from one of its Gale diagrams.

Theorem 1.3.3. Let P be a d-polytope on n = d + γ + 1 vertices v1,...vn, and let A = (a1,...,an) be a Gale diagram of P , where the ai are vectors in Rγ. Then I [n] is the index set of a face F of P , that is, the set of vertices ⊆ of F is exactly vi : i I , if and only if the set A′ = aj : j / I is positively spanning{ for its∈ linear} span. { ∈ }

Figure 1.5 displays a Gale diagram in R2, where we indicate how many times a vector appears in the configuration by a label next to it. We see that the corresponding polytope has 4 2 = 8 vertices, and so its dimension is d = 8 3 = 5. Theorem 1.3.3 implies· that two vertices that correspond to the two− copies of a vector in Figure 1.5 do not form an edge of the polytope. γ For d 0, γ 0 we define d to be the class of d-polytopes on d+γ +1 ≥ ≥ γ P γ vertices. Further define d to be the subclass of d of simplicial polytopes. All the polytopes in S γ have γ-dimensional GaleP diagrams. Pd

1.3.6 Polytope Constructions If P and P are two polytopes of dimensions d and e, then P P denotes 1 2 1 × 2 the product of P1 and P2, obtained by taking the cartesian product of P1 and P2. This is a (d + e)-dimensional polytope. If P2 = I is a line segment,

17 1. The Basics

2 2

Figure 1.5: Gale diagram of a 5-polytope on 8 vertices with 4 disjoint missing edges.

then the product P1 I is also called the prism over P1. Iterating the prism with I = [ 1, 1] starting× from a point, we get a translation of the standard d-cube. − By P1 P2 we denote the join of two polytopes P1 and P2. This operation was apparently∗ first introduced in [105]; see also [51, Exercise 4.8.1]. Geometrically, the join of polytopes P1 and P2 is obtained by placing them in skew affine spaces of dimensions dim(P1) and dim(P2), respectively, and taking the convex hull. Combinatorially, the faces of P P are the joins of faces of P and P . 1 ∗ 2 1 2 The dimension of P1 P2 is dim(P1) + dim(P2) + 1. The join specializes∗ to the operation of taking a pyramid over a polytope. If P isa(d 1)-polytope in a hyperplane of Rd, and a is a point outside of − that hyperplane, then pyra(P ) := (conv P a )= P a is the pyramid over P with apex a and base P . We will∪ { also} simply∗ { write} pyr(P ) for a pyramid over P , and we call a polytope that is a pyramid pyramidal. Furthermore, we denote by pyrk(P ) the k-fold pyramid over P , that is, the polytope obtained by iterating the pyramid operation k times. A (d+1)-fold pyramid over the empty set is a d-simplex. For faces F P and G P we denote by (P ,F ) (P ,G) the subdirect ⊆ 1 ⊆ 2 1 ⊕ 2 sum of the polytopes P1 and P2 with respect to F and G [77]. This operation can be realized geometrically by placing P1 and P2 in affine spaces that intersect exactly in one point x0 that is a relative interior point of F and G. The combinatorial description is given by the following proposition.

Proposition 1.3.4 (McMullen [77]). The faces of (P ,F ) (P ,G) fall 1 ⊕ 2 18 1.3. Polytopes into two categories. A face is either given by

(i) F G , where F F F and G G G, or 1 ∗ 1 1 ∩ ⊂ 1 ∩ ⊂ (ii) (F ,F ) (G ,G), where F F and G G . 1 ⊕ 1 ⊆ 1 ⊆ 1 We call the special subdirect sum (P ,v ) (P , P ), where v is a vertex 1 0 ⊕ 2 2 0 of P1, a vertex sum. If P2 is a line segment, a vertex sum is called a vertex split. We write P P := (P , P ) (P , P ) and call this the direct sum of 1 ⊕ 2 1 1 ⊕ 2 2 the polytopes P1 and P2; see [51, Exercise 4.8.4]. Combinatorially, the boundary of P1 P2 is the join of the boundaries of P and P , that is, the proper faces of⊕ P P are the joins of proper 1 2 1 ⊕ 2 faces of P1 and P2. The dimension of P1 P2 is dim(P1) + dim(P2). A special case of the direct sum is the⊕ sum of a polytope P and an interval I, which yields the bipyramid bipyr P = P I. We call P a base of the bipyramid. The two vertices of I are the apexes⊕ of the bipyramid. Iterating the bipyramid with I = [ 1, 1] starting from a point, we obtain a translation of the standard d-crosspolytope.−

1.3.7 Polytopal Complexes Definition 1.3.5 (Polytopal complex). A polytopal complex in Rd is a finite collection of polytopes, called the faces of , in Rd such thatC C (i) the empty set is in , ∅ C (ii) if P and F P is a face of P , then F , and ∈ C ⊆ ∈ C (iii) the intersection P Q of polytopes P, Q is a face of both P and Q, that is, the complex∩ satisfies the intersection∈ C property.

The dimension of a polytopal complex is the maximum over the dimensions of its faces.

We write ( ) for the set of vertices of , that is, ( )= P (P ). V C C V C ∪ ∈C V Special cases of polytopal complexes are the complex of all faces of a polytope and the boundary complex (P ) of a polytope, that is, the complex of all proper faces of a polytope. B Let be a polytopal complex. The face poset ( ) of is the set of facesC of partially ordered by inclusion. Two polytopalF C complexesC are combinatoriallyC isomorphic if their face posets are isomorphic.

19 1. The Basics

Figure 1.6: The vertex ﬁgure at the pulled vertex is isomorphic to the link.

A subcomplex of a polytopal complex is a subset of its faces that is again a polytopal complex. A polytopal complex has certain distinguished subcomplexes that we define in the following. d Let be a polytopal complex in R . The k-skeleton skelk( ) is the polytopalC complex of faces of of dimension at most k, that is, C C skel ( ) := F : F , dim(F ) k . k C { ∈ C ≤ } Other important subcomplexes are the star, antistar, and the link1. They are defined in the following definition.

Definition 1.3.6 (Star, antistar, link). Let be a polytopal complex and let F be a face of . We define C C the star of F by • star (F ) := G : there is H such that F H and G H , C { ∈ C ∈ C ⊆ ⊆ } the antistar of F by ast (F ) := G : F G , • C { ∈ C 6⊆ } and the link at F by link (F ) := ast (F ) star (F ). • C C ∩ C The link of a vertex in the boundary complex of a d-polytope is a polytopal complex that is combinatorially isomorphic to the boundary complex of a (d 1)-polytope. It is isomorphic to the vertex figure after “pulling” − the vertex; see Figure 1.6. If is a set of faces of a polytopal complex, we denote by = FR ⊆d the C underlying space of . || F || ∪ F ⊂ F

1There seems to be no consensus on the meaning of the term link in the polytope literature. For example, Billera & Björner define this term to mean what we call face figure [16, p. 421]. We use it in the same way as Ziegler [118, p. 237].

20 Part I

Connectivity of Polytope Skeleta

Chapter 2

Reﬁnement Homeomorphisms of Polytopes

A refinement homeomorphism between polytopes P and Q is a homeomorphism φ : P Q that maps any face of P to a union of faces of Q; see also Definition 2.1.1.→ Refinement homeomorphisms of polytopes are useful for a number of purposes. In this thesis, they are used to determine the connectivity of incidence graphs of polytopes in Section 3.3, to prove a lower bound on the linkedness of general polytopes in Section 5.3, and to prove Perles’ Skeleton Theorem; see Chapters 7 and 8, in particular Sections 7.2 and 8.4. The basic statement needed for these applications, Grünbaum’s theorem that every d-polytope is a refinement of the d-simplex [50], is proved in this chapter. Our version asserts that one may preassign a flag of faces as principal faces of the refinement, as was proposed by Grünbaum [51, Exercise 11.1.3], and certain technical conditions that we need in Chapter 3 and Chapter 7. (A principal face is a face such that the preimage in ∆d is a face.) Because of the usefulness of this theorem some authors have asked for extensions. The following conjecture was raised for example in Grünbaum’s book in a stronger version [51, Exercise 11.1.4], and by Jockusch & Prabhu; see [56, Conjecture 3].

Conjecture 2.0.1 (disproved [69]). Let P be a d-polytope and v1,v2 be distinct vertices of P . Then there is a reﬁnement homeomorphism from the d-simplex to P , such that v1 and v2 are principal faces of the reﬁnement.

23 2. Refinement Homeomorphisms of Polytopes

This conjecture, if it were true, would have implied solutions to a number of problems or conjectures that we discuss in the following. The first one is the restriction of Conjecture 2.0.1 to the 1-skeleton. Problem 2.0.2 (Gallivan, Lockeberg, and McMullen [46]). Let P be a d-polytope. Does G(P ) contain, for every x, y V (P ), a refinement of ∈ Kd+1 in which x and y are both principal? The answer to this question is affirmative in the special case of simplicial polytopes. This was shown by Larman & Mani [66]. Gallivan, Lockeberg & McMullen [46] have shown that, in general, one may not preassign three arbitrary vertices of P as principal vertices of a Kd+1- refinement. Precisely, they have proven that (a) if d = 3 one may preassign three vertices as principal, but not four, and (b) for all d 4, there ≥ is a d-polytope P with distinct vertices x, y, z that cannot all be principal in any Kd+1-refinement in G(P ). The second problem is Lockeberg’s conjecture. To state it concisely we need the following definition of strong chains. Definition 2.0.3. Let P be a d-polytope and 0 k d.A strong k-chain ≤ ≤ C in P is a union of k-faces C = F1 Fm such that Fi Fi+1 is a (k 1)-face for all i 1,...,m 1 . ∪···∪ ∩ − ∈ { − } Conjecture 2.0.4 (Lockeberg’s conjecture [69]). Let P be a d-polytope, v ,v V (P ), and let d = d + +d for positive integers d ,...,d . Then 1 2 ∈ 1 ··· r 1 r there are strong di-chains Ci, for i = 1,...,r, such that Ci Cj = v1,v2 , for i = j. ∩ { } 6 This conjecture is also mentioned by McMullen [48, Problem 57], in a survey by Kalai [63, Conjecture 19.5.4], and by Jockusch & Prabhu [56, Conjecture 1]. Lockeberg’s conjecture is a strengthening of Balinski’s theorem: If we set di = 1 for all i = 1,...,r = d, then the statement reduces to d- connectedness. Furthermore, a strong di-chain is di-connected. Thus Locke- berg’s conjecture implies that one may choose the d independent paths from v1 to v2 on the boundaries of the di-chains, i = 1,...,r. In contrast to this, it is not difficult to see that Lockeberg’s conjecture is a weakening of Conjecture 2.0.1. See Figure 2.1(a) for an example of strong chains. The third problem whose solution would have followed from a positive solution to Conjecture 2.0.1 is a problem by Prabhu [93]; see also Jockusch & Prabhu [56, Conjecture 2], and Ziegler [118, Exercise 3.14]. For the statement we need the following definition of k-paths.

24 v2 v2

v1 v1

(a) Two strong 2-chains and a strong 1- (b) Two 2-paths between the vertices v1 chain that intersect in the vertices v1 and and v2. These paths are disjoint in the v2. Lockeberg’s conjecture asserts that we sense of Deﬁnition 2.0.5 as they do not can ﬁnd a subcomplex similar to this one share a 2-face. in every 5-polytope.

Figure 2.1: Strong k-chains and disjoint k-paths.

Deﬁnition 2.0.5. Let P be a d-polytope, v1,v2 V (P ), and 0 k d. A k-path between v and v is a strong k-chain F ∈ F F such≤ ≤ that 1 2 1 ∪ 2 ∪···∪ m v1 V (F1), v2 V (Fm). Two k-paths are said to be disjoint if they have no ∈k-face in common.∈

See Figure 2.1(b) for two 2-paths between vertices v1 and v2. It is important to observe that the notion of disjointness of k-paths is very diﬀerent from the disjointness of strong chains in Lockeberg’s conjecture; see Figure 2.1. In that respect, the following problem is more in line with the material in Chapters 3 and 4.

d Question 2.0.6 (Prabhu [93]). Are there k disjoint k-paths between any two vertices v and v of any d-polytope? 1 2 ¡ ¢ The answer is aﬃrmative for the d-simplex. Thus we could infer the general case indeed from Conjecture 2.0.1 by taking a reﬁnement with principal vertices v1 and v2. Jockusch & Prabhu [56] have shown that a natural geometric approach to Question 2.0.6 fails.

25 2. Refinement Homeomorphisms of Polytopes

In Section 2.2 we present counterexamples to Conjecture 2.0.1. We thereby rule out this approach to the aforementioned problems. The examples are due to Lockeberg [69], who was a PhD student of McMullen. There are two reasons why these examples are included here: First, the explanation given here why no such refinement exists is simpler than the one given by Lockeberg. Second, so far these examples have only appeared in Lockeberg’s dissertation, which is not easily available (McMullen was so kind to copy the relevant chapters for us. Thanks!). I believe there is only one reference to them in the literature, see McMullen [78, 3B17], but even there the examples themselves are not given. We then consider refinements of centrally symmetric polytopes related to the following conjecture by Grünbaum [51].

Conjecture 2.0.7 (Grünbaum [51, Exercise 11.1.5, p. 205]). It may be conjectured that for each d there exists a finite family Pi 1 i n(d) of centrally symmetric d-polytopes such that for each centrally{ | symmetric≤ ≤ d}- polytope P the complex (P ) is a refinement of at least one of the complexes (P ), for 1 i n(d)B. B i ≤ ≤ It was remarked in the second edition of Grünbaum’s book, see [51, p. 224b], that this conjecture may be related to the “3d-conjecture” by Kalai [61].

Exercise 11.1.5 may be related to the conjecture by Kalai that if a d- polytope is centrally symmetric, then it must have at least 3d non-empty faces.

Kalai conjectures that equality is achieved only by the Hanner polytopes that can be generated from an interval by taking products and dualization (which includes the cubes and the cross polytopes). One may speculate that this also provides the ﬁnite family needed for exercise 11.1.5.

We reformulate this remark as the following conjecture.

Conjecture 2.0.8 ((disproved)). Every centrally symmetric d-polytope is a reﬁnement of some d-dimensional Hanner polytope.

Clearly, if P is a reﬁnement of a polytope Q, then the f-vector of P is componentwise at least as large as the f-vector of Q. Recent examples by Sanyal, Werner & Ziegler [99] imply that Conjecture 2.0.8 is false for all d 4. ≥ 26 2.1. Simplex Refinements of Polytopes

2.1 Simplex Reﬁnements of Polytopes

Definition 2.1.1 (Refinement homeomorphism, principal face). Let and be polytopal complexes. A refinement homeomorphism is a home- KomorphismC φ : such that the image of a face of is a union of faces of Q. K → C K A refinement homeomorphism between polytopes is a homeomorphism φ : P Q such that the image of a face of P is a union of faces of Q. A →principal face of a refinement homeomorphism φ is a face F of Q such 1 that φ− (F ) is a face of P .

The following theorem is a strengthening of a theorem by Gr¨unbaum [50]; see also [51, Section 11.1, p. 200] [51, Exercise 11.1.3].

Theorem 2.1.2. Let P be a d-polytope, and

= F 1 F0 F1 F2 Fd 1 Fd = P ∅ − ⊂ ⊂ ⊂ ⊂ · · · ⊂ − ⊂ a complete ﬂag of faces. Then there is a reﬁnement homeomorphism φ : ∆d P such that the following statements hold for every i 1, 0,...,d 1 : → ∈ {− − }

(i) The face Fi is a principal face of the reﬁnement.

1 (ii) The image of an ℓ-face of ∆ , with i ℓ d, that contains φ− (F ), d ≤ ≤ i which is a face by (i), contains only ℓ-faces of P that contain Fi.

Furthermore, the principal vertices φ(V (∆)) are aﬃnely independent.

Proof. The proof proceeds by induction on the dimension d. For d = 0 the statement clearly holds. Let d 1, take the vertex v := F0 and consider the vertex ﬁgure P/v. By induction,≥ the statement holds for P/v and the ﬂag

= F0/v F1/v F2/v Fd 1/v P/v ∅ ⊂ ⊂ ⊂ · · · ⊂ − ⊂

Let v0 be a vertex of ∆d and let φv : ∆d 1 = ∆d/v0 P/v be a re- − → finement homeomorphism that satisfies (i) and (ii) in P/v for the quotient of the flag. This implies that there is a refinement homeomorphism from star∆d (v0) = ∆d to pyrv(P/v), and a radial projection emanating from v yields a refinement homeomorphism from ∆d to starP (v)= P . To prove (i) and (ii) for this refinement, we need the following property of radial projection.

27 2. Refinement Homeomorphisms of Polytopes

F1 F2

Figure 2.2: A reﬁnement homeomorphism as constructed in the proof shown for the star in the boundary of a 3-polytope. The diﬀerently shaded areas show the images of the facets of ∆3 incident to the vertex that maps to F0.

(A) Let F be a face of P that contains v. Then the image of a face

pyrv(F/v) of pyrv(P/v) under radial projection from v is the face F .

If Fi/v is a principal face of φv, then (A) implies that Fi is a principal face of φ, for every i 0,...,d 1 . This shows (i), as the statement for i = 1 is trivial. ∈ { − } − We now prove (ii). Fix i 1,...,d and let δ be an ℓ-face of ∆d 1 ∈ {− } that contains φ− (F ). The statement is trivially true if i = 1, in which i − case Fi = . Otherwise, (ii) holds by induction in P/F0 = P/v for the refinement ∅φ . That is, the image φ (δ/v ) in P/v contains only (ℓ 1)- v v 0 − faces that contain Fi/v. The radial projection of such a face then contains Fi by (A). By induction, we can assume that the principal vertices of φv in P/v are affinely independent. Consequently, the principal vertices of φ in P are affinely independent.

See Figure 2.2 for an illustration of the main features of the refinement homeomorphism constructed in the proof. It is not true that the principal vertices are affinely independent in every refinement homeomorphism from the d-simplex to a d-polytope, and examples illustrating this exist already in dimension 3: The regular 3- crosspolytope for instance can be expressed as a refinement of the 3-simplex

28 2.1. Simplex Refinements of Polytopes in which the equatorial vertices, that is, the four vertices with last coordi- nate zero, are principal.

Definition 2.1.3 (Rooted refinement). Let P be a d-polytope and v a vertex of P . We say that a refinement homeomorphism with principal vertices U v is rooted at v if U is a subset of the neighbors of v in G(P ). ∪ { } Corollary 2.1.4. Let P be a d-polytope and v a vertex of P . Then there is a refinement homeomorphism φ : ∆ P that is rooted at v. d → Proof. This follows from Theorem 2.1.2 (ii) for ℓ = 1.

Corollary 2.1.5 (Gr¨unbaum [50]). Let P be a d-polytope, v V (P ) a vertex of P , and G = G(P ) the graph of P . Then G contains a subdivision∈ of Kd+1 rooted at v.

Proof. The image of the graph of ∆d, which is a Kd+1, under a reﬁnement homeomorphism as in Theorem 2.1.2 is a subdivision of Kd+1 rooted at the vertex v.

As another exemplary application of Gr¨unbaum’s theorem we derive the existence of complementary ﬂags in polytopal face lattices.

Corollary 2.1.6. Every flag has a complementary flag: For every flag

= F 1 F0 F1 F2 Fd 1 Fd = P ∅ − ⊂ ⊂ ⊂ ⊂ · · · ⊂ − ⊂ there is a ﬂag

= G 1 G0 G1 G2 Gd 1 Gd = P ∅ − ⊂ ⊂ ⊂ ⊂ · · · ⊂ − ⊂ such that Fi Gd i 1 = and Fi Gd i 1 is not contained in a facet, that ∩ − − ∅ ∪ − − is, Fi and Gd i 1 are complements in the face lattice of P , for every i with − − 1 i d. − ≤ ≤

Proof. Let φ : ∆d P be a reﬁnement homeomorphism such that Fi is a principal face for every→ i 1,...,d . Let ∈ {− }

τ 1 = τ0 τ1 τ2 τd 1 τd = ∆d − ∅ ⊂ ⊂ ⊂ ⊂ · · · ⊂ − ⊂ 1 be the corresponding flag in ∆ , that is, τ = φ− (F ) for every i 1,...,d . d i i ∈ {− } Take the (unique) complementary flag in ∆d. The image of this flag under φ contains a flag in P that is complementary to F .

29 2. Refinement Homeomorphisms of Polytopes

2.2 Lockeberg’s Counterexamples

Let φ : ∆d P be a refinement homeomorphism, where P is a d-polytope. We can imagine→ that this refinement amalgates facets of P into strongly connected polytopal (d 1)-complexes—the images of the facets of ∆d under the refinement. (Lockeberg− calls such a complex a pseudofacet [69].) We will interpret this amalgation as a coloring of the facets: Let m := fd 1 be the number of facets, let F1,...,Fm be the facets of P , and let − τ , . . . , τ be the facets of ∆ . Define a (surjective) map c : [m] [d + 1] 1 d+1 d → by setting c(i) := j if φ(τi) contains Fj. Thus, two facets Fj and Fk get the same color i, that is, c(j) = c(k) = i, if and only if φ(τi) contains both Fj and Fk. The intersection of the polytopal complexes corresponding to two (or more) color classes is a polytopal complex in P . It is the image of the intersection of the corresponding facets in ∆d under the refinement map. Thus we can also deduce from this coloring the principal vertices (or more general the principal faces, but we do not need this): Denote by I(v) [m] the indices of the facets that contain the given vertex v.A vertex⊆v is principal for the refinement φ if and only if c(I(v)) contains exactly d colors. In particular, if v is a vertex with c(I(v)) = d, then there is no other vertex w with c(I(v)) = c(I(w)). A violation| | of this condition is the main contradiction we derive in the proof of the following theorem by Locke- berg [69].

Theorem 2.2.1 (Lockeberg [69, Theorem 6.1]). Let d 6. Then there is a d-polytope P on d + 3 facets that has (necessarily nonadjacent)≥ vertices v and v that cannot both be principal in a reﬁnement φ : ∆ P . 1 2 d →

Proof. Let P ∆ be the 6-polytope on 9 vertices given by the Gale diagram in Figure 2.3 with vertex set [9] = 1, 2, 3, 4, 5, 6, 7, 8, 9 . By Theorem 1.3.3, { } the complements of the sets 4, 5, 6 and 7, 8, 9 correspond to facets G1 ∆ { } { } and G2 of P , respectively. ∆ 3 Let P be the polar of P . We claim that the vertices v1 := G1 and 3 v2 := G2 , that is, the vertices of P that correspond under polarity to the facets G1 and G2, cannot both be principal in a simplex reﬁnement of P . Suppose φ : ∆ P is a reﬁnement homeomorphism with both v and d → 1 v2 principal. Let F1,F2,...,F9 be the facets of P , such that Fi corresponds to the vertex i under polarity, for every i 1,..., 9 . Let c : [9] [7] be a coloring of the facets of P as described above.∈ { } →

30 2.2. Lockeberg’s Counterexamples

4 7

1 3

58 6 9

Figure 2.3: A Gale diagram of P ∆. The polar polytope P is a polytope that has two vertices that cannot be prescribed as principal vertices in a simplex reﬁnement of P . In contrast to every other Gale diagram in this thesis, here the little number next to a vector is a label and not the multiplicity of that vector.

∆ Since G1 and G2 are both simplex facets of P , the vertices v1 and v2 are simple vertices of P . Then no two facets incident to v1, or to v2, can have the same color. Thus, without loss of generality, we can assume c(1) = 1, c(2) = 2, and c(3) = 3. Furthermore, there is a pair i, j with i 4, 5, 6 and j 7, 8, 9 such that c(i) = c(j) and neither c({i) nor} c(j) appears∈ { as} a color∈ of { any other} facet. 6 Up to symmetry, we can assume that i = 4. For j we cannot have j = 7: If j = 7, then the facets F1, F4 and F7 must intersect in a 3-face of P , as in this case all three of them would be principal faces of the reﬁnement. Consequently, the vertices 1, 4, 7 would have to lie in a common 2-face of P ∆. From the Gale diagram of P ∆ we infer that the smallest face that contains 1, 4, 7 as vertices has as its complement the set 3, 5, 8 . But this is a ridge of P ∆, which is of dimension 5. Thus, up to symmetry{ } we can assume that j = 8. Without loss of generality we set c(4) := 4 and c(8) := 7. We are left, up to symmetry, with two possibilities of colors for the remaining facets F5,F6,F7, and F9. Case (i). Suppose c(5) = c(7) = 5 and c(6) = c(9) = 6. Consider the facet of P ∆ given by 4, 5, 9 . Let w be the vertex of P that corresponds { } to this facet. Then

I(v )= 1, 2, 3, 7, 8, 9 1 { } I(w)= 1, 2, 3, 6, 7, 8 , { }

31 2. Refinement Homeomorphisms of Polytopes and we have c(I(v )) = 1, 2, 3, 5, 6, 7 = c(I(w)), 1 { } so both v1 and w are principal with the same set of colors, which is a contradiction. Case (ii). Suppose c(5) = c(9) = 5 and c(6) = c(7) = 6. Let u and w be the vertices of P that correspond to the facets of P ∆ given by 1, 6 and 1, 9 . Then { } { } I(u)= 2, 3, 4, 5, 7, 8, 9 { } I(w)= 2, 3, 4, 5, 6, 7, 8 { } and we have c(I(u)) = 2, 3, 4, 5, 6, 7 = c(I(w)). { } Again, this is a contradiction. d 6 For d 7, let Q := pyr − (P ). It is clear that the d 6 pyramidal ≥ − vertices must be principal in any refinement φ : ∆d Q, since any one of them has exactly one facet that does not contain it.→ Let F be the (d 7)- simplex on the pyramidal vertices. Then any refinement of Q in which− v1 and v2 are principal induces a refinement of P in which v1 and v2 are principal via the isomorphism P ∼= Q/F . A corresponding statement for pairs of higher dimensional faces is much simpler to prove. For example, the bipyramid over a triangle has two edges that cannot both be principal in a simplex refinement. By taking pyramids, we can get examples for pairs of k-faces with k 2; compare the much stronger statement in [69, Theorem 6.4]. ≥ The polytope constructed for Theorem 2.2.1 has a simple direct description without Gale diagrams; see Section 9.3.2 where we meet it again. Lockeberg gave more counterexamples to the existence of refinement homeomorphisms with certain properties [69]: For d 3, he has shown that the prism over a (d 1)-simplex is a • counterexample≥ to a statement proposed by Grünbaum− [51, Exercise 11.1.4]. For p,q 2, the polytope ∆ ∆ has a set of three vertices that • ≥ p × q are not principal for any Kd+1-subdivision in G(∆p ∆q); see [69, Proposition 5.1] and [46]. × For every d 4, there exists a simple d-polytope P with d + 4 facets • and v ,v ≥V (P ), such that P is not a refinement of ∆ with v 1 2 ∈ d 1 and v2 principal. A polytope with this property can be constructed by taking the polar of a polytope that was obtained from Cd(d + 3)

32 2.3. Refinements of Centrally Symmetric Polytopes

by addition of a vertex in suitable position (such a polytope appears as an example of a neighborly 4-polytope that is not cyclic in [51, p. 124]). As for positive results, Lockeberg has shown the following: If P is a d-polytope on d + 2 facets, then for any two vertices of P • there is a reﬁnement homeomorphism φ : ∆d P with these two vertices principal [69, Theorem 6.5]. → If P is a d-polytope with d 4, 5 and P has d + 3 facets, then • ∈ { } for any two vertices a reﬁnement of ∆d with these two vertices as principal vertices exists [69, Chapter 7]. Thus, with respect to the number of facets, the results and counterexamples are best possible. A polytope with the Gale diagram in Figure 2.3 is nonsimplicial, as it has facets on d + 1 = 7 vertices. This suggests the following question.

Problem 2.2.2. Let P be a simplicial d-polytope, and let v ,v V (P ). 1 2 ∈ Is there a reﬁnement homeomorphism φ : ∆d P with v1 and v2 both principal? →

This question has an aﬃrmative answer if we restrict it to the 1-skeleton, that is, every simplicial d-polytope contains a subdivision of Kd+1 with two prescribed principal vertices. This was shown by Larman & Mani [66]; see also Gallivan, Lockeberg & McMullen [46].

2.3 Reﬁnements of Centrally Symmetric Polytopes

We make a very short trip into the land of centrally symmetric polytopes that fits the topic of this chapter. Centrally symmetric polytopes, that is, polytopes that are fixed by a central symmetry, do not play a role in the rest of this thesis. The following is immediate from the definition of refinement homeomorphisms.

Proposition 2.3.1. Let and be polytopal complexes and let φ : be a reﬁnement homeomorphism.K C Then the f-vector of is componentwiseK → C C larger than that of , that is, fi( ) fi( ), with equality in all components if and only if andK are combinatoriallyK ≤ C isomorphic. C K Indeed, for polytopes there are even stricter conditions. Because the face lattice of a d-polytope is coatomic, the existence of a reﬁnement home-

33 2. Refinement Homeomorphisms of Polytopes omorphism between two polytopes on the same number of facets implies that these two polytopes are combinatorially isomorphic.

Deﬁnition 2.3.2 (Hanner polytope). A Hanner polytope is any polytope that can be obtained by taking products and dualizing, starting from a line segment.

We show here that the family of Hanner polytopes is not the finite family asked for in [51, Exercise 11.1.5] if d 4, that is, Conjecture 2.0.8 is false for d 4. However, Grünbaum remarks≥ [51, Exercise 11.1.5], and this ≥ was shown, for example, by Barnette [12], that Conjecture 2.0.8 is true in dimension d = 3: Every centrally symmetric 3-polytope is the refinement of a 3-cube or a 3-crosspolytope. Furthermore, the refinement can be chosen to be compatible with the central symmetry.

Theorem 2.3.3. There are centrally symmetric d-polytopes for d 4 that are not reﬁnements of d-dimensional Hanner polytopes, that is,≥ Conjec- ture 2.0.8 is false for d 4. ≥ Proof. For d 5 this follows from recent results by Sanyal, Werner, and Ziegler [99]. Speciﬁcally,≥ they give examples of centrally symmetric polytopes for all d 5 for which no Hanner polytope with componentwise smaller f-vector≥ exists. These are counterexamples to a conjecture by Kalai [61, Conjecture B]. In dimension d = 4, another example from [99] provides the example we seek. Let

P := [ 1, +1]4 x R4 : 2 x + x + x + x 2 . − ∩ { ∈ − ≤ 1 2 3 4 ≤ }

This polytope has f-vector f(P ) = (14, 36, 32, 10) [99] (the entries f0 = 14 and f3 = 10 are obvious). In dimension 4, there are four Hanner polytopes, as summarized in the following table, see [99], where I := [0, 1] denotes the unit interval. polytope f-vector ∆ ∆ (C3 I) (10, 28, 30, 12) C∆ ×I (12, 30, 28, 10) 3 × C4 (16, 32, 24, 8) ∆ C4 (8, 24, 32, 16) If P is a refinement of any of them, say H, then the f-vector of P is componentwise larger than that of H. Thus, the only candidate for H is C∆ I. However, this polytope and the polytope P both have 10 3 × 34 2.3. Refinements of Centrally Symmetric Polytopes facets, but are not combinatorially isomorphic. Therefore, a refinement homeomorphism cannot exist. The following conjecture by Grünbaum [51, p. 216] would follow if every simplicial centrally symmetric d-polytope is a refinement of the d- crosspolytope, according to a result by Halin [53].

Conjecture 2.3.4 (Grünbaum [51, p. 216]). The graph of every centrally symmetric simplicial d-polytope contains a refinement of the complete graph on 3d/2 vertices. ⌊ ⌋ Larman & Mani [66] have proven that every centrally symmetric simplicial 4-polytope contains a refinement of the complete graph on 6 vertices.

Chapter 3

Incidence Graphs of Polytopes

Balinski’s theorem [5] is a fundamental theorem on graphs of polytopes. It states that the graph of every d-polytope is d-connected. It was reproved by several other authors; see Barnette [6] [11], and Brøndsted & Maxwell [32]. There are various ways to generalize Balinski’s theorem, as was done by Athanasiadis [3], Barnette [8] [10], Holt & Klee [54], Klee [64], Larman & Rogers [67], Perles & Prabhu [90], and Sallee [98]. Bj¨orner [22] studied higher topological connectivity of polytopes. In this and the following chapter we are interested in the generalization proposed by Sallee [98] and Athanasiadis [3]. Both consider the graph of a polytope as a special case of the incidence graph on the faces of a polytope. The (k,ℓ)-incidence graph Gk,ℓ(P ) is the graph that has as vertices the k-faces of a given polytope P , and two vertices in this graph are connected by an edge if and only if the corresponding k-faces lie in a common ℓ-face. As a special case, we obtain the graph of a d-polytope for k = 0 and ℓ = 1. Balinski’s theorem implies that this graph is d-connected. It is also possible to derive from Balinski’s theorem that the (d 2,d 1)- incidence graph is d-connected. This is the line graph of the dual− graph− of the polytope. Let d and k be nonnegative integers that satisfy 0 k d 1, let P be ≤ ≤ − a d-polytope, and let Gk(P )= Gk,k+1(P ) be the (k, k + 1)-incidence graph of P . Athanasiadis [3] has shown that

d, if k = d 2, κ(Gk(P )) − ≥ (k + 1)(d k), otherwise. ½ − 37 3. Incidence Graphs of Polytopes

This generalizes Balinski’s theorem to higher incidence graphs of polytopes. Athanasiadis result is best possible, as there are d-polytopes that attain the bound for all k and d. Adding to this result, we determine in this chapter the connectivity of (k,ℓ)-incidence graphs with ℓ 2k + 1. We show that in this case the ≥ connectivity of the incidence graph Gk,ℓ(P ) of a d-polytope satisfies d κ(G (P )) . k,ℓ ≥ k + 1 µ ¶ Again, this bound is best possible, as there are d-polytopes that attain the bound for all d,ℓ and k with d 1 ℓ 2k + 1. We also briefly discuss a question− ≥ by≥ Athanasiadis, which asks which (k,ℓ)-incidence graphs are graphs of polytopes. Question 3.0.1 (Athanasiadis [4]). Which (k,ℓ)-incidence graphs are polytopal, that is, (0, 1)-incidence graphs of other polytopes, not necessarily of the same dimension? We give a twofold answer for special choices of k and ℓ. We show that (a) in every dimension d 3 there are infinitely many (1, 2)-incidence graphs which are polytopal,≥ and that (b) there are infinitely many nonpolytopal (0, 2)- and (1, 2)-incidence graphs of 3-polytopes. The main tool for the construction of the examples is the truncating operation by Paffen- holz & Ziegler [87].

3.1 Balinski’s Theorem for Polytopes

In this section we reprove Balinski’s theorem. Our version of this theorem combines statements found in papers by Perles & Prabhu [90], and Holt & Klee [54]. Deﬁnition 3.1.1. Let P be a d-polytope in Rd and let c be a linear function. We denote by minc(P ) and maxc(P ) the sets of points of P for which c attains its minimum and maximum, respectively. The function c is generic with respect to P if for no two vertices u and v of P the value c(u) equals the value c(v). By Gc(P ) we denote the graph of P directed according to the values of c, that is, (u, v) is a directed edge of Gc(P ) if and only if c(u) < c(v) and (u, v) G(P ). ∈ Theorem 3.1.2 (see Bondy & Murty [30, Theorem 11.6]). Let G be a directed graph, and x and y two vertices of G. If for every set C ⊆ 38 3.1. Balinski’s Theorem for Polytopes

V (G) x, y with C d 1 there is a directed path from x to y in G C, then there\ { are} d vertex-disjoint| | ≤ − directed paths from x to y. \ The following theorem combines statements by Perles & Prabhu [90], and Holt & Klee [54]. The proof closely follows Holt & Klee’s proof [54]. Theorem 3.1.3 (“Monotone Balinski”). Let P be a d-polytope in Rd and x, y (P ). Let c be a linear function that is generic with respect to ∈ V P such that minc(P )= x and maxc(P )= y. Let U (P ) x, y be a subset of the vertices with dim aff(U) = e. Then the following⊆ V \ holds: { } (i) If e = d 1 and U is contained in some facet F of P , then there is a monotone− path in G (P ) U from x to y. c \ (ii) If e d 2, then there are at least d e 1 monotone paths in G (P ) ≤ − − − c from x to y that only intersect in their endpoints x and y. Proof. If d = 2 the statement is obvious. For d 3 we proceed by induction. We first make a couple of assumptions on≥ the position of P in Rd. We assume that x = 0 is the origin. We also assume that the linear function d c is given by c(p) = pd, for p = (p1,...,pd) R , and that c(y) = 1. We consider c as a row vector it has the form c =∈ (0, 0,..., 0, 1). We define a linear transformation on Rd by

p p + c(p)(e y). 7→ d −

Then y y + c(y)(ed y)= ed. This transformation leaves the values of c invariant,7→ since −

c(p)= c(p)+ c(p)(c(e ) c(y)) = c(p + c(p)(e y)). d − d − =0

Thus, we can assume that y |= ed ={z (0,...,} 0, 1). Case (i). Assume we have dim aﬀ(U)= e = d 1 and that U is contained in some facet F of P . Let a be a vector that is− normal to F and denote by π the orthogonal projection of Rd onto the plane spanned by cT and a. Then π(x) and π(y) are two vertices of the 2-dimensional polytope π(P ). There is a path (v0,e1,v1,...,en,vn) in G(π(P )) that avoids the edge π(F ) and that is monotone with respect to cπ = (0, 1) = e2. It remains to show that this path lifts to a directed path in P that avoids F . This will be done below. Case (ii). Now, assume we have that e d 2. Let U ′ be a set of ≤ − vertices in V (U x, y ) with U ′ d e 2. Then for S = U U ′ \ ∪ { } | | ≤ − − ∪ 39 3. Incidence Graphs of Polytopes

we have that dim aff U U ′ e +(d e 3)+1 = d 2. According to ∪ ≤ − − − Theorem 3.1.2 it suffices to show that there is a directed path in Gc(P ) from x to y that avoids S. d + Let J = p R : pd 1 = 0 , J be the positive half-space (pd 1 > 0) { ∈ − } − bounded by J, and J − the negative half-space bounded by J. Denote by Φ the orthogonal projection to the first d 1 coordinates. Then the projection − Φ(S) is contained in a (d 2)-dimensional affine subspace H of J. Let H0 be the subspace parallel to− H through the origin. By applying a further transformation that leaves e fixed, we can assume that H = p Rd : d 0 { ∈ pd 1 = pd = 0 and that either S J or S J −. − } ⊆ ⊆ + Suppose that S J − and that P does not intersect J . Then J P is a ⊂ ∩ face F of P that contains x and y and a monotone path in Gc(F ). Clearly, this path avoids S. + Now, either S J − and P intersects J or we have that S J. In the latter case we can⊂ assume that P intersects J +, otherwise we reflect⊂ in J. Let π denote the projection to the plane spanned by ed 1 and ed. Then − π(x) and π(y) are vertices of a 2-dimensional polytope π(P ) that are the maximum and minimum with respect to cπ = (0, 1) = e2. There is a path (v0,e1,v1,...,vn 1,en,vn) in the boundary of π(P ) that avoids π(J −) and − thus S. It remains to show that the path v0,e1,v1,...,en,vn constructed in the two cases above can be lifted to a monotone path in P that avoids S. 1 The preimage π− (ei) is a face Fi. Since in both cases the projection preserves the values of c, that is, c(p)= cπ(π(p)), there are unique vertices ui 1 and ui of Fi such that π(ui 1) = vi 1 and π(ui) = vi, with v0 = x − − − and vn = y. Every Fi contains a monotone path connecting ui 1 and ui in − Gc(P ) that clearly avoids S. The union of these paths is the desired path from x to y. The following corollary can be derived by much simpler methods. For example, the proof in [118, Theorem 3.14] also yields the following statement.

Corollary 3.1.4. Let P be a d-polytope and let C (P ) be a set of vertices whose aﬃne span is at most d 2 dimensional.⊆ V Then the graph − G(P ) C is connected. \ Proof. Let x, y (P ) be two vertices of P and let H and H be nonpar- ∈ V x y allel supporting hyperplanes with Hx P = x and Hy P = y . Let H be a third hyperplane with H P = ∩ and {H } H H∩ = H { }H . Let ∩ ∅ ∩ x ∩ y x ∩ y π be a projective transformation that sends H to inﬁnity. Then π(Hx) and π(H ) are parallel hyperplanes, and, if c (Rd) denotes a vector normal y ∈ ∗ 40 3.2. Higher Incidence Graphs

to π(Hx) and π(Hy), we have that minc(π(P )) = x and minc(π(P )) = y. By Theorem 3.1.3 with U = C, there is a monotone path from x to y.

Corollary 3.1.5 (Balinski’s theorem [5]). Let G be the graph of a d- polytope. Then G is d-connected.

Balinski’s theorem is best possible in the sense that for every f d + 1 0 ≥ there is a d-polytope on f0 vertices that is not (d + 1)-connected. For instance, take a stacked polytope on f0 vertices, that is, a polytope obtained from the simplex by iteratively stacking a d-simplex onto a facet f0 d 1 times. Such a polytope has a vertex of degree d, and so the graph− either− has exactly d + 1 vertices, or there is a separating set of size d.

3.2 Higher Incidence Graphs

Deﬁnition 3.2.1. Let P be a d-polytope and let 0 k ℓ d. The ≤ ≤ ≤ (k,ℓ)-incidence graph Gk,ℓ(P ) is the graph on the k-faces of P in which two nodes are connected by an edge if they are contained in a common ℓ-face of P . We write Gk(P ) for the (k, k + 1)-incidence graph of P .

A very simple example of a (k,ℓ)-incidence graph is the graph Gk,d, which is the complete graph on the k-faces of a polytope, that is, it is the complete graph on fk vertices. The graph Gk,k is the complement of the graph Gk,d. An obvious but useful property is that Gk,ℓ is always a subgraph of Gk,ℓ′ for all ℓ ℓ′ d, with equality for ℓ + 1 ℓ′ d 1 if P is ℓ′-simplicial. We deﬁne≤ ≤ ≤ ≤ −

κ (d) := min κ(G (P )) : P is a d-polytope . k,ℓ { k,ℓ } Sallee [98] obtained various results on certain types of connectivities related to (k,ℓ)-incidences in polytope skeleta, including bounds on the connectivity of Gk,ℓ. He proved, see [98, p. 495], that

ℓ ℓ + 1 (d 1 ℓ) + κ (d) − − k k + 1 ≤ k,ℓ µ ¶ µ ¶ ℓ k d − d k k + 1 min , − . ≤ k + 1 i i ( i=1 ) µ ¶ X µ ¶µ ¶ For k = 0, the bounds are equivalent to Balinski’s theorem. For larger k the upper and the lower bound do not coincide.

41 3. Incidence Graphs of Polytopes

G F v

Wv Figure 3.1: The basic idea behind Athanasiadis’ proof.

d The bipyramid over the (d 1)-simplex gives the upper bound of k+1 − ℓ k d k k+1 in the above inequalities, whereas the upper bound of − − is i=1 i i¡ ¢ given by the d-simplex. In the first case, the separating set consists of all ¡ ¢¡ ¢ the k-faces of the base of the bipyramid. In the second case,P the separating set consists of all neighbors of a given k-face. It is plausible that the upper bound is best possible, so one may conjecture that κk,ℓ(d) indeed equals the upper bound. This conjecture is supported by Athanasiadis’ recent result [3] that the graph Gk(P )= Gk,k+1(P ) is m (d)-connected, where for k d 1 we define k ≤ − d, k = d 2 m (d) := k (k + 1)(d k), otherwise.− ½ − Further support is given by our result in the next section that the connec- d tivity of Gk,ℓ(P ) is k+1 if ℓ 2k + 1. As a preparation for the next≥ section and for Chapter 4, where we prove ¡ ¢ a conjectured generalization of Athanasiadis’ result [3, Conjecture 6.2], we review the basic idea of Athanasiadis’ proof. One can find similar ideas in Sallee’s work; see for example the first lemma in [98, Section 6]. Suppose that we are given two k-faces F and G of a polytope P . How can we connect them by a path or a walk in Gk,k+1(P )? Here is one way, as is illustrated in Figure 3.1: Take a path W in the ordinary graph of P from a vertex in F to a vertex in G. For every edge of this path, choose a k-face that contains this edge. Two k-faces that contain a vertex v of the path can be connected by a walk Wv in Gk,k+1(P ) whose faces all contain the vertex v. This follows by induction, as a walk in Gk 1,k(P/v) corresponds to such − a walk. The walks Wv now add up to a walk that connects F and G. If we want to extend this idea to prove a higher connectivity of the graph Gk,ℓ, we have to choose the path W with more care. Suppose that we want to prove that Gk,ℓ(d) is f(k,ℓ,d)-connected. By definition, it suffices

42 3.3. Connectivity of Incidence Graphs to show that, for every set of forbidden k-faces of size at most f(k,ℓ,d) 1, − the graph Gk,ℓ without these forbidden faces is still connected. This suggests the following two conditions on W : d 1 Every edge of W lies in less than k−1 forbidden k-faces. As there • d 1 − are at least − faces of dimension k that contain a ﬁxed edge, this k 1 ¡ ¢ condition implies− that every edge has a free k-face, that is, a face that ¡ ¢ does not lie in the set of forbidden faces. Every vertex of W does lie in at most f(k 1,ℓ 1,d 1) of the • forbidden k-faces. Then it is possible to apply− induction.− Walks− in the graph Gk 1,ℓ 1(P/v) of the vertex ﬁgure P/v at a vertex v correspond − − to walks in Gk,ℓ(P ) that contain the vertex v. This is the basic idea. It is put to work in the next section with d some variations, where we we prove that (k,ℓ)-incidence graphs are k+1 - connected if ℓ 2k+1, and in the next chapter. There we prove a conjecture ¡ ¢ by Athanasiadis≥ on (k, k + 1)-incidence graphs of Cohen-Macaulay regular cell complexes with intersection property. It remains an open problem to determine the minimal connectivities for (k,ℓ)-incidence graphs of polytopes for k + 2 ℓ 2k. It is likely that the ideas and methods of this and the next chapter≤ ≤ yield better bounds than the ones proved by Sallee [98] also in this case.

Problem 3.2.2. What is the minimum connectivity κk,ℓ(d) of the (k,ℓ)- incidence graphs for k + 2 ℓ 2k? ≤ ≤ From what we know the answer could be

κk,ℓ(d) = min κ(Gk,ℓ(∆d)), κ(Gk,ℓ(bipyr(∆d 1))) . { − }

3.3 Connectivity of Incidence Graphs

In this section, we determine the connectivity of G (P ) for ℓ 2k + 1. If k,ℓ ≥ P = ∆d, then Gk,ℓ(P ) is the complete graph on the k-faces of ∆d, and we have κ(G )(P ) d+1 1. k,ℓ ≥ k+1 − ¡ ¢ Deﬁnition 3.3.1 (Good and bad vertices). Let P be a d-polytope and d 0 k d 1. Let be a set or multiset of k-faces of size less than k+1 . ≤ ≤ − E d 1 We call a vertex v good if it is incident to at most − 1 elements of k ¡ ¢ (counting multiplicities if is a multiset), and otherwise−bad. E E ¡ ¢

43 3. Incidence Graphs of Polytopes

3.3.1 Neighborhoods in Incidence Graphs Lemma 3.3.2. Let P be a d-polytope and ℓ 2k + 1. Then the minimum degree δ of the (k,ℓ)-incidence graph satisﬁes≥

δ(G (P )) δ(G (∆ )). k,ℓ ≥ k,ℓ d Proof. Let F be a k-face of P and let φ : ∆ P be a reﬁnement as in d → Theorem 2.1.2 such that F is a principal face (choose any ﬂag that contains the face F ). Let τ ∆ with π(τ) = F and let τ ′ be a neighbor of τ in ⊆ d Gk,ℓ(∆d) via the ℓ-face λ. By Theorem 2.1.2, every ℓ-face in φ(λ) contains the k-face F . Thus, every k-face in φ(τ ′) φ(λ) is a neighbor of F in Gk,ℓ(P ). Since φ is bijective, the statement⊆ follows.

Lemma 3.3.3. Let P be a d-polytope, d k 0, and let be a multiset d ≥ ≥ E of k-faces of P of size less than k+1 . Then the bad vertices span at most a (d 2)-dimensional aﬃne subspace of Rd. − ¡ ¢ Proof. Suppose their span would be at least (d 1)-dimensional. Then there is a set U of d aﬃnely independent bad vertices.− Any k-face in E contains at most k + 1 of the vertices in U. But then double counting the elements of yields that E d 1 (k + 1) d − , | E | ≥ k µ ¶ and we have the contradiction d d 1 d − = , | E | ≥ k + 1 k k + 1 µ ¶ µ ¶ which implies the statement.

Lemma 3.3.4. Let d ℓ 2k + 1 and a multiset of k-faces of the d- ≥ ≥ d E simplex ∆d of size less than k+1 . Let F be a k-face of ∆d that is not in . Then F contains a good vertex, or there is a k-face that is a neighbor of E ¡ ¢ F in Gk,ℓ(∆d) that contains a good vertex. Proof. Since ℓ 2k + 1, we have that the graph G (∆ ) is the complete ≥ k,ℓ d graph on the k-faces of ∆d. The d+1 vertices of ∆d are aﬃnely independent, thus Lemma 3.3.3 implies that there are at most d 1 bad vertices in (∆d). − d V Let v (∆d) be one of the good vertices. There are k k-faces incident to the∈ good V vertex v, so there is a k-face G with v G and G / . ∈ ¡ ¢ ∈ E 44 3.3. Connectivity of Incidence Graphs

Lemma 3.3.5. Let d ℓ 2k + 1 and a set of k-faces of a d-polytope P of size less than d .≥ Let≥F be a k-faceE that is not in . Then F contains k+1 E a good vertex, or there is a k-face that is a neighbor of F in G (P ) that ¡ ¢ k,ℓ contains a good vertex.

Proof. Let φ : ∆d P be a reﬁnement homeomorphism as in Theo- rem 2.1.2 such that F→is a principal face. Deﬁne a multiset ′ of the k-faces of the d-simplex as follows. One copy E of a k-face τ ∆d is in ′ for every k-face of P in φ(τ) . Since φ is ⊆ E d ∩ || E || bijective, we have that ′ = 1. | E | | E | ≤ k+1 − By Lemma 3.3.4, there is a k-face τ ′ ∆ that is not in ′ and that ¡ ¢⊆ d E contains a good vertex v′. Then v := φ(v′) is a good vertex in P . Let λ be an ℓ-face that contains both τ and τ ′. By Theorem 2.1.2 (ii), the polytopal complex φ(λ) contains only ℓ-faces that contain F . Then every k-face in φ(τ) is a neighbor of F , and none of them lies in , as E there is no copy of τ ′ in ′. Thus there is a k-face in φ(τ ′) with the desired properties that containsE the good vertex v.

3.3.2 Free Faces at Edges Lemma 3.3.6. Let P be a d-polytope and suppose that d 2k + 1. Assign a red label to some of the k-faces of P , and assign≥ a green label to some of the k-faces of P such that each type of label is assigned at most d d k+1 + k 1 times (faces may have two labels of diﬀerent colors). Then there− is a (2k + 1)-face that has one k-face that does not have a ¡ ¢ ¡ ¢ red label and one k-face that does not have a green label. Proof. We can assume that all faces either have a red label or a green label, otherwise we pick an unlabeled k-face and any (2k + 1)-face that contains that face. There are d + 1 d d d d = 1 (3.1) k + 1 − k + 1 − k 1 k − k 1 ≥ µ ¶ µ ¶ µ − ¶ µ ¶ µ − ¶ faces that only have a red label, as we have d 2k + 1. Let F be one of them. All faces that lie in (2k + 1)-faces that≥ contain F must also have a red label, otherwise we have the desired result. By Lemma 3.3.2, the d+1 number of such k-faces is at least as many as in ∆d and this is k+1 . By the calculation given in Equation (3.1) at least one of them has only a green ¡ ¢ label. Recall the idea discussed in the last section that we want to apply to prove the connectivity. Because of the huge number of forbidden k-faces,

45 3. Incidence Graphs of Polytopes we cannot necessarily always find a free k-face at every edge of a path in the graph of a polytope, even if the two incident vertices are good. The following lemma, however, shows that, if all the k-faces of an edge with good endpoints are forbidden, then there is an edge in Gk,ℓ that connects a free k-face incident to one endpoint of the edge to a free k-face incident to the other endpoint of the edge. Lemma 3.3.7. Let P be a d-polytope, d ℓ 2k + 1 3, and let be a d ≥ ≥ ≥ E subset of the k-faces of size less than k+1 . Furthermore, let e = vw be an edge of P such that both endpoints v and w of e are good. Then ¡ ¢ (i) there is a k-face G that is not in that contains e, or E (ii) there is an ℓ-face G that has a free k-face incident to v and a free k-face incident to w. Necessarily e is contained in G. Proof. We assume that (i) does not hold, and prove (ii) under this assumption. We want to apply Lemma 3.3.6 on the (k 1)-faces of the edge figure P/e at e. In the following we thus describe a labeling− of these faces. First, we label the (k 1)-faces of P/v and of P/w. A(k 1)-face of P/v gets a green label, if the− corresponding k-face in P lies in .− Otherwise, it gets no label at all. Likewise, a (k 1)-face of P/w getsE a red label, if the corresponding k-face in P lies in −. Otherwise, it gets no label at all. E Write v′ for the vertex e/w in P/w and w′ for the vertex e/v in P/v. We now label the (k 1)-faces of P/e by “transporting” the labeling from − the vertex figures P/v and P/w to P/e. A(k 1)-face F of P/e gets a − green label, if, under the natural isomorphism P/e ∼= (P/v)/w′, all the (k 1)-faces of the k-face of P/v that corresponds to F have a green label according− to the labeling in P/v. As we also have a natural isomorphism between P/e and (P/w)/v′, the same procedure can be applied to get a labeling with red labels: A(k 1)- face F of P/e gets a red label, if all the (k 1)-faces of the k-face of−P/w that correponds to F have a red label according− to the previous labeling. We now count the number of faces in P/e that have a green label. By symmetry, the same arguments apply for the number of faces with red labels. Consider a (k 1)-face of P/e and suppose it has a green label. Let F be the corresponding− k-face of P/v. Then every (k 1)-face of F has a green label in P/v. Furthermore, since F corresponds− to a face in P/e, the face F contains the vertex w′ of P/v. Then there is a (k 1)-face F ′ − in ast(w′,F ) with a green label that does not lie in any other k-face of P/v that contains the vertex w′.

46 3.3. Connectivity of Incidence Graphs

This shows that the number of (k 1)-faces with green labels of P/e is at most the number of elements of −incident to v minus the number of E elements of that contain the edge e. Since v isE a good vertex the number of faces of incident to v is less d 1 E than −k . Furthermore, since we assume that (i) does not hold, all k-faces that contain e are in . ¡ ¢ Thus the number ofE k-faces in P/v all of whose (k 1)-faces have green labels is bounded by − d 1 d 1 d 2 d 2 − − = − + − . k − k 1 k k 2 µ ¶ µ − ¶ µ ¶ µ − ¶ The same bound holds for the number of faces with red labels. We can therefore apply Lemma 3.3.6. This yields that there is a face of dimension 2(k 1) + 1 in P/e that has a (k 1)-face that does not have a green label and− a (k 1)-face that does not− have a red label. This face corresponds to an ℓ-face− of P that has a k-face incident to v that is not in and a k-face incident to w that is not in . E E 3.3.3 Proof of the Connectivity Theorem 3.3.8. Let d ℓ 2k + 1 and let P be a d-polytope. Then the ≥ ≥ d (k,ℓ)-incidence graph Gk,ℓ(P ) is k+1 -connected. Proof. We prove the statement by¡ induction¢ on k. If k = 0 the statement follows from Balinski’s theorem, Corollary 3.1.5, as G0,1 is a subgraph of G0,ℓ for all ℓ 1 on the same set of vertices. Thus, let ≥k 1 and let be a subset of the k-faces of cardinality less d ≥ E than k+1 . We have to show that Gk,ℓ(P ) is connected, that is, for every pair F,G of k-faces not in there is a path\E in G (P ) . ¡ ¢ k,ℓ By Lemma 3.3.5, everyEk-face not in has a neighbor\ E that is not in that contains a good vertex. Thus, weE can assume that F and G both Econtain a good vertex. Let v be a good vertex of F and w a good vertex of G. By Lemma 3.3.3, the bad vertices aﬃnely span at most a (d 2)- dimensional subspace of Rd. Thus, by Corollary 3.1.4 there is a walk −W in G(P ) that consists only of good vertices and that connects v to w. Let u W be a vertex of this walk. Since u is a good vertex, the ∈ d 1 number of k-faces in that contain u is at most −k 1. Furthermore, since ℓ 2k + 1 we haveE ℓ 1 2(k 1) + 1. By the induction− hypothesis, ¡ ¢ any two≥ (k 1)-faces of P/u− that≥ correspond− to faces that are not in can − E be connected in Gk 1,ℓ 1(P/u) by a walk that avoids the (k 1)-faces that − − correspond to k-faces in . − E 47 3. Incidence Graphs of Polytopes

This path translates into a path of k-faces that contains u and avoids the set . Thus, any two k-faces that contain a vertex u of the walk W can E be connected by a path in Gk,ℓ(P ) . The statement then follows from Lemma 3.3.7. \ E The following lemma shows that the bound is best possible. Lemma 3.3.9 (Sallee [98]). For every d 2 and 2k + 1 ℓ d 1 ≥ ≤ ≤ −d there is a d-polytope P such that Gk,ℓ(P ) can be separated by a set of k+1 k-faces. ¡ ¢ Proof. Let P := bipyr(∆d 1) be the bipyramid over the (d 1)-simplex with − d − apexes x and y. Clearly, the faces of dimension k in ∆d 1 separate k+1 − the k-faces of P that contain x from the k-faces of P that contain y. ¡ ¢ Corollary 3.3.10. Let 2k + 1 ℓ d 1. Then κ (d)= d . ≤ ≤ − k,ℓ k+1 ¡ ¢ 3.4 Polytopality of Incidence Graphs

We now consider a question on polytopality of incidence graphs. To avoid confusion: We do not want to decide whether a given graph is a (k,ℓ)- incidence graph of a d-polytope. Instead, we want to realize a given (k,ℓ)- incidence graph as the (0, 1)-incidence graph of a different polytope. Thus, this is our definition of polytopality. Definition 3.4.1 (Polytopal). A graph G is d-polytopal if there is a d- polytope P with G(P )= G. If a graph G is d-polytopal for some d, we say that G is polytopal. It was observed by Athanasiadis [4] that there are indeed (k,ℓ)-incidence graphs that are also (0, 1)-incidence graphs, that is, that are polytope in the sense of Definition 3.4.1 Take, for example, the d-simplex ∆ on d + 1 vertices. If ℓ 2k + 1, d ≥ then any two k-faces F1 and F2 have a common ℓ-face G that contains them d+1 both. Thus, in this case Gk,ℓ(∆d) is the complete graph on k+1 vertices. This graph is d -polytopal for every 4 d d+1 1. If ℓ = 2k = d 1, ′ ′ k+1 ¡ ¢ any two k-faces F and G of G (∆ ≤) are≤ connected− by an edge, unless− k,ℓ d ¡ ¢ (F ) (F ) = 2k +2= d + 1. Thus, for every k-face F there is exactly | V 1 ∪ V 2 | one k-face G that is not a neighbor of F . The graph Gk,ℓ is Km E, where d+1 \ m := k+1 and E denotes a perfect matching in Km. This is the graph of a crosspolytope of suitable dimension. ¡ ¢ In this section, we discuss this question in more detail and look at some interesting examples.

48 3.4. Polytopality of Incidence Graphs

(a) The graph of the 3-crosspolytope (b) Example of a polytope with a non- is the (1, 2)-incidence graph of the 3- polytopal (0, 2)-incidence-graph. The simplex. graph G0,2 clearly is not 4-connected, but has a K6-subgraph.

Figure 3.2: Polytopal and nonpolytopal incidence graphs.

3.4.1 Polytopal Incidence Graphs To construct inﬁnitely many polytopal (1, 2)-incidence graphs we look at Paﬀenholz & Ziegler’s truncatable polytopes [87].

Deﬁnition 3.4.2 (Truncatable Polytopes). If P is a d-polytope, then P is called truncatable if for every edge e there is a relative interior point ve such that the polytope T (P ) := conv ve : e E(G(P )) is obtainable by truncating all vertices of P , that is, by{ cutting∈ oﬀ all vertices} by suitable hyperplanes.

The following proposition is “geometrically clear.”

Proposition 3.4.3. Let P be a 2-simplicial truncatable d-polytope. Then we have that G0,1(T (P )) ∼= G1,2(P ).

See Figure 3.2(a) for a realization of T (∆3), which is an octahedron, as a truncation of ∆3. According to the following theorem by Paﬀenholz & Ziegler [87], truncatable polytopes can be constructed by ﬁnding realizations such that all edges of the polytope are tangent to a sphere.

Theorem 3.4.4 (Paﬀenholz & Ziegler [87]). Let P be a d-polytope for d 3 such that all edges of P are tangent to a (d 1)-sphere. Then P is truncatable.≥ −

49 3. Incidence Graphs of Polytopes

The case of dimension 2 is not interesting: Every 2-polytope can be realized with all edges tangent to a 1-sphere. This is trivial. In dimension 3, the analogous statement holds but is far from trivial. Theorem 3.4.5 (Koebe-Andreev-Thurston Theorem; see [118, The- orem 4.12]). Every 3-polytope can be realized with all edges tangent to some 2-sphere. This theorem is proved by non-linear methods, for example, the variational principles by Bobenko & Springborn [27]; see also Ziegler [120]. For 3-polytopes we can consequently describe exactly when a (k,ℓ)- incidence graph is polytopal. This is done in the following theorem. Theorem 3.4.6. Let P be a 3-polytope and k ℓ+1 with k 0, 1, 2 and ℓ 1, 2, 3 . Then the following hold: ≤ ∈ { } ∈ { } (i) If k = 0, then Gk,ℓ is 3-polytopal if and only if P is ℓ-simplicial.

(ii) If k = 1, then Gk,ℓ is 3-polytopal if and only if P is simplicial.

(iii) If k = 2, then Gk,ℓ = G2,3 = Kf2 is 3-polytopal if and only if P is the 3-simplex. Proof. Let k = 0. If ℓ = 1, then G trivially is 3-polytopal, so assume ℓ 2, 3 . If ℓ = 3, P cannot have more than 4 vertices, so P must be the ∈ { } 3-simplex. If ℓ = 2 and P is 2-simplicial, then G0,2 = G2,3. Otherwise, P has an n-gon face for some n 4. Then G0,2 clearly has more edges than a triangulated planar graph and≥ thus is not planar. This shows (i). Suppose k = 1. If P is simplicial, then G1,2(P ) ∼= G0,1(T (P )). If P is not simplicial, then G1,2 is not planar for the same reason as G0,2 is not planar. Thus (ii) follows.

Finally, if k = 2 we have Gk,ℓ = Kf2 . Since every 3-polytope that is not the simplex has more than 4 faces of dimension 2. Thus (iii) is proved. For higher dimensions, “edge tangency” may fail. However, this property is not necessary for truncatability. We have the following theorem by Paffenholz & Ziegler [87]. Theorem 3.4.7 (Paffenholz & Ziegler [87, Theorem 3.5]). Let P be a stacked d-polytope and d 3. Then P has a realization that is truncatable. ≥ This yields an infinite family of polytopal (1, 2)-incidence graphs of simplicial polytopes in fixed dimension. Theorem 3.4.8. For every d 3 there is an infinite family of simplicial d-polytopes with polytopal (1, 2)≥-incidence graphs.

50 3.4. Polytopality of Incidence Graphs

3.4.2 Nonpolytopal Incidence Graphs We exploit two conditions of polytope graphs to show that there are inﬁ- nite series of nonpolytopal (0, 2)- or (1, 2)-incidence graphs of 3-polytopes, namely that (a) a graph of a 3-polytope does not have a K5 subgraph, and that (b) a graph of a d-polytope with d 4 is at least d-connected by Balinski’s theorem. ≥

Theorem 3.4.9. There is an infinite number of combinatorial types of 3- polytopes such that neither G nor G is d-polytopal for any d N. 0,2 1,2 ∈ Proof. Consider the (0, 2)-incidence graph G of the polytope given in Fig- ure 3.2(b). The subgraph that corresponds to the 5-gon face is a K5. Thus, G is not the graph of a 3-polytope. However, it is also not the graph of a polytope of larger dimension, as it is clearly not 4-connected. The same polytope also yields an example of a (1, 2)-incidence graph that is not the graph of a polytope. Starting from this example, we clearly can construct an infinite family of nonplanar (0, 2)- or (1, 2)-incidence graphs that are at most 3-connected by stacking onto triangular faces. For higher dimensions, finding nonpolytopal incidence graphs might be an intractable problem, as every graph is an induced subgraph of a 4- polytope; see [63, Proposition 20.2.3].

Chapter 4

Athanasiadis’ Conjecture on Incidence Graphs

Athanasiadis conjectured that his result on (k, k + 1)-incidence graphs that we briefly discussed in the last chapter would extend to a certain class of cell complexes, Cohen-Macaulay regular cell complexes with intersection property [3, Conjecture 6.2]. We prove his conjecture by establishing the connectivity of (k, k + 1)- incidence graphs of a subclass of Barnette’s graph manifolds [8]. It is then easy to extend this result to weakly normal d-graph complexes, which we define in Section 4.3. These are a combinatorial generalization of Cohen- Macaulay regular cell complexes with intersection property. The reader acquainted with Athanasiadis’ proof might have a feeling of déjà-vu in this chapter: The proof given here is basically Athanasiadis’ proof adapted to the more general setting. In the end I have tried to preserve as much of this parallelism as possible, although originally I had not planned to do so. I do not know if the result on (k,ℓ)-incidence graphs with ℓ 2k + 1 that we proved in Chapter 3 also generalizes to the setting of this≥ chapter. In the proof of this result we had used the fact that every d-polytope is a refinement of the d-simplex, and I do not know how to generalize this to graph manifolds in a meaningful way.

53 4. Athanasiadis’ Conjecture on Incidence Graphs

4.1 Regular Cell Complexes

The definitions and notation for regular cell complexes are from [23, Sec- tion 4.7]. Definition 4.1.1 (Regular cell complex). A regular cell complex is a C finite collection of balls σ in a Hausdorff space = σ σ that satisfies k C k ∈C the following conditions: S (i) The empty set is a member of , ∅ C (ii) the interiors σ◦ partition , and k C k (iii) the boundary ∂σ of every σ is the union of some members of , for all σ . C ∈ C We call the balls σ the cells or faces of . The dimension of a regular cell complex is the maximal dimension of oneC of its faces. Faces of dimension 0 and 1 are called the vertices and edges of , re- C spectively. Denote by f0( ) the number of vertices of . We write ( ) for the set of vertices of , andC G( ) for the graph of , thatC is, the regularV C cell complex given by theC verticesC and edges of . C If is d-dimensional, then the faces of dimensionC d are called facets, and the facesC of dimension d 1 are called ridges. If T is a topological space− and T = then is called a regular cell decomposition of T . The set of all faces kσ C k orderedC by inclusion forms a poset, the face poset ( ). ∈ C A regular cell complexF C is Cohen-Macaulay if its face poset ( ) is a Cohen-Macaulay poset; seeC [21, Section 11]. We only need theF followingC two properties of Cohen-Macaulay cell complexes. They are pure, that is, every face lies in a facet, and they are strongly connected, that is, every two facets can be connected by a sequence of facets in which two consecutive facets intersect in a ridge. A regular cell complex is strongly regular if it satisfies the intersection property, that is, the intersection of two faces is a single face.

4.2 Athanasiadis’ Conjecture

Given a strongly regular cell complex , we denote by Gk( ) the graph on the k-cells of in which two k-cells areC connected by an edgeC if they lie on C a common (k + 1)-cell. Thus, we have G0( )= G( ). Athanasiadis conjectured the followingC extensionC of his result on the connectivity of (k, k + 1)-incidence graph of polytopes.

54 4.2. Athanasiadis’ Conjecture

Conjecture 4.2.1 (Athanasiadis [3, Conjecture 6.2]). Let be a d- dimensional Cohen-Macaulay strongly regular cell complex. ThenC the graph G ( ) is k C

(k + 1)(d k)-connected if 0 k d 3 and • − ≤ ≤ −

d-connected if k = d 2. • −

(For k = d 1, the generalization is false: Consider, for example, the complex consisting− of two d-simplices glued together along a common facet.)

The main difficulty in adapting the proof is that face figures of regular cell complexes are not regular cell complexes, in general. Consider, for instance, the double suspension of the Poincaréhomology 3-sphere [35] [92]. To circumvent this problem we consider Conjecture 4.2.1 in greater generality in the setting of Barnette’s graph manifolds [8]. We have a similar problem here; a face figure of a graph manifold is not necessarily a graph manifold. The punchline will be however that the class of graph manifolds that is closed under taking vertex figures is still large enough to cover Athanasiadis’ conjecture. There are some technical subtleties though, as discussed in Section 4.3. With the right setup and definitions we have that the strong components of a vertex figure of a graph manifold are graph manifolds. The only technical problem then is to control the different components of the vertex figures. This is done using the notion of strong walks, which is introduced in Section 4.4.2. A strong walk is, basically, a walk in the graph of a graph manifold along a strong chain of facets, that is, along a sequence of facets such that consecutive facets intersect in a ridge. Because a strong chain cannot jump between strong components in a face figure, a strong walk “selects” strong components in a consistent way. There is an alternative approach that might turn out to be more elegant: A normalization procedure described by Stanley [106, p. 83], Goresky & MacPherson [47, p. 151], Björner [19], and Kalai [59], turns a simplicial pseudo-manifold into a normal one, that is, one in which links of faces are connected. This technique works well in connection with lower bound theorems; see Kalai [59] and Tay [111]. I do not know how to make it work for the connectivity problem, though. The problem is that disjoint paths in an incidence graph of the normalized manifold are not necessarily disjoint in the manifold we started with (but see [10]).

55 4. Athanasiadis’ Conjecture on Incidence Graphs

4.3 Graph Manifolds

Graph manifolds were devised by Barnette as a tool to study graph properties of manifolds [8].

Definition 4.3.1 (Face structure, graph manifold, pseudo-graph manifold). A d-face structure is a graph, together with a collection of r-face structures, for 2 rM d 1, called the faces of (we alsoC call a face), inductively− defined≤ ≤ as− follows. A ( 2)-face structureM is the emptyM set. A ( 1)-face structure is single vertex −v with faces v and .A 0-face structure− is an edge e = uv with faces e, u, v, and . ∅ For d 1, the collection of faces of the d-face structure∅ is a collection of≥ r-face structures, forC 2 r d 1 that satisfies the followingM conditions: − ≤ ≤ −

(1) If F is a face of , and F ′ is a face of F , then F ′ (closed under∈ taking C faces). M ∈ C

(2) Every face F of is a face of a (d 1)-face structure of (pure). M − M If F is an r-face structure and F , we say that F is an r-face of . The 0-, 1-, (d 2)-, and (d 1)-faces∈ of C are the vertices, edges, ridgesM, and facets of − , respectively.− We denoteM the vertices of by ( ). We say thatM M V M M (3) has the intersection property if for all F,G the intersection F G is a face of F and of G; ∈ C ∩

(4) is strongly connected if for every two facets , ′ there is a sequence F F ∈ C

= , , , ,..., , = ′ F F 0 R1 F 1 R2 Rn F n F

of facets and ridges, called a strong chain, such that i i+1 = i+1, for all i = 0,...,n 1; F ∩F R − (5) has the pseudo-manifold property if each ridge of is a face of exactly two facets of . M M A weak d-graph manifold is a d-face structure that satisﬁes (3) and (5), and such that every r-face of , for 2 r d 1, is again a weak r-graph manifold. For example,M a weak− 1-graph≤ ≤ manifold− is a collection of disjoint cycles and all their faces. A d-graph manifold is a weak d-graph manifold that additionally satisﬁes (4), and such that every r-face, 2 r d 1, is again an r-graph manifold. − ≤ ≤ − 56 4.3. Graph Manifolds

face structure (1, 2)

weak graph manifold (3, 5)

pseudo-graph manifold graph complex (faces are graph manifolds) (3, 4)

weakly normal graph manifold graph complex

weakly normal normal graph complex graph manifold

normal graph manifold

Figure 4.1: Inclusion relations among some of the structures deﬁned in this section.

A d-pseudo-graph manifold is a weak d-graph manifold such that every r-face, 2 r d 1, is an r-graph manifold. If a−d-face≤ structure≤ − satisﬁes (3) and (4), and every r-face of , for 2 r d 1, is anMr-graph manifold, we say that is a d-graphM complex− ≤. ≤ − M A d-graph complex such that

(6) each ridge of is contained in at most two facets of M M is a d-graph manifold with boundary.

The definitions of graph manifold and pseudo-graph manifold coincide with the definitions given by Barnette [8]. The defined structures are related by their inclusions in Figure 4.1. There already the classes of weakly normal and normal graph manifolds appear. We will define these classes, as well as the corresponding graph complexes, below.

57 4. Athanasiadis’ Conjecture on Incidence Graphs

Definition 4.3.2 (Face figure, cf. [8, Proof of Theorem 7]). Let be a d-graph manifold, and F a (k 1)-face of , for 0 k d. The faceM − M ≤ ≤ figure of at F , denoted /F , is the (d k 1)-face structure obtained in the followingM way: M − −

(i) For every k-face in that contains F there is a vertex in /F . M M (ii) Inductively, a collection of r-faces of /F , for 1 r d 3, determines an (r + 1)-face of /v ifM and only if− the≤ corresponding≤ − M (r +k +1)-faces of are exactly the (r +k +1)-faces of an (r +k +2)- face of that containM F . M If F is a vertex, we call /F a vertex figure, and if F is an edge, we call /F an edge figure. M M The poset of faces of a graph manifold is a graded lattice, with as 1ˆ and as 0.ˆ Every interval in this lattice is the face poset of a weakM graph manifold∅ and face figures are upper intervals in this lattice. A subtle mistake seems to appear in Barnette’s work [8, Proof of The- orem 7] in connection with vertex figures of graph manifolds. He claims that a vertex figure “clearly” is a pseudo-graph manifold [8, p. 67] and concludes that the maximal strongly connected components of the vertex figure, called the strong components, are graph manifolds. Indeed, it follows from properties (1), (2), (3), and (5) that these also hold for face figures. Thus, by induction a face figure of a graph manifold (or more generally, of a weak graph manifold) is a weak graph manifold. However, a face of a face figure may fail to be a graph manifold. Thus, Barnette’s claim that a vertex figure is a pseudo-graph manifold is incorrect. To support our point, we construct an explicit example of a graph manifold in which the unique strong component of a vertex figure is not a graph manifold. Given a graph manifold we construct a graph manifold pyr( ), called the pyramid over , inM the following way: The graph of pyr( M) is the join of the graph of Mwith an additional vertex v , that is, it consistsM M 0 of the graph of and all the edges between v0 and vertices of the graph of . The pyramidM over a ( 2)-graph manifold, that is, the empty set, is the M − single vertex v0 with v0 and as its faces. For higher dimensions, all the faces of are faces of pyr( ∅ ), and for every face F of , the pyramid pyr(F ) isM a face of pyr( ).M If is a d-graph manifold, thenM pyr( ) is a (d + 1)-graph manifold.M M M The 2-graph manifold in Figure 4.2(a) (after a figure by Barnette [10]) is our starting point forM our counterexample. The vertex figure at v is a

58 4.3. Graph Manifolds

(a) A pinched torus. The vertex figure at (b) The vertex figure at v of the pyramid v consists of two disjoint cycles. over the pinched torus. The face F consists of two disjoint cycles. disjoint union of two triangles, which is a weak graph manifold, and the strong components of it are graph manifolds. So, it is indeed a pseudo- graph manifold. However, if we take ′ as the pyramid over , then the M M vertex figure at v of ′ is the weak graph manifold in Figure 4.2(b) (where M v0 is the vertex of ′ /v that corresponds to the edge from v to the pyramid apex). M What is strange about this weak 2-graph manifold? The two disjoint cycles that form the base of the two pyramids with apex v0 form a single face F , according to Definition 4.3.2. In particular, this vertex figure has only one strong component, as a strong chain can “jump” via F from one pyramid to the other. Barnette’s proof of [8, Theorem 7] by induction via vertex figures is rendered invalid by this problem. The result of this theorem that every d-graph manifold contains a subdivision of Kd+1 can however be proved for weak d-graph manifolds with exactly the same proof. This implies the result for d-graph manifolds. So, in this case, it is just a matter of choosing the right generality for the induction. For our methods in this chapter it is however crucial that the strong components of a face figure are graph manifolds. Thus we define the class of weakly normal graph manifolds as follows.

Definition 4.3.3. A d-graph manifold is weakly normal if either d 1 or d 2 and every strong component of every vertex figure is again a wea≤ kly normal≥ graph manifold. It is normal if either d 1 or d 2 and every vertex figure is a normal graph manifold. ≤ ≥

59 4. Athanasiadis’ Conjecture on Incidence Graphs

By this deﬁnition, all 2-graph manifolds are weakly normal, but not all are normal. For example, the pinched torus in Figure 4.2(a) is weakly normal, but not normal. Another example is given by the collection of graphs of the faces of a d-polytope. They form a normal (d 1)-graph manifold. −

Deﬁnition 4.3.4. A d-graph complex is weakly normal if every face of it is a weakly normal graph manifold. It is normal if every face of it is a normal graph manifold.

The generality of normal graph manifolds is still suﬃcient for a proof of Athanasiadis’ conjecture. Nevertheless, we prove the more general result for weakly normal graph complexes.

Definition 4.3.5 (Antistar, star, link). Let be a weak graph manifold and F a face of . M Define the d-faceM structure ast(F, ), the antistar of at F , tobe the collection of all faces of that doM not contain F . M M Define the d-face structure star(F, M), the star of at F , to be the collection of all faces that are contained in a face thatM contains F . Finally, define the link of at F by link(F, M) := ast(F, M) star(F, M). This is a (d 1)-face structure.M ∩ −

4.4 Basic Properties of Graph Manifolds

d+2 In this section, we (a) prove that a d-graph manifold has at least k+2 faces of dimension k, (b) introduce the notion of strong walk, and (c) prove two ¡ ¢ variants of Balinski’s theorem for graph manifolds.

4.4.1 Face Numbers of Graph Manifolds Lemma 4.4.1. Let be a weak d-graph manifold with d 0 and let u, v be vertices of . ThenM there is a facet that contains u≥but not v. In particular, theM antistar of any vertex contains at least one facet.

Proof. If is a weak 0-graph manifold the statement clearly holds. Let be a weakM d-graph manifold with d 1, and let u, v ( ). Let Mbe a facet that contains u. If does not≥ meet v, we are done.∈ V M Otherwise,F by induction, there is a ridge F in (a facet of the facet ) that contains R F F u but not v. By the pseudo-manifold property (5), there is a facet ′, distinct from , that contains . By the intersection property (3) we haveF F R ′ = . Thus, ′ is a facet that contains u but not v. F∩F R F 60 4.4. Basic Properties of Graph Manifolds

Lemma 4.4.2. Let d 2 and be a weak d-graph manifold. Then has at least d+2 faces≥ of − dimensionM k, for all 2 k d. M k+2 − ≤ ≤ Proof. We prove¡ ¢ the statement by induction on the dimension. The cases d = 2, 1 and k = 2,d are trivial. Let− − be a weak d−-graph manifold, where d 0, and let 1 k d 1. Let v beM a vertex of and choose a facet ≥in ast(v, −), which≤ ≤ exists− M F M d+1 according to Lemma 4.4.1. By induction, the facet has at least k+2 faces of dimension k. These are k-faces of that do notF contain v. Additionally, M d+1¡ ¢ any strong component of the vertex ﬁgure at v has at least k+1 faces of dimension k 1. These correspond to k-faces of that contain v. Thus, ¡ ¢ in total has− at least M M d + 1 d + 1 d + 2 + = k + 2 k + 1 k + 2 µ ¶ µ ¶ µ ¶ faces of dimension k.

d+2 Corollary 4.4.3. A d-graph manifold has at least k+2 faces of dimension k, for all 2 k d. − ≤ ≤ ¡ ¢ 4.4.2 Strong Walks in Graph Manifolds If and are strong chains in a graph manifold, and the last facet of C1 C2 C1 coincides with the first facet of 2 we write 1 2 for the concatenation of and , that is, for the strongC chain that firstC C traverses and then . C1 C2 C1 C2 Definition 4.4.4 (Strong walk). Let be a weakly normal graph manifold. A strong walk in is a pair (W,M χ) consisting of a walk M

W =(v0,e1,v1,...,en,vn) in G( ) and a map χ that associates to every i 0,...,n a strong chain χ(i) suchM that ∈ { }

v for every facet in χ(i), and • i ∈F F the last facet of χ(i) equals the ﬁrst facet of χ(i + 1) for every i • ∈ 0,...,n 1 , that is, the concatenation χ := χ(0)χ(1) χ(n) is a {strong chain− } in . C ··· M

A strong walk (W, χ) with W =(v0,e1,v1,...,en,vn) determines for every i 0,...,n a strong component of /vi and for every i 1,...,n a strong∈ { component} of /e . M ∈ { } M i 61 4. Athanasiadis’ Conjecture on Incidence Graphs

F n 0 F i v V i W

=( ,..., ) Cχ F 0 F m

Figure 4.2: A strong walk (W, χ) with the strong component induced at i. Vi

We denote the strong component of /vi that contains the quotient of the first facet of χ(i) with v by forM every i 0,...,n . Then /v i Vi ∈ { } M i contains every quotient of a facet of χ(i) with vi, as χ(i) is a strong chain of facets that all contain vi. Furthermore, for every i 1,...,n we have that the first facet of χ(i) ∈ { } contains the edge ei, because the first facet of χ(i) must coincide with the last facet of χ(i 1) and consequently must contain both vi 1 and vi. We − denote the strong− component of /e that contains the quotient of the M i first facet of χ(i) with ei by i for every i 0,...,n . It is important that the verticesE e /v ∈( { /v ) and} e /v ( /v ) i i ∈ V M i i+1 i ∈ V M i both lie in i and that a k-face of i corresponds to a (k + 1)-face in i 1 − and to a (kV+ 1)-face in . E V Vi Definition 4.4.5 (Concatenation of strong walks). Let (W, χ) and (W ′, χ′) be two strong walks in such that M the last vertex of W and the first vertex of W ′ coincide, • the last facet of and the first facet of ′ coincide. • Cχ Cχ Suppose that the lengths of W and W ′ are n and n′, respectively. The concatenation of (W, χ) and (W ′, χ′) is the strong walk

(W, χ)(W ′, χ′) :=(WW ′, χ˜), where WW ′ is the usual concatenation of walks in graphs, and whereχ ˜ is deﬁned by χ(i), for i 0,...,n ∈ { } χ˜(i) := χ(i)χ (0), for i = n  ′ χ′(i n), for i n + 1,...,n + n′ .  − ∈ { }  62 4.4. Basic Properties of Graph Manifolds

We describe how to construct a strong walk from a vertex v to a vertex w if we are given a strong chain =( , , ,..., , ) with v C F 0 R1 F 1 Rm F m ∈F 0 and w . Let w := v, w a vertex of for i 1,...,m 1 , and ∈ F m 0 i Ri ∈ { − } wm = w. Choose a walk Wi from wi to wi+1 in i for every i 0,...,m , and let W = W W W . F ∈ { } 0 1 ··· m Write Wi as (v0,e1,v1,...,en,vn), that is, we have v0 = wi and vn = wi+1. Deﬁne a strong chain (Wi, χi) by setting χi(j) = i for every j 0,...,n 1 . If i = m, we set χ (n)=( , , ), otherwiseF χ (n) =∈ { − } 6 i F i Ri F i+1 i i = m. F ThenF (W, χ) :=(W , χ )(W , χ ) (W , χ ) 0 0 1 1 ··· m m is a strong walk such that the walk W connects the vertices v and w.

4.4.3 Balinski’s Theorem for Graph Manifolds Lemma 4.4.6 (Barnette [8, Lemma 2]). The antistar of any vertex of a d-graph manifold is a d-graph manifold with boundary. In particular, it is strongly connected. Theorem 4.4.7 (Barnette [8]). Let be a d-graph manifold. Then the graph of is (d + 1)-connected. M Furthermore,M if ( ) is a set of vertices of cardinality = d that separates a strongS ⊆ chain, V M then is contained in a facet of .| S | S M Proof. The graph manifold has at least d+2 vertices by Corollary 4.4.3. Let be a subset of theM vertices of of cardinality at most d. Let S M u and choose v,w (M) . Let and ′ be facets of ast(u, ) with ∈ S ∈ V \S F F M v and w ′. These exist by Lemma 4.4.1. By Lemma 4.4.6, there ∈ F ∈ F is a strong chain of facets = 0, 1, 1,..., n, n = ′ in ast(u, ). By induction, the graph ofF everyF R isFd-connected.R F Furthermore,F everyM F i intersection i+1, i 0,...,n 1 ,isa(d 2)-graph manifold and thus has at least dRvertices,∈ { by Corollary− } 4.4.3. − Thus, if we choose vertices w1,...,wn with wi ( i) and set w := v and w := w, there are walks W ,...,W∈ V R \ Ssuch that 0 n+1 0 ⊆F 0 n ⊆F n Wi connects wi to wi+1. Thus, there is a walk W = W W W starting in v and ending in w 0 1 ··· n that avoids . Clearly, if a strong chain ′ , ′ , ′ ,..., ′ , ′ is separated S F 0 R1 F 1 Rn F n by , then either separates the graph of one facet ′ and we have ′ , S S F i S⊆F i or all vertices in lie in some intersection ′ = ′ ′ . S Ri+1 F i ∩F i+1 Lemma 4.4.8. Let d 1 and be a d-graph manifold. Then G( ) ( ) is connected for≥ any − facet Mof . M \ V F F M 63 4. Athanasiadis’ Conjecture on Incidence Graphs

Proof. We prove the statement by induction. The case d = 1 is trivial. Let be a facet of and u ( ) a vertex of . Let− v,w be two F M ∈ V F F vertices in G( ) ( ). By Lemma 4.4.1, there are facets 1 and 2 with v andM w\V F that avoid u (possibly = ). Clearly,G G ∈ G1 ∈ G2 G1 G2 G1 and 2 lie in ast(u, ). By Lemma 4.4.6 there is a strong chain of facets =G , , ,...,M , = that connects and in ast(u, ). G1 F 0 R1 F 1 Rn F n G2 G1 G2 M By induction, G( i) ( i ) is connected for every i 0,...,n , as we can choose aF facet\V thatF ∩F contains . Furthermore,∈ { for every} F i ∩F i 1,...,n we have i = , as i is not contained in . Thus, there is∈ a { walk in G} ( ) (R )\ that F 6 connects∅ R v,w. F M \V F Similarly one shows the following lemma that the dual graph of a graph manifold, that is, the graph on the set of facets in which two facets are connected if they share a common ridge, is at least 2-connected. Lemma 4.4.9. Let d 0, be a d-graph manifold, and ˜ a facet of . Then for any two facets≥ M= ˜ and = ˜ there is a strongF chain fromM to that avoids ˜ . F 6 F G 6 F F G F 4.5 Connectivity of Graph Manifold Skeleta

Deﬁnition 4.5.1. Let be a d-graph manifold and let 1 k ℓ d. M − ≤ ≤ ≤ The (k,ℓ)-incidence graph Gk,ℓ( ) is the graph on the k-faces of in which two k-faces are connectedM by an edge if they are contained inM a common ℓ-face of . We write Gk( ) for the (k, k + 1)-incidence graph of . M M M In this section we prove the following theorem. Theorem 4.5.2. Let d 1 and 1 k d 1. Let be a weakly normal d-graph manifold,≥ and − let G−( ≤) be≤ the (−k, k + 1)-incidenceM graph k M of . Then the graph G ( ) is M k M (d + 1)-connected if k = d 2, and • − (k + 2)(d k)-connected, if 1 k d 3 or k = d 1. • − − ≤ ≤ − − We deﬁne the functionm ˜ (d) for 1 k d 1 as k − ≤ ≤ − d + 1, k = d 2 m˜ (d) := k (k + 2)(d k), otherwise.− ½ − Thus, Theorem 4.5.2 states that the graph G ( ) of a weakly normal k M d-graph manifold ism ˜ k(d)-connected.

64 4.5. Connectivity of Graph Manifold Skeleta

The case k = d 1 directly follows from Corollary 4.4.3: A d-graph − manifold has at least d + 2 facets, and Gd 1( ) is the complete graph on − M the set of facets of . The case k = 1 is Theorem 4.4.7. The proof of theM other cases is split− into the next three sections: We prove the special case k = d 2 in Section 4.5.1. • The other case is proved by induction− on k. The base case of the • induction is Balinski’s theorem for graph manifolds. However, the case k = 0 needs slightly different arguments than the case k 1, and we start with this case in Section 4.5.2. ≥ In Section 4.5.3 we finish the induction for all 1 k d 3. • ≤ ≤ − 4.5.1 The Case k = d 2. − Lemma 4.5.3 (Barnette [8, Lemma 1]). Let d 1, and let ≥ =( , , ,..., , ) C F 0 R1 F 1 Rn F n be a strong chain of distinct facets in a d-graph manifold such that each facet of contains a fixed (d 3)-face H. Then there is a strong chain of distinct facetsC −

′ =( , , ,..., , , , ,..., , = ) C F 0 R1 F 1 Rn F n Rn+1 F n+1 Rn+k F n+k F 0 such that each facet of ′ contains H. Such a strong chain is called a strong cycle. C The proof of the following lemma is, at its core, identical to Athanasiadis’ proof in the polytope case [3].

Lemma 4.5.4. Let be a d-graph manifold. Then the graph Gd 2( ) is − (d + 1)-connected. M M Proof. We can assume that d 1, as the other cases are trivial. Let be a subset of the set≥ of (d 2)-face of of cardinality at most d, and letS F,G be two (d 2) faces of− that areM not in . − M S A strong chain =( 0, 1, 1,..., n, n) from a facet 0 that contains F to a facet C thatF containsR F G naturallyR F correspondsF to a walk in F n Gd 2( ). − M Suppose that for some i 1,...,n the ridge i is a member of . Because of the cardinality of ∈and { by Corollary} 4.4.3,R there is a (d 3)-faceS S − H of i such that i is the only face in that contains the face H. We then constructR a strongR cycle of facets asS in Lemma 4.5.3 around the face H. This yields a strong chain of facets from i 1 to i that avoids the − ridge and, by the choice of H, any elementF of . F Ri S 65 4. Athanasiadis’ Conjecture on Incidence Graphs

In the proof we have gotten a glimpse of the idea of the strong walks: Lemma 4.5.3 essentially says that the strong components of the face figure at a (d 3)-face are 1-graph manifolds. Because the walk in Lemma 4.5.4 is a strong− chain, we stay in one strong component of the face figure at a (d 3)-face at every forbidden face we have to avoid. − 4.5.2 The Case k = 0. Let be a weakly normal graph manifold and let be a subset of the edgesM of . S M Let v be a vertex of and v one of the strong components of /v. Define s(v, ) to be theM numberM of edges in that correspond to verticesM Mv S in the component v. Let e = uv beM an edge of , choose a strong component of M Mu /u and a strong component v of /v, and define t(e, u, v) := sM(u, )+ s(v, ). M M M M Mu Mv Let (W, χ) be a strong walk with W =(v0,e1,v1,...,en,vn). As defined in Section 4.4.2, let 0,..., n be the strong components of the vertex figures along W inducedV by theV strong walk (W, χ). We then write s(i) for s(vi, i) and t(i) for t(ei, i 1, i). V V − V Definition 4.5.5 (Connecting walk, good walk). Let (W, χ) be a strong walk in a weakly normal graph manifold with M

W =(v0,e1,v1,...,en,vn), and =( , , , ,..., , ). Cχ F 0 R1 F 1 R2 Rm F m If F ,G are faces of with v F and v G , we 1 1 ∈ M M 0 ∈ 1 ⊆ F 0 n ∈ 1 ⊆ F m say that the strong walk (W, χ) connects F1 and G1. We say that the strong walk (W, χ) is good (in the case k = 0), if it satisﬁes the following two conditions:

(i) s(i) d 1, for 0 i n and ≤ − ≤ ≤ (ii) t(i) d, for all 1 i n with e . ≤ ≤ ≤ i ∈ S The following lemma shows that a good walk that connects two k-faces (here, k = 0) yields the desired walk in G ( ). k M Lemma 4.5.6. Let d 3, be a weakly normal d-graph manifold and a subset of the edges of≥ cardinalityM less than 2d. Let F and G be edges of S M that are connected by a good walk. Then there exists a walk in G0( ) that connects F and G. M \ S

66 4.5. Connectivity of Graph Manifold Skeleta

W =(v0,e1,v2,e2,...,v5)

a 2 G v0 b2 F e4 e2 v5 a4 b4

Figure 4.3: The choice of edges along a good walk in . In this example, the M edges e and e are the only edges of W that are members of . For all other 2 4 S edges we have ai = bi = ei (i = 1, 3, 5). Furthermore, a0 = F and b6 = G.

Proof. Let (W, χ) be a good walk that connects F and G with W = (v0,e1,v1,...,en,vn). Let 0,..., n and 1,..., n be the components of the vertex figures and edgeV figuresV alongE the walkE (W, χ), respectively, that are induced by the walk (W, χ). For every i 1,...,n , the graph ∈ { } manifold i isa(d 2)-graph manifold and therefore has at least d vertices, by CorollaryE 4.4.3.− Let J 1,...,n be the set of indices of edges of W such that e ⊆ { } j ∈ S for every j J. For every∈ j J we choose two edges a and b in in the following ∈ j j M way: By Definition 4.5.5 (ii), there is a vertex in j that corresponds to a 1-face H in such that the two edges of H thatE intersect e are not in . M j S Let bj be the edge of H that intersects ej in vj 1, and let aj be the edge − of H that intersects ej in vj. The edges aj and bj are neighbors in G0( ), because they both lie in the 1-face H. M For every j 1,...,n J we set a = b = e . Finally, let a = F and ∈ { } \ j j j 0 bn+1 = G. For 0 i n, observe that a and b are edges of which share ≤ ≤ i i+1 M \ S vi as a common endpoint; compare Figure 4.3. Furthermore, for every i 0,...,n , the vertices ai/vi and bi+1/vi both lie in . The graph manifold∈ { is} of dimension d 1 and therefore Vi Vi − d-connected. By Definition 4.5.5 (i), there are at most d 1 vertices in i that correspond to elements in . The graph G( ) minus− these verticesV is S Vi therefore connected by Theorem 4.4.7. Thus, ai and bi+1 can be connected by a walk in G0( ) . Therefore, F = a0 and G = bn+1 can also be connected in G ( M) \ S. 0 M \ S It remains to show that a good walk always exists. This is established by the following lemma.

67 4. Athanasiadis’ Conjecture on Incidence Graphs

Lemma 4.5.7. Let d 3, be a weakly normal d-graph manifold and a subset of the edges of≥ Mof cardinality less than m˜ (d) = 2d. For anyS M k two edges F,G of there is a good walk that connects F and G. M \ S

Proof. We call a vertex v of bad if it is an endpoint of at least (d + 2)/2 edges in . Otherwise we callM it good. Call an edge bad if it intersects at least d +S 1 edges in , otherwise good. If(W, χ) is a strong walk in and W avoids bad vertices,S then (W, χ) is a good walk. The number pMof bad vertices satisﬁes p (d + 2)/2 2 2(2d 1). Since d 3 we have p d, that is, there⌈ are at most⌉ ≤ d |bad S | ≤vertices.− ≥ ≤ If there are facets that contain all bad vertices let ˜ be one of them. Otherwise, let ˜ be any facet. F Let be aF facet with F and a facet with G and = ˜ F ∈ F G ∈ G F 6 F and = ˜ . Furthermore, let = ( = 0, 1, 1,..., n, n = ) be a strongG 6 chainF that connects Cand F andF thatR avoidsF ˜R. ThisF existsG by Lemma 4.4.9. Then, for any goodF vertexG v in and anyF good vertex w in , there is a good walk from v to w. F G Thus, it suﬃces to show that we can begin a good walk at one of the endpoints of F and end a good walk at one of the endpoints of G. Suppose F = ab and G = xy. Let

a be the strong component of /a that contains /a, • M be the strong component of M/b that contains F/b, • Mb M F x be the strong component of /x that contains /x, • M be the strong component of M /y that contains G /y. • My M G We distinguish in the following two cases. Case (i). Consider first the edge F , and suppose that one of the endpoints of F , without loss of generality a, satisfies s(a, a) d 1. The set has at most 2d 1 elements and F / , thus we mustM have≥ −s(b, ) S − ∈ S Mb ≤ d 2. Thus there is an edge HF that corresponds to a vertex in b that does− not belong to . Simple counting shows that one of the endpointsM of S HF is good and that HF is good. If we assume that one of the endpoints of G, say x, satisfies s(x, ) Mx ≥ d 1, we analogously derive the existence of an edge HG at y with the desired− properties. Case (ii). Suppose that s(a, a) d 2 and s(b, b) d 2. Then each of a and b is an endpoint ofM one edge≤ of− that correspondsM ≤ − to a vertex in or , respectively, that is not in M. We can further assume that Ma Mb S both a and b are bad. Suppose first that there are distinct vertices a′ and b′ of and that correspond to edges of that connect a to a′ and b to Ma Mb M b′. Counting shows that at least one of a′, b′ is good. In this case, we have

68 4.5. Connectivity of Graph Manifold Skeleta

the existence of a good edge HF that is incident to F , that corresponds to a vertex in either or , depending which of the vertices a′ and b′ is Ma Mb good, and that has a good endpoint. Otherwise, both a and b must be connected to a vertex c of by edges that correspond to vertices in and , respectively, and thatM are not in Ma Mb . Furthermore, we have s(a, a)= s(b, b)= d 2. Then s(c, c) 1 Sfor any strong component Mof /c andMc is a good− vertex. In thisM case,≤ Mc M let HF be the edge from a to c. Then HF is good. Similarly, if we assume that s(x, ) d 2 and s(y, ) d 2, we Mx ≤ − My ≤ − derive the existence of an edge HG with the desired properties. Let q a, b be the common vertex of F and H , let q x, y 1 ∈ { } F 2 ∈ { } be the common vertex of G and HG, let q1 be the strong component of /q that contains the facet /q , and letM be the strong component M 1 F 1 Mq2 of /q that contains the facet /q . M 2 F 2 Let ′′ be a facet with H ′′ and ′′ /q and ﬁnd a strong F F ∈ F F 1 ∈ Mq1 chain from to ′′ that avoids F˜ such that every facet of contains C1 F F C1 q . Let ′′ be a facet with H ′′ and ′′ /q M and ﬁnd a strong 1 G G ∈ G G 2 ∈ q2 chain from to ′′ that avoids F˜ such that every facet of contains C2 G G C2 q2. Both 1 and 2 can be chosen to avoid F˜ by Lemma 4.4.9. C C 1 1 1 ˜ ˜− The chain := 1 1− 2 2− (where for a chain the chain is the strong chain onC theC sameC C set C ofC facets traversed in theC oppositeC direction) is a strong chain of . It supports a good walk (W, χ˜) that connects F and G. M

4.5.3 The Case k 1. ≥ Let be a weakly normal graph manifold and let k 1. Let be a subset of theMk-faces of of cardinality less thanm ˜ (d)=(≥k + 2)(dS k). M k − Let (W, χ) be a strong walk with W = (v0,e1,v1,...,en,vn), and let V ,..., , ,..., be the strong components of the vertex figures and 0 Vn E 1 E n edge figures, respectively, along W that are induced by (W, χ). For i 0,...,n , define s(i) to be the number of k-faces in that ∈ { } S correspond to (k 1)-faces in i. For i 1,...,n− , define Vt(i) to be the number of k-faces in that correspond∈ to { (k 2)-faces} in . S − E i Definition 4.5.8 (Good walk). We say that a strong walk (W, χ) is good if it satisfies the following two conditions:

(i) s(i) < m˜ k 1(d 1) for all i 0,...,n , and − − ∈ { } (ii) t(i) < d for all i 1,...,n . k ∈ { } ¡ ¢ 69 4. Athanasiadis’ Conjecture on Incidence Graphs

As in the case k = 0, the existence of a good walk establishes the existence of the desired walk in G ( ). k M Lemma 4.5.9. Let d 4, k 1, be a weakly normal d-graph manifold ≥ ≥ M and a subset of the k-faces of of cardinality less than m˜ k(d)=(k + 2)(d S k). M Let− F and G be two k-faces of and suppose that (W, χ) is a good walk that connects F and G. ThenM there exists a walk in G ( ) that k M \ S connects F and G. Proof. Let (W, χ) be a good walk that connects F and G with

W =(v0,e1,v1,...,en,vn) =( , , ,..., , ). Cχ F 0 R1 F 1 Rm F m

Set G0 := F and Gn+1 := G. Let i and i be the strong components of /v and /e , respectively, thatV are inducedE by (W, χ). For every i M i M i choose a k-face Gi in with Gi/ei i. One sees that such a face exists, using conditionM (ii) \ of S Definition 4.5.8∈ E and Corollary 4.4.3. Now, vi is a vertex of Gi and Gi+1. Furthermore, Gi/vi and Gi+1/vi both lie in . By induction and by Definition 4.5.8 (i), there is a walk in Vi Gk 1( i) that avoids the (k 1)-faces of i that correspond to faces in − V − V S and that connects Gi/vi and Gi+1/vi. Thus, also F = G0 and G = Gn+1 can be connected by a walk in G ( ). k M Before we can show that a good walk always exists, we need one more lemma. This lemma is the combinatorial analogon to a simple geometric statement: If any (k + 2) points of a set in Rd lie in a k-flat, that is, are affinely dependent, then the whole set lies in a k-flat. Lemma 4.5.10. Let U ( ) be a subset of the vertices of a graph ⊆ V M U manifold and k 2. Suppose that for every U ′ k+2 there is a (k 1)-faceM that contains≥ U. Then there is a single (k 1)∈-face that contains ¡ ¢ U.− −

Proof. Suppose there is a set U ′ of cardinality k + 1 such that the smallest face F that contains U ′ is of dimension k 1. By assumption, U ′ v is − ∪ { } contained in a (k 1) face F ′ and necessarily F ′ = F , by the intersection − property (otherwise F ′ F is a smaller face than F that contains U ′). ∩ If no such set as U ′ exists, then the smallest face that contains a set U ′′ U of cardinality k + 1 is of dimension less than k 1 for any set U ′′. Thus,⊆ by induction the statement follows, where for k−= 2 only the ﬁrst case applies.

70 4.5. Connectivity of Graph Manifold Skeleta

Finally, we show that a good walk always exist.

Lemma 4.5.11. Let d 4, k 1, and let be a weakly normal d-graph ≥ ≥ M manifold and a subset of the k-faces of of cardinality less than m˜ k(d)= (k +2)(d k)S. For any two k-faces F,GMof there is a good walk that connects F− and G. M \ S

Proof. To prove the lemma we distinguish the cases k = 1 and k 2. Case k = 1. The set is a subset of the 1-faces of ,m ˜ (d)≥ = 3d 3, S M k − andm ˜ k 1(d 1) = 2d 2. Call an edge or vertex of bad if it is contained − in at least d−or 2d 2− elements of , respectively. M We show that − S (a) there are at most two bad edges, (b) if v,w are two distinct bad vertices, then u and v are connected by a bad edge, and (c) if v is a bad vertex and e is a bad edge, then v is a vertex of e. Suppose there are three bad edges e ,e ,e . Then for i 1, 2, 3 the 1 2 3 ∈ { } edge ei is incident to at least d edges. Any two of these three edges share at most one 1-face of . Thus, S 3d 3. | S | ≥ −

This contradicts that

that contain U ′. That is, for every subset of U of size k + 2 the vertices of this subset all lie in a common (k 1)-face of . By Lemma 4.5.10, there − M is a single (k 1)-face H that contains U. − Let ′ be a facet that contains the face H. Choose facets and F ∈ M F with F and G with = ′ and = ′, and let G ∈F ∈ G F 6 F G 6 F =( = , , ,..., , = ) C F F 0 R1 F 1 Rm F m G be a strong chain between and that avoids ′—this exists, as the dual graph is at least 2-connectedF by LemmaG 4.4.9. F Let v F and w G with v,w / U. These exist because U is contained ∈ ∈ ∈ in a (k 1)-face of . The intersection of ′ with every i of the strong chain −is at most aM (d 3)-graph manifold,F thus for everyR there is a C − Ri vertex not in U. Also, the graph of every minus ′ is connected, F i F ∩F i by Lemma 4.4.8. Thus, there is a walk W from v to w that visits the facets of consecutively and that avoids the vertices in U. C This also concludes the proof of Theorem 4.5.2.

4.6 Proof of Athanasiadis’ Conjecture

Theorem 4.6.1. Let be a weakly normal (d + 1)-graph complex, and let G ( ) be the (k, k +M 1)-incidence graph, for 1 k d 2. Then k M − ≤ ≤ − d + 1, k = d 2 κ(Gk( )) − M ≥ (k + 2)(d k), otherwise ½ − Proof. Recall that d + 1, k = d 2 m˜ (d) := k (k + 2)(d k), otherwise,− ½ − and let be a set of k-faces of cardinality less thanm ˜ k(d). The Sd-faces of are weakly normal d-graph manifolds. Let F,G be two k-faces of ,M let and be two facets of such that F and M F G M ⊆ F G . Choose a strong chain =( = 0, 1, 1,..., n, n = ) from ⊆to G . C F F R F R F G F G Then k( i) is at leastm ˜ k(d)-connected, for every i 0,...,n by G F d+1 ∈ { } Theorem 4.5.2. Furthermore, every i has at least k+2 faces of dimension k. Since we have R ¡ ¢ d + 1 (k + 2)(d k), for k d 3, k + 2 ≥ − ≤ − µ ¶ there is at least one k-dimensional face in i that is not in . Thus we can connect F and G by a walkR in G ( ). S k M 72 4.6. Proof of Athanasiadis’ Conjecture

As was remarked in Section 4.3, because of technical diﬃculties with vertex ﬁgures the methods of this chapter do not generalize in an obvious way to the whole class of graph manifolds.

Conjecture 4.6.2. Let be a (d + 1)-graph complex, and let Gk( ) be the (k, k + 1)-incidence graph,M for 1 k d 2. Then M − ≤ ≤ − d + 1, k = d 2 κ(Gk( )) − M ≥ (k + 2)(d k), otherwise ½ − Conjecture 4.6.2 is probably as general as one can get. Naatz [83] defines face structures (not to be confused with our face structures) that generalize Barnette’s graph manifolds and proves Balinski’s theorem for them. His definition has the advantage that every d-connected graph supports a face structure of rank d [83, Proposition 3.2.3]. This also means that his face structures are far away from resembling any geometric situation. For example, there are face structures in which the faces of rank one can have arbitrarily many vertices. Of course, the choice of graph manifold as the structure in which Conjec- ture 4.6.2 is stated is rather arbitrary. It would probably be more natural (and the definitions less convoluted) to state this conjecture for graded relatively complemented lattices with some additional properties (strongly connectedness, pseudo-manifold property); compare with the concept of abstract polytopes [51] [62]. Solely for historical reasons, I prefer graph manifolds to other terminology.

Lemma 4.6.3. Let be a d-dimensional Cohen-Macaulay regular cell complex with intersectionC property. Then the graphs of the cells of form a normal d-graph complex. C

Proof. Let be the d-graph complex formed by the graphs of the cells of M the cell complex . Then satisfiesC (1), as a “cell of a cell of is a cell of ”; it satisfies (3) by the intersectionM property of ; the complexC is stronglyC connected and pure by Cohen-Macauliness [3],C that is, the complexC satisfies (4) and (2); finally, every face F of satisfies the pseudomanifoldM property (5) as the boundary of the correspondingM cell of is a cell decomposition of a sphere; compare [23, p. 204]. Furthermore, becauseC every vertex figure of a manifold is a homology sphere, see Cairns [34, Chapter 7] or Munkres [82, 63], and thus strongly connected, the complex is normal. §

This ﬁnally proves Athanasiadis’ conjecure, Conjecture 4.2.1.

73 4. Athanasiadis’ Conjecture on Incidence Graphs

Corollary 4.6.4 (Conjecture 6.2 [3]). For any d-dimensional Cohen- Macaulay regular cell complex with intersection property, the graph G ( ) C k C is

(k + 1)(d k)-connected if 0 k d 3, • − ≤ ≤ − d-connected if k = d 2. • − Proof. This follows from Lemma 4.6.3 and Theorem 4.6.1.

74 Chapter 5

Linkages in Polytope Graphs

Linkages are a very important concept in graph theory. They play a major role in the theory of minors, and they are in a very strong sense related to connectivity; see Bollob´as & Thomason [28], Kostochka [65], Larman & Mani [66], Robertson & Seymour [95], Thomas & Wollan [113], and Thomason [114]. In this chapter, we consider linkages in graphs of polytopes. These were ﬁrst studied by Larman & Mani [66] and later by Gallivan [44] [45]. The Handbook of Discrete and Computational Geometry states the following question by Larman & Mani [66] as a problem:

Question 5.0.1 (Larman & Mani [66], Kalai [63, Problem 20.2.6]). Let G be the graph of a d-polytope and k = d/2 . Is it true that for every ⌊ ⌋ two disjoint sequences (s1,...,sk) and (t1,...,tk) of vertices of G there are k vertex-disjoint paths connecting si to ti, i = 1,...,k?

This question asks, rephrased in customary graph theory language, whether the graph of every d-polytope is d/2 -linked. We can answer in the aﬃrmative in dimensions⌊ ⌋ d 5: This is trivial ≤ in dimensions d = 0, 1, 2. In dimension d = 3 a polytope is 2-linked if and only if it is simplicial and otherwise 1-linked; see Figures 5.1(a) and 5.1(b) for illustrations of these cases. Every 4-polytope and every 5-polytope is 2-linked—this follows from the characterization of 2-linked graphs by Sey- mour [100], Shiloach [102], and Thomassen [115]; results by Jung [58]; or a geometric argument similar to those employed in some proofs of Balinski’s

75 5. Linkages in Polytope Graphs

t1 t2 t1

s2 s1 s2 (a) Simplicial 3-polytopes are 2-linked. (b) Every path from s1 to t1 disconnects s2 and t2.

Figure 5.1: Linkages in simplicial polytopes and 3-dimensional polytopes.

theorem, as in [118, Theorem 3.14]. In higher dimensions the situation is quite diﬀerent: Question 5.0.1 has a negative answer in dimensions 8, 10, and d 12. Even when k is chosen ≥ as 2(d + 4)/5 (which is strictly smaller than d/2 for all d 22), d- polytopes⌊ are not⌋ necessarily k-linked. Indeed, polytopes⌊ ⌋ with this≥ property were already discovered in the 1970s by Gallivan [44, Theorem 7, p.46] and later published by McMullen [78] and Gallivan [45]. We denote by k(d) the largest integer such that every d-polytope is k(d)- linked. Gallivan’s examples show that k(d) (2d + 3)/5 , and Larman & Mani [66] have proven a lower bound of (d +≤ 1) ⌊ /3 . We improve⌋ this lower bound marginally to (d + 2)/3 in Section⌊ 5.3. Of⌋ course, this improvement is irrelevant for asymptotic⌊ questions,⌋ but it implies exact values for k(d) in dimensions 7, 10, and 13. In the special case of simplicial polytopes, a precise answer was given by Larman & Mani [66]. They have shown that every simplicial d-polytope is (d + 1)/2 -linked. The stacked polytopes show that this bound cannot be⌊ improved.⌋ A closer look at Gallivan’s examples made it apparent that minimal linkedness of d-polytopes on f0 = d + γ + 1 vertices does depend on γ, at least if γ is small. In that respect, linkedness behaves diﬀerently than

76 5.1. Linkages connectivity in the polytope case. We therefore introduce a new parameter k(d, γ) that measures minimal linkedness of d-polytopes on d + γ + 1 vertices. We determine k(d, γ) for polytopes on at most (6d + 7)/5 vertices in Section 5.4 and analyze the combinatorial types of polytopes with linkedness exactly k(d, γ). Among the combinatorial types that meet the lower bound, Gallivan’s polytopes are in some sense the canonical ones, in some cases even unique: If f0 is odd there is only one combinatorial type with linkedness k(d, γ) among all polytopes on f0 vertices. This type is given by an iterated pyramid over a join of quadrilaterals. Because of the special combinatorial structure of these polytopes they are even projectively unique. We complement this result by showing that, if f0 is even, there are “many” combinatorial types of polytopes with minimal linkedness k(d, γ).

5.1 Linkages

Deﬁnition 5.1.1 (k-linked, linkage). Let G =(V,E) be a graph. The graph G is k-linked if V 2k and for every choice of 2k distinct | | ≥ vertices s1,...,sk,t1,...,tk there exist k vertex-disjoint paths L1,...,Lk such that Li joins si and ti for i = 1,...,k. If we write s = (s1,...,sk) and t = (t1,...,tk), we call the paths L1,...,LK an (s, t)-linkage. We denote by k(G) the largest integer k such that G is k-linked.

Deﬁnition 5.1.2 (Linkage parameters). We say that a polytope P is k-linked if the graph G(P ) is k-linked and deﬁne the following parameters for general polytopes:

k(P ) := k(G(P )), k(d, γ) := min k(P ) : P γ , { ∈Pd} k(d) := min k(d, γ). γ

For simplicial polytopes we deﬁne:

k (d, γ) := min k(P ) : P γ , S { ∈ Sd} kS(d) := min kS(d, γ). γ

We call k(d) the minimal linkedness of d-polytopes and kS(d) the minimal linkedness of simplicial d-polytopes.

77 5. Linkages in Polytope Graphs

In this chapter we use the following convenient notation for subpaths of a path L = v0v1 ...vk (found in [38]),

Lvi := v0 ...vi,

vjL := vj ...vk,

vjLvi := vj ...vi, where j i for i, j 0,...,k . ≤ ∈ { }

5.2 Simplicial Polytopes and 3-Polytopes

We determine the minimal linkedness of simplicial polytopes by reproving a theorem by Larman & Mani [66]. From this result, we derive an exact criterion for the linkedness of 3- polytopes.

5.2.1 Linkedness of Simplicial Polytopes It is almost trivial that a simplicial d-polytope is d/2 -linked; compare Lemma 5.4.1. To get the tight bound of (d + 1)/2 ⌊ proved⌋ by Larman & Mani [66] we have to work harder. ⌊ ⌋

Lemma 5.2.1. Let P be a simplicial d-polytope and v be a vertex of P . Then v is connected to every vertex of link (P )(v) by an edge. B Proof. The set of vertices in the link consists of precisely the vertices in facets incident to v, the vertex v excluded. Since P is simplicial, all these facets are simplices and thus have complete graphs.

The following theorem was shown by Larman & Mani [66]. Their proof is similar to ours. See Figure 5.2 for an illustration of the 4-dimensional case.

Theorem 5.2.2 (Larman & Mani [66]). Let d 2, γ 0. Then ≥ ≥ d + 1 k (d, γ)= k (d)= . S S 2 ¹ º Proof. We begin by proving the lower bound. Clearly, if d = 2 then kS(d, γ)= kS(d) = 1. If d = 3, let s = (s1,t1) and t = (s2,t2), where s1,s2,t1,t2 are distinct vertices of a simplicial 3-polytope P .

78 5.2. Simplicial Polytopes and 3-Polytopes

t1′

s2′ t2

t1′′

s1′ = s1′′

s2 s1

Figure 5.2: The link Q = link (P )(t2) in a 4-polytope P . The path from s2 to B Q extends to a path from s2 to t2. The paths from s1 and t1 to Q can be joined in R = linkQ(s2′ ).

The graph G(P ) is 3-connected by Balinski’s theorem. Thus by Menger’s theorem there exist disjoint paths S , T ,S from s ,t ,s to t , such that 1 1 2 { 1 1 2} 2

S1 link (P )(t2)= s1′ , T1 link (P )(t2)= t1′ , S2 link (P )(t2)= s2′ . ∩ B { } ∩ B { } ∩ B { } These conditions can be assumed by Lemma 5.2.1. Since Q := link (P )(t2) is combinatorially isomorphic to the boundary B complex of a 2-polytope and thus its graph is 2-connected, there is a path

S1′ in Q from s1′ to t1′ that avoids s2′ . Consequently, the paths

L1 :=(S1s1′ )S1′ (t1′ T1) and L2 := S2 are an (s, t)-linkage in P . Let d 4, and let P be a simplicial d-polytope. Let k = (d + 1)/2 , ≥ ⌊ ⌋ and let s = (s1,...,sk) and t = (t1,...,tk), where s1,...,sk,t1,...,tk are distinct vertices of P . We have to construct an (s, t)-linkage in G(P ). Consider Q := link (P )(tk), the link of P at tk. Then Q is combinato- B rially isomorphic to the boundary complex of a vertex ﬁgure at tk, that is, to the boundary complex of a simplicial (d 1)-polytope. Since d 2k 1 and by Balinski’s theorem,− there exist 2k 1 vertex- ≥ − − disjoint paths S1,...,Sk and T1,...,Tk 1 that connect the vertices s1,...,sk − and the vertices t1,...,tk 1 to tk. − Since P is simplicial we can assume that each of the paths hits Q exactly once by Lemma 5.2.1. Let s1′ ,...,sk′ and t1′ ,...,tk′ 1 be the intersection − 79 5. Linkages in Polytope Graphs vertices, that is,

s′ = S Q and t′ = T Q { i} i ∩ { j} j ∩ for each i = 1,...,k and j = 1,...,k 1. − Let R := link (s′ ). Since G(Q) is at least (2k 2)-connected, we Q k − ﬁnd vertex-disjoint paths S1′ ,...,Sk′ 1 and T1′,...,Tk′ 1 in Q that connect − − s1′ ,...,sk′ 1 and t1′ ,...,tk′ 1 to sk′ . We can assume that these paths hit − − R exactly once with intersection vertices s1′′,...,sk′′ 1 and t1′′,...,tk′′ 1 by Lemma 5.2.1. − − The polytope R is simplicial and (d 2)-dimensional. Therefore it is d 1 − d 1 −2 -linked by induction, and we have k 1 = −2 . Let L1′′,...,Lk′′ 1 ⌊ ⌋ − ⌊ ⌋ − be an (s′′, t′′)-linkage for s′′ = (s1′′,...,sk′′ 1) and t′′ = (t1′′,...,tk′′ 1) in R. Then − − (S s )(S s )L (t T )(t T ), 1 i k 1 L = i i′ i′ i′′ i′′ i′′ i′ i′ i i S , i =≤k ≤ − ½ k is an (s, t)-linkage in P . To prove the upper bound observe that any stacked polytope on d+γ+1 vertices has a separating set of vertices of size d 2 d+1 . ≤ ⌊ 2 ⌋ Larman & Mani [66] have proven Theorem 5.2.2 in more generality. They deﬁne a d-simplicial graph as follows: A 2-simplicial graph is a cycle. For d 3, a graph G is d-simplicial if it is connected and the graph induced by the≥ neighbors N(v) of any vertex v contains a (d 1)-simplicial graph − d+1 with N(v) as vertices. They have shown that a d-simplicial graph is 2 - linked. It is easy to see that the graph of a simplicial d-polytope is d- ¥ ¦ simplicial (induction on d by taking vertex links). The proof given above can be adapted to this situation, based on the fact that a d-simplicial graph is d-connected. This was shown in [66].

5.2.2 Linkedness of 3-Polytopes In order to write down an exact statement for linkedness of 3-polytopes we determine the parameter K(d, γ) := max k(P ) : P γ , { ∈Pd} that is, the maximal linkedness of a d-polytope on d + γ + 1 vertices. It is easily observed that 1 , d = 1, 2 K(d, γ)= 2 , d = 3  d+γ+1 , d 4.  2 ≥  ¥ 80 ¦ 5.3. Minimal Linkedness of Polytopes

This is trivial for d = 1, 2, while for d = 3 the statement follows from planarity. For d 4 there are polytopes with graph K for any γ 0— ≥ d+γ+1 ≥ take the d-dimensional cyclic polytope on d + γ + 1 vertices, for instance. As a consequence, we can precisely describe the linkedness of a given d-polytope when d 3 in the following corollary to Theorem 5.2.2. ≤ Corollary 5.2.3. If P is a 1- or 2-polytope, then k(P ) = 1. If P is a 3-polytope, then k(P ) 2 and k(P ) = 2 if and only if P is simplicial. ≤

5.3 Minimal Linkedness of Polytopes

In this section we provide lower and upper bounds on k(d) for general polytopes in arbitrary dimension d. We prove an upper bound on k(d, γ) that is independent of γ.

5.3.1 Lower Bound on Minimal Linkedness There are general results in graph theory that provide a link1 between connectivity and linkages. A k-linked graph is at least (2k 1)-connected, because a k-linked graph cannot have a separating set of size− 2k 2. The proof that a highly connected graph− is also highly linked is nontriv- ial. The most recent result of that type is by Thomas & Wollan [113], who have shown that every 10k-connected graph is k-linked. Balinski’s theorem thus implies that every d-polytopal graph is at least d/10 -linked. The bound of d/10 can be greatly improved by looking ⌊ ⌋ ⌊ ⌋ at large complete topological minors of polytope graphs. These exist by Corollary 2.1.5. In fact, Larman & Mani [66] have shown that every 2k-connected graph that contains a K3k as a topological minor is k-linked. Robertson & Sey- mour [95] have proven the much stronger statement that one may replace “topological minor” by “minor” in the previous statement. Consequently, every d-polytope is (d + 1)/3 -linked. However, already in dimension 4 this bound is not tight.⌊ It is easy⌋ to see by a geometric argument that every 4-polytope is 2-linked (as was remarked in the introduction, this also follows from the characterization of 2-linked graphs by Sey- mour [100], Shiloach [102], and Thomassen [115], or results by Jung [58]).

1or: a connection

81 5. Linkages in Polytope Graphs

We improve Larman & Mani’s bound slightly by considering rooted topological minors of d-polytopes. The proof of the following lemma is a variation of an argument by Diestel [38, pp. 70–71]. Lemma 5.3.1. Let G = (V,E) be a 2k-connected graph. Suppose that for every vertex v of G the graph G contains a subdivision of K3k 1 rooted at − v. Then G is k-linked.

Proof. Let s = (s1,...,sk) and t = (t1,...,tk), where s1,...,sk,t1,...tk are distinct vertices of G. Let K be a subdivision of K3k 1 rooted at vertex − tk with principal vertices U := U ′ tk , where U ′ N(tk). Since G t is (2k 1)-connected∪ { } there exist⊆ 2k 1 disjoint paths \ { k} − − S1,...,Sk and T1,...,Tk 1 in G that avoid tk such that Si joins si to U ′, − for i = 1,...,k, and Ti joins ti to U ′, for i = 1,...,k 1. Moreover, we assume that the paths have been chosen such that they do− not have interior vertices in U ′ (and thus also not in U) and that their total number of edges outside of E(K) is minimal. Let W = v1,...,vk,w1,...,wk 1 be the vertices of these paths in U ′, { − } where vi is in Si and wi is in Ti. We then have a partition of U into sets tk , W and W ′ := U ′ W with W ′ = k 1. Let u1,...,uk 1 be the { } \ | | − − vertices in W ′ U. We call these vertices free. ⊆ Since the path Sk joins sk to a neighbor of tk the path Lk := SkTk joins sk and tk, where Tk is the path that consists of the single edge from the vertex sk to the vertex tk. Now ﬁx some i 1,...,k 1 and let M be the path in K from the ∈ { − } i free vertex ui to vi and Ni be the path in K from ui to wi. Since the paths S1,...,Sk, T1,...,Tk 1 were chosen minimal with respect to their number − of edges outside of K and ui is a free vertex, the paths Sj are disjoint from M for j = i, and they are disjoint from N for all j = 1,...,k. Similarly, i 6 i the paths Tj are disjoint from Ni for j = i, and they are disjoint from Mi for all j = 1,...,k 1. Hence we can join6 v to w via the free vertex u . − i i i Denote by si′ the intersection vertex of Si and Mi that is closest to ui, and by ti′ the intersection vertex of Ti and Ni that is closest to ui. We then get pairwise disjoint paths

(S s )(s M )(N t )(t T ), 1 i k 1 L = i i′ i′ i i i′ i′ i i S T , i =≤k ≤ − ½ k k such that Li joins si and ti, that is, an (s, t)-linkage. See Figure 5.3 for an illustration of the proof. Theorem 5.3.2. Every d-polytope is (d + 2)/3 -linked. ⌊ ⌋ 82 5.3. Minimal Linkedness of Polytopes

s1 v1

W tk

t1 vk u1 sk W ′

Figure 5.3: Illustration of the proof of Lemma 5.3.1 with k = 2.

Proof. Let P be a d-polytope. We set k := (d + 2)/3 . For d 2, we then have d 2k and d + 1 3k 1. ⌊ ⌋ ≥ Therefore,≥ by Corollary≥ 2.1.5,− the graph G(P ) contains, at every vertex v, a K3k 1 subdivision rooted at v. By Balinski’s theorem, G(P ) is at least − 2k-connected. Lemma 5.3.1 implies that the graph of P is k-linked.

5.3.2 Upper Bound on Minimal Linkedness Theorem 5.3.3. Let d 2 and γ 1. Then the minimal linkedness of d-polytopes on d + γ + 1 ≥vertices satisﬁes≥

k(d, γ) d/2 . ≤ ⌊ ⌋ Proof. For d = 2 the assertion clearly is true. Let d 3 and γ 1. To prove the statement we have to construct a d-polytope≥ on d + γ +≥ 1 vertices with k(P ) d/2 . Let Q be a 3-polytope on 4 + γ vertices≤ that ⌊ ⌋ has a square facet. For instance, for γ = 1 take the pyramid over a square and for γ > 1 stack this pyramid γ 1 times over triangular facets. − d 3 Let P := pyr − (Q), the (d 3)-fold pyramid over Q. Then P is a − d-polytope and has d + γ + 1 vertices. We claim that P is not ( d/2 + 1)-linked. To see this let s1,t1,s2,t2 be the vertices of a square facet⌊ of Q⌋ (in that order around the facet). Then, by planarity, these cannot be linked in G(Q). Additionally, with m = (d 3)/2 there are exactly 2m vertices in (P ) (Q) if d is odd and exactly⌊ − 2m ⌋+ 1 if d is even. We choose distinct V \V 83 5. Linkages in Polytope Graphs

vertices s3,...,sm+2,t3,...,tm+2 arbitrarily from the set (P ) (Q) and, if d is even, we let s be the last vertex left in (P ) V (Q)\V and choose m+3 V \V tm+3 arbitrarily from (Q) s1,s2,t1,t2 . This set of d/2 +V 1 pairs\ { of vertices} cannot be linked in P . Therefore k(P ) d/2 . ⌊ ⌋ ≤ ⌊ ⌋ In the special case γ = 0 we trivially have k(d, γ)= (d + 1)/2 , as the d-simplex is (d + 1)/2 -linked. ⌊ ⌋ Theorem⌊ 5.3.3 implies⌋ that k(d) d/2 . We improve this bound sig- niﬁcantly in the next section. ≤ ⌊ ⌋

5.4 Linkages in Polytopes with Few Vertices

We now study linkedness of d-polytopes that have only few vertices more than the minimum of d + 1. One remark on the usage of the term “few vertices” is in order: It is used here in a slightly diﬀerent sense than usual. It does not mean that γ = f0 d 1 is considered as a constant. However, γ will not be larger than d in− this− section. For γ (d+2)/5, we precisely determine the value of k(d, γ) and analyze polytopes≤ that attain the value of k(d, γ).

5.4.1 Lower Bound for Polytopes with Few Vertices Linkedness of a graph is a local property in the following sense: If a graph is highly connected, then a k-linked subgraph ensures k-linkedness for the whole graph. The precise statement is the following lemma.

Lemma 5.4.1. Let G =(V,E) be a 2k-connected graph and G′ a subgraph of G that is k-linked. Then G is k-linked.

Proof. Let s = (s1,...,sk) and t = (t1,...,tk), where s1,...,sk,t1,...,tk are distinct vertices in G. Since G is 2k-connected, there exist 2k vertex disjoint paths S1,...,Sk and T1,...,Tk such that Si connects si to G′ and Ti connects ti to G′. We choose the paths such that each contains only one vertex from G′. Let s′ = G′ S and t′ = G′ T , for i = 1,...,k. { i} ∩ i { i} ∩ i Since G′ is k-linked there exists an (s′, t′)-linkage L1′ ,...,Lk′ in G′ for s′ =(s1′ ,...,sk′ ) and t′ =(t1′ ,...,tk′ ). Then L = S L′ T , 1 i k i i i i ≤ ≤ is an (s, t)-linkage in G.

84 5.4. Linkages in Polytopes with Few Vertices

We obtain a lower bound on linkedness of polytopes with few vertices by ﬁnding a highly-linked subgraph in the graph of P . This highly-linked subgraph is a complete subgraph: the graph of a simplex face of high dimension.

Lemma 5.4.2 (see Marcus [72], Kalai [62]). Let P be a d-polytope on d + γ + 1 vertices with d γ. Then P has a (d γ)-face that is a simplex. ≥ − Proof. The statement is true for every 2-polytope on 3 + γ vertices, γ 0. Let P be a d-polytope, d 3. Choose a facet F , which is of dimension≥ ≥ d′ = d 1. Suppose that F has d′ + γ′ + 1 vertices, where 0 γ′ γ. − ≤ ≤ By induction, F has a simplex face S of dimension dim(S)= d′ γ′ = − d 1 γ′ = d (γ′ + 1). If γ γ′ + 1, then dim(S) d γ and we are done.− − − ≥ ≥ − If γ = γ′, then (P ) (F ) = v and P = pyr (F ), that is, P is a V \V { } v pyramid over F with apex v. Hence pyrv(S) is a face of P and a simplex of dimension dim S +1= d γ. − Lemma 5.4.2 also follows from the Blumenthal–Robinson Theorem [26], see Theorem 10.2.1, by considering Gale diagrams: A Gale diagram of a polytope on f0 = d + γ + 1 vertices must contain a positive basis of size n where γ + 1 n 2γ. This positive basis corresponds to a complement of ≤ ≤ a simplex face of size f0 n d γ + 1. For “large” γ, the bound− given≥ − in Lemma 5.4.2 can be slightly improved; see Marcus [72].

Lemma 5.4.3. Let d 2, and d γ 0. Then ≥ ≥ ≥ d γ + 1 k(d, γ) − . ≥ 2 ¹ º Proof. In the special case γ = 0 the assertion is trivially true. For d 2, γ 1 it follows from Lemma 5.4.1 and Lemma 5.4.2, since 2 (d γ +≥ 1)/2 ≥ d γ + 1 d, and the graph of a d-polytope is at least d⌊-connected,− by⌋ Balinski’s ≤ − theorem.≤

5.4.2 Upper Bound for Polytopes with Few Vertices To prove a good upper bound on the number k(d, γ) we have to ﬁnd a polytope P on d + γ + 1 vertices with small k(P ). For γ (d + 2)/5 the lower bound of Lemma 5.4.3 can be attained. ≤ The examples we describe here were ﬁrst discovered in this context by Gallivan [44] [45]. He constructed them using Gale diagrams.

85 5. Linkages in Polytope Graphs

Deﬁnition 5.4.4. For integers n, m 0 and j , k ,...,j , k 1 deﬁne ≥ 1 1 m m ≥

(n, j1, k1,...,jm, km) := ∆n 1 (∆j1 ∆k1 ) (∆jm ∆km ) G − ∗ ⊕ ∗ · · · ∗ ⊕ and

(n, m) := (n, 1,..., 1) = ∆n 1 ¤ ¤ . G G − ∗ ∗ · · · ∗ 2m times m times | {z } | {z } The parameters d and γ for (n, m) can be determined by observing G that (n, m) has 4m + n vertices, so d + γ +1=4m + n, and dimension dim( G(n, m)) = n 1 + 3m. Therefore we have G − d = n 1 + 3m, and (5.1) − γ = m. (5.2) We consider the complement graph G( (n, m)) of G( (n, m)) to ex- amine the linkedness of the polytopes (n,G m). It is easyG to see that the graph of the join of two polytopes correspondsG to the join of the graphs of the polytopes. Thus, the complement graph G( (n, m)) has the following form. G

G( (n, m)) : G ······

2m edges n pyramidal vertices | {z } | {z } The reason for the low linkedness of (n, m) is that there are few vertices that can be used on a “detour” for a linkageG between the 2m pairs that are not connected by an edge. To determine the linkedness of (n, m), we determine the linkedness of G graphs of type Kp M, where M is a matching. Obviously, the graphs of G( (n, m)) are of\ this type. G

Lemma 5.4.5. Let G = Kp M, where M is a matching of size q p/2, that is, G consists of q disjoint\ edges and p 2q isolated vertices. Then≤ the linkedness of G is −

p , if p 3q 1 3 ≤ − k(G) = p q ( − , if p 3q 1. ¥ 2 ¦ ≥ − ¥ ¦ 86 5.4. Linkages in Polytopes with Few Vertices

Proof. Observe that the connectivity of G is p, if q = 0 κ(G) = p 1, if q 1, ½ − ≥ and that G has a complete subgraph of size p q. By Lemma 5.4.1, we thus have − κ(G) p q k(G) min , − . (5.3) ≥ 2 2 ½¹ º ¹ º¾ Let p 3q 1. Then choose all q pairs of nonconnected vertices and ≥ p 3q−+2 additional − 2 pairs arbitrarily from the vertices of full degree. Then these cannot⌊ be linked⌋ in G and the linkedness satisﬁes p 3q p q k(G) q + − = − . ≤ 2 2 ¹ º ¹ º For q 1, we have (p q)/2 κ(G)/2 , and for q = 0, we have κ(G)/≥2 = p/2 , and⌊ thus− Equation⌋ ≤ ⌊ 5.3 implies⌋ that also k(G) (p ⌊q)/2 . ⌋ ⌊ ⌋ ≥ ⌊ − Let⌋ p 3q 1. Then choose p/3 +1 of the q nonedges of G as pairs. These cannot≤ be− linked in G, as there⌊ ⌋ are at most p 2( p/3 + 1) p/3 − ⌊ ⌋ ≤ ⌊ ⌋ additional vertices. Clearly, any p/3 pairs of vertices of G can be linked, as any such pair needs at most one⌊ additional⌋ vertex to be connected, and if a pair is a nonedge, then every other vertex can be used. For p = 3q 1, both terms yield the same value. − Remark 5.4.6. For every p 3q 2, 3q 1, 3q, 3q + 1 we have that the terms p/3 and (p q)/2 yield∈ { the− same− value. Thus the} case distinction in Lemma⌊ ⌋ 5.4.5⌊ can− be made⌋ at any of these boundary cases. We have chosen the value p = 3p 1 to be consistent with Gallivan [45] in the statement of Corollary 5.4.10− below. Lemma 5.4.7. Let n, m 0 be integers. The linkedness of (n, m) is ≥ G given by 4m+n , if n 2m 1 k( (n, m)) = 3 ≤ − G ( 2m+n , if n 2m 1. ¥ 2 ¦ ≥ − If we use substitutions (5.1) and¥ (5.2),¦ this evaluates to d+γ+1 , if d + 2 5γ k( (n, m)) = 3 ≤ d γ+1 G ( ¥ − ¦ , if d + 2 5γ. 2 ≥ ¥ ¦ 87 5. Linkages in Polytope Graphs

Proof. This follows from Lemma 5.4.5, as the number of vertices of (n, m) is 4m + n and the number of nonedges is 2m. G Example 5.4.8. Let d = 8 and γ = 2. Then n = 3 and m = 2 and we obtain the 8-polytope on 11 vertices

P := (3, 2) = ∆ ¤ ¤ = pyr3(¤ ¤). G 2 ∗ ∗ ∗ The complement of the graph of P consists of 4 disjoint edges and 3 isolated vertices. Obviously, G(P ) is not 4-linked. The polytope P is the smallest known example of a polytope that is not d/2 -linked. ⌊ ⌋ In combination with Lemma 5.4.3 we obtain the following result. Theorem 5.4.9. Let d 2 and 0 γ (d + 2)/5. Then ≥ ≤ ≤ d γ + 1 k(d, γ)= − . 2 ¹ º Choosing γ = (d+2)/5 , we obtain Gallivan’s examples, and the bound of the last theorem⌊ implies⌋ the following bound on k(d) ﬁrst given by Gal- livan [45]. Corollary 5.4.10 (Gallivan [45]). The minimal linkedness of d-polytopes satisﬁes k(d) (2d + 3)/5 . ≤ ⌊ ⌋

One can construct polytopes with f0 = 3 d/2 1 vertices that are not d/2 -linked. If d is even let ⌊ ⌋ − ⌊ ⌋ P := ∆ ¤ ¤ C∆ C∆ . 2 ∗ ∗ ∗ 3 ∗ · · · ∗ 3 m times

Then d = 4m + 8, f0 = 6m + 11 and k(P| ) = 2{zm + 3.} For d odd let

P := ∆ ¤ ¤ ¤ C∆ C∆ . 4 ∗ ∗ ∗ ∗ 3 ∗ · · · ∗ 3 m times

Then d = 4m + 13, f0 = 6m + 17 and k(P|) = 2m{z+ 5. } For large f0 the methods used in this section to prove upper bounds fail. This suggests the following question Problem 5.4.11. Are all d-polytopes on at least 3 d/2 vertices d/2 - linked? Weaker: Is there some N(d), such that every⌊d-polytope⌋ on at⌊ least⌋ N(d) vertices is d/2 -linked? ⌊ ⌋ 88 5.4. Linkages in Polytopes with Few Vertices

Figure 5.4: Is this a subgraph of the complement graph of some 8-polytope on 12 vertices?

We know of only one obstruction for d-polytopes to not be d/2 -linked, the obstruction exploited in this section: The polytopes (n, m⌊) have⌋ many missing edges and not enough vertices to route all pathsG around the missing edges. If a polytope has 3 d/2 or more vertices, there has to be a diﬀerent obstruction if it is not⌊ d/⌋2 -linked. Regarding Problem 5.4.11, it would be interesting to know if⌊ the⌋ graph in Figure 5.4 is a subgraph of the complement graph of an 8-polytope on 12 vertices. The complement of this graph is 8-connected, has at every vertex a subdivision of K9 rooted at that vertex, and is not 4-linked. It would also be interesting to determine the linkedness of cubical polytopes, that is, of polytopes whose facets all are (d 1)-dimensional cubes. − Question 5.4.12. Is every cubical d-polytope d/2 -linked? ⌊ ⌋ Start with the d-cube. It this d/2 -linked? If yes, then by Lemma 5.4.1, every cubical d-polytope is at least⌊ ⌋(d 1)/2 -linked. In particular, this would imply a positive answer to Question⌊ − 5.4.12⌋ for odd dimensions d. Also, a cubical polytope has at least 2d vertices by a result by Blind & Blind [25], so a positive answer to Question 5.4.12 would also follow from a positive answer to Problem 5.4.11 (if, in the weaker version, we have N(d) 2d, of course). ≤ 5.4.3 Polytopes that Meet the Lower Bound The main theorem of this section is Theorem 5.4.16. It states that polytopes that meet the lower bound of Lemma 5.4.3 are

unique, if f0 is odd, and • of rather restricted combinatorial type, otherwise. • Furthermore, a lower bound on the number of combinatorial types in the latter case is proven. The “road map” to Theorem 5.4.16 is the following: We prove a concise characterization of polytopes that have small facet complements in The- orem 5.4.14. We have not seen this characterization in work of others,

89 5. Linkages in Polytope Graphs but suspect that it is known. This characterization implies the well-known characterization of d-polytopes on d + 2 vertices [51, pp. 97–101]. This theorem already imposes severe conditions on the combinatorial types of polytopes that meet the lower bound. The combinatorial type is then even more restricted by Lemma 5.4.15, which analyzes the graphs of the types that appear in Theorem 5.4.14. The proof of Theorem 5.4.16 then mainly consists of the construction of combinatorial types for the case when f0 is even. We consider polytopes with small facet complements, that is, polytopes in which every facet almost contains all vertices. If the size of every facet complement is one, then clearly the polytope in question is the simplex. The following lemma is the next step: What combinatorial types are possible if for every facet there are at most 2 vertices that are not contained in it?

Lemma 5.4.13. Let P be a d-polytope such that every facet F of P satisﬁes (P ) (F ) 2. | V Then\VP is| of ≤ the form

(n, j1, k1,...,jm, km) = ∆n 1 (∆j1 ∆k1 ) (∆jm ∆km ) G − ∗ ⊕ ∗ · · · ∗ ⊕ where k ...,k , j ,...,j 1, and d = n 1+ j + k + . . . + j + k + m. 1 m 1 m ≥ − 1 1 m m Proof. The property (P ) (F ) 2 implies that the hypergraph of facet-complements, that| V is, the\V hypergraph| ≤

G (P ) :=( (P ), W (P ) : (P ) W is the vertex set of a facet ) cofacet V { ⊆ V V \ } is a graph (with no parallel edges, but possibly with loops). The edges of Gcofacet(P ) are in bijection with the facets of P . Since the combinatorial type of a polytope is determined by the vertex-facet incidences, the combinatorial type of Gcofacet(P ) determines the combinatorial type of P . For Q = (n, j1, k1,...,jm, km) the graph Gcofacet(Q) is a disjoint union of n copies ofG the graph that consists of one single vertex and one single loop, and complete bipartite graphs Kj1,k1 ,...,Kjm,km . Thus we have to show that Gcofacet(P ) is of this type. It is easy to see that loops can only occur at isolated vertices and that there are no vertices of degree 1 in Gcofacet(P ) (we follow the convention that loops contribute two edges to the degree count). Then it suﬃces to check the following two properties of Gcofacet(P ):

(i) The graph Gcofacet(P ) does not have odd cycles (except loops at isolated vertices).

90 5.4. Linkages in Polytopes with Few Vertices

(ii) Whenever there is a path v1v2v3v4 of length 3 in Gcofacet(P ), then v1v4 is also an edge of Gcofacet(P ). In fact, (i) follows from (ii) and the absence of triangles, as any larger odd cycle together with (ii) would imply the existence of a triangle. We now show that Gcofacet(P ) does not have triangles. Suppose there is a triangle with vertices v1,v2,v3 and edges corresponding to facets F1,F2,F3 with (F )= (P ) v ,v , (F )= (P ) v ,v , and (F )= (P ) V 1 V \ { 2 3} V 2 V \ { 1 3} V 3 V \ v ,v . Let F ′ be the face F F = F F = F F . Then clearly { 1 2} 1 ∩ 2 1 ∩ 3 2 ∩ 3 F = pyr (F ′), F = pyr (F ′), and F = pyr (F ′). Thus dim F ′ = d 2 1 v1 2 v2 3 v3 − and P/F ′ is a 1-polytope on 3-vertices, a contradiction. (In face lattice terms, this is a contradiction to the diamond property.) Finally, we show that a path v1v2v3v4 of length 3 implies the existence of the edge v1v4. The edges of the path v1v2v3v4 correspond to facets F1,F2, and F with (F )= (P ) v ,v , (F )= (P ) v ,v , and (F )= 3 V 1 V \ { 1 2} V 2 V \ { 2 3} V 3 (P ) v ,v . Let F ′ = F F F . Then clearly F ′ has dimension d 3, V \{ 3 4} 1 ∩ 2 ∩ 3 − as only 4 vertices of P do not lie on F ′. The facets F , F and F are of dimension d 1 and each of them contains 1 2 3 − exactly two more vertices than F ′. We conclude that pyrv1 (F ′), pyrv2 (F ′), pyrv3 (F ′), and pyrv4 (F ′) are all faces of P . Thus, P/F ′ is a 2-polytope on 4 vertices, which implies that F4 := pyrv4 (pyrv2 (F ′)) is also a facet of P with (F )= (P ) v ,v . V 4 V \ { 1 4} The proof is purely combinatorial, so one may ask to what extent the statement generalizes. We have used the diamond property of the face lattice. This is a stronger condition than relatively complementedness, and indeed, the statement is false for graded relatively complemented lattices; see for example the lattice in Figure 8.5(c). We can deﬁne a subclass of the graded relatively complemented lattices by requiring the diamond property and the property that every interval of length 3 on 4 atoms looks like the lattice of a 4-gon. The proof generalizes to this class (true?). With Lemma 5.4.13, the following theorem is easily proven. Theorem 5.4.14. Let d 2, let 0 γ d 2, and let P be a d-polytope on d + γ + 1 vertices. Then≥ the following≤ ≤ are− equivalent: (i) Every facet F of P satisﬁes (P ) (F ) 2. | V \V | ≤ (ii) With m = γ, the polytope P is of the form

(n, j1, k1,...,jm, km) = ∆n 1 (∆j1 ∆k1 ) (∆jm ∆km ) G − ∗ ⊕ ∗ · · · ∗ ⊕

with suitable parameters ji and ki.

91 5. Linkages in Polytope Graphs

(iii) The polytope P does not have a simplex face of dimension d γ + 1. − Proof. If (P ) (F ) 2 for every facet F of P , then by Lemma 5.4.13 | V \V | ≤ the polytope P is of the form (n, j1, k1,...,jm, km). Clearly, we have that m = γ. G Now, suppose P is an iterated pyramid over a join of sums of simplices. Let S be a simplex face of P of maximal dimension. Then S is the join of

∆n 1 with facets from each factor ∆ji ∆ki . A facet of this sum in turn is − obtained by leaving out a vertex from⊕ each of the two simplices. Hence, S has n + j + k + . . . + j + k = d m +1 = d γ + 1 1 1 m m − − vertices and therefore dimension d γ. Finally, if P does not have a simplex− face of dimension d γ + 1, then (P ) (F ) 2 for every facet F . Otherwise, suppose there− is a facet F| Vwith\V (P )| ≤ (F ) 3. Then γ(F ) γ 2, and by Lemma 5.4.2 the facet F |has V a simplex\V | face ≥ of dimension ≤ −

(d 1) γ(F )= d (γ(F )+1) d γ + 1, − − − ≥ − in contradiction to the hypothesis.

Theorem 5.4.14 contains the classiﬁcation of d-polytopes on d + 2 vertices; see Gr¨unbaum [51, pp. 97–101]: No d-polytope on d + 2 vertices contains a simplex d-face. Thus, all polytopes on d + 2 vertices are of type (n, j1, k1,...,jm, km) with m = γ = 1. G As for Lemma 5.4.2, there is also a Gale duality proof of Theorem 5.4.14 (this is left as an exercise for the reader). So far, we have only excluded large simplex faces. The following lemma analyzes the situation if we also exclude large subgraphs.

Lemma 5.4.15. Let P be a d-polytope on d + γ + 1 vertices. Suppose that the graph G(P ) does not have a Kd γ+2-subgraph. Then P is of the form −

(n, m) = ∆n 1 ¤ ¤, G − ∗ ∗ · · · ∗ m times with n = d 3γ + 1 and m = γ. | {z } −

Proof. Since P does not have a Kd γ+2-subgraph, P does not have a simplex − face of dimension d γ + 1. Thus, by Theorem 5.4.14, P is of the form (n, j , k ,...,j , k −). G 1 1 m m 92 5.4. Linkages in Polytopes with Few Vertices

To show that j1 = k1 = . . . = jm = km = 1 observe that the graph

is the complete graph K if j, k 2 j+k+2 ≥ G(∆ ∆ ) contains a K if j 2, k =1 or j = 1, k 2 j⊕ k  j+k+1 ≥ ≥  is a 4-cycle if j = k = 1. Furthermore, in a join P Q every vertex of P deﬁnes an edge with every ∗ vertex of Q. Suppose now that ji 2 or ki 2 for some i. Then G(P ) contains a complete graph on ≥ ≥ n + j + k + . . . + j + k +1+ . . . + j + k = d m +2 = d γ + 2 1 1 i i m m − − vertices, which contradicts the hypothesis.

Theorem 5.4.16. Let d 2, and γ 0 with d 3γ 1. Set n := ≥ ≥ ≥ − d 3γ +1 0, and m := γ. Let P be a d-polytope on f0 = d+γ +1 vertices with− linkedness≥ k(P )= (d γ + 1)/2 . ⌊ − ⌋ If f0 is odd, then P is of type

(n, m) = ∆n 1 ¤ ¤ . G − ∗ ∗ · · · ∗ m times

If f0 is even, then there are at least|2d {z 3γ }+ 1 possibilities for the combinatorial type of P . More precisely, exactly− one of the following cases applies:

(i) The polytope P is of type (n, m). G (ii) We have m 1 and P is of type ≥ (n k, 1,..., 1, k + 1), G − 2m 1 times − with 1 k n. In this case| there{z } are exactly n = d 3γ + 1 possibilities≤ for≤ the type of P . −

(iii) We have m 2 and P has a facet F of type ≥ ∆ ¤ ¤ . n+4 ∗ ∗ · · · ∗ m 2 times − In particular, k(F ) = k(P ). In| this{z case} there are at least d 1 possibilities for the combinatorial type of P . −

93 5. Linkages in Polytope Graphs

Proof. If P is a d-polytope with d 2 on d + γ + 1 vertices and k(P ) = (d γ + 1)/2 , then clearly d > γ ≥ 0, as every such polytope is at least ⌊ − ⌋ ≥ 1-linked. Let f be odd, that is, d γ is even. If k(P ) = (d γ + 1)/2 , then 0 − ⌊ − ⌋ G(P ) cannot have a Kd γ+2-subgraph, and by Lemma 5.4.15 the polytope − P is of type (n, m) with m = γ and n = d + γ + 1 4γ = d 3γ + 1. Let f beG even, that is, d γ is odd. Under this assumption,− − the graph 0 − of P cannot have a Kd γ+3-subgraph, as it would otherwise be higher linked − than (d γ + 1)/2 . We distinguish the following three cases: ⌊ − ⌋ (1) The graph G(P ) does not have a Kd γ+2-subgraph. − (2) The graph G(P ) does have a Kd γ+2-subgraph, but P does not have a − (d γ + 1)-dimensional simplex face. (3) The− polytope P has a (d γ + 1)-simplex face. − These are all possible cases, and clearly they exclude each other. Case (1). In this case, Lemma 5.4.15 implies that P is of type (n, m). Case (2). Under the assumptions in (2), Theorem 5.4.14 impliesG that the polytope P is of type (n′, j , k ,...,j , k ) for m = γ 1 and suitable G 1 1 m m ≥ parameters n′, j1, k1,...,jm, km. The complement graph G(P ) consists of some number q of disjoint edges and f0 2q isolated vertices. Since G(P ) does not have a Kd γ+3 subgraph − we must− have f q d γ + 2, that is, 0 − ≤ − q 2γ 1. ≥ − The only polytope of type (n′, j1, k1,...,jm, km) with 2γ disjoint edges, which is the maximum possibleG value for q, is (n, m). However, (n, m) G G does not have a (d γ + 1)-simplex face and thus we have q = 2γ 1. This − − is only achievable if, up to symmetry, j1 = k1 = = jm 1 = km 1 = 1, ··· − − jm = 1 and 2 km d 3γ + 2, that is, the first m 1 factors of the join are quadrilaterals≤ and≤ the− last factor is a bipyramid.− Thus P is of type (n k, 1,..., 1 , k + 1), G − 2m 1 times − for 1 k n = d 3γ + 1. This| gives{z d} 3γ + 1 different combinatorial types≤ for the≤ different− choices of k. − Case (3). In the final case, the polytope P has a (d γ + 1)-simplex face but not a (d γ + 2)-simplex face. − Then for every− facet F of P we have (P ) (F ) 3. Otherwise there | V \V | ≤ is a facet F ′, which is of dimension d 1, that has γ(F ′) γ 3. That is, − ≤ − F ′ contains a simplex face of dimension (d 1) (γ 3) = d γ + 2, in contradiction to the hypothesis. − − − −

94 5.4. Linkages in Polytopes with Few Vertices

By Theorem 5.4.14, there exists a facet F with (P ) (F ) = 3, since otherwise P would not have a (d γ + 1)-simplex| V face.\V This| facet must − satisfy (F ) (F ′) 2 for every facet F ′ of F . Otherwise, we could | V \V | ≤ show with a similar calculation as before that there is a ridge F ′ of P with a simplex face of dimension at least d γ +2. This implies that m = γ 2. By Lemma 5.4.1, we have k(F ) −k(P ). But also ≥ ≤ (d 1) (γ 2) + 1 d γ + 2 k(F ) − − − = − = k(P ), ≥ 2 2 ¹ º ¹ º by Lemma 5.4.3 since d γ + 2 is odd. Furthermore, (d 1) γ(F ) = d 1 (γ 2) = d γ +− 1 is even. Thus applying (i) yields− that− F is of − − − − type (n′,m′) with n′ = d 3γ +6 and m′ = γ 2, as G − − dim( (n′,m′)) = n′ 1 + 3m′ = d 3γ +5+3γ 6= d 1. G − − − − To get the d 1 diﬀerent combinatorial types we modify (n, m). Let − G v0,v1 be the vertices of one of the missing edges of one of the quadrilaterals. Let Q := conv( ( (n, m)) v0,v1 ). Let F be a k-face of Q that has a quadrilateral 2-face,V G for 2 \ {k d}. Such a face exists since m = γ 2. Take the subdirect sum R :=(≤ Q,≤ F ) (∆ , ∆ ). Then the k-face F ∆≥lies ⊕ 1 1 ⊕ 1 in a facet of R whose complement has size 3, as F ∆1 has a (k 1)-face that has a complement of size 3 in F ∆ . ⊕ − ⊕ 1 For p = f and q = 2m, the polytope (n, m) has linkedness 0 G p q d γ + 1 k( (n, m)) = − = − . G 2 2 ¹ º ¹ º This implies that p 3q 2 by Lemma 5.4.5; compare Remark 5.4.6. Since d γ + 1 is≥ even,− also p q is even. The constructed polytope has p vertices and− exactly q 1 disjoint− missing edges. Lemma 5.4.5 implies that the linkedness of G(−R) is p (q 1) p q − − = − = k( (n, m)), 2 2 G ¹ º ¹ º since p q +1 is odd and p 3(q 1) 1. − ≥ − − As we have shown, for γ (d+2)/5 and f0 odd, polytopes that meet the lower bound are combinatorially≤ unique. Due to their special combinatorial type, an iterated pyramid over a join of quadrilaterals, these polytopes are projectively unique; see McMullen [77]. See Figure 5.5 for an illustration of minimal linkedness of 25-dimensional polytopes.

95 5. Linkages in Polytope Graphs

k ∆25 13 simplicial polytopes ∆22 ¤ 12 ∗ d/2 ⌊ ⌋ 11 10 ∆19 ¤ ¤ 9 ∗ ∗ d+2 ⌊ 3 ⌋ 8 ∆ ¤ ¤ ¤ 16 ∗ ∗ ∗

01234567 8 9 γ

Figure 5.5: Linkedness of polytopes in dimension 25. Shown in this graph are (a) the minimal linkedness of simplicial polytopes, (b) the minimal linkedness of polytopes on few vertices (in dimension 25 known up to γ = 5), (c) the unique extremal examples for γ = 0, 1, 3, 5, and (d) the best known upper and lower bounds for general polytopes.

In the case when f0 is even, the number of combinatorial types that are tight for the bound increases at least linearly in d. It is a consequence of Perles’ Skeleton Theorem [89] [62] that the number of graphs of polytopes on d + γ + 1 vertices, however, is bounded by a function of γ; see Part II.

5.5 Minimal Linkedness in Small Dimensions

Theorem 5.3.2 and Corollary 5.4.10 imply the values for the minimal linkedness k(d) as displayed in Table 5.1. In particular, we obtain exact values in dimensions 7, 10, and 13. The value k(8) = 3 follows from Larman & Mani’s old lower bound [66] and Gallivan’s upper bound [45].

d 1 234 5 678910 k(d) 1 1 1 2 2 2,3 3 3 3,4 4

d 11 12 13 14 15 . . . k(d) 4,5 4,5 5 5,6 5, 6, 7 . . .

Table 5.1: Possible values of k(d) in dimensions 1 d 15. ≤ ≤

96 5.5. Minimal Linkedness in Small Dimensions

The value k(6) is the first open value of k(d) and it seems to be a difficult problem to determine it. Our analysis of polytopes with few vertices in Theorem 5.4.16 shows that k(6, 0) = k(6, 1) = k(6, 2) = 3. We have also verified enumeratively that k(6, 3) = 3; beyond that we do not know much.

Problem 5.5.1. Determine k(6): Either show that all 6-polytopes are 3- linked, or give an example of a 6-polytope P with k(P ) = 2.

Thomas & Wollan [113] have shown that every 6-connected graph with at least 5f0 14 edges is 3-linked. For polytopes this implies that the graph obtained− from the graph of a 6-polytope by triangulating all 2-faces arbitrarily is 3-linked, according to the Lower Bound Theorem for general polytopes by Kalai [59]. Thus, all 2-simplicial 6-polytopes, that is, polytopes with only triangular 2-faces, are 3-linked. To determine k(d) in general it suﬃces to look at iterated pyramids over unneighborly polytopes, that is, polytopes in which every vertex lies on an edge of the complement graph. This was remarked by McMullen [78]. Indeed, suppose that a polytope P has a vertex of full degree that is not pyramidal. Then the pyramid over the link of that vertex yields the combinatorial type of a polytope that is at most as linked as P , as the resulting graph is a subgraph of the graph of P on the same number of vertices. However, the class of unneighborly polytopes contains the class of simple polytopes (except for the simplex) and not even for them the exact values of minimal linkedness are known. Indeed, the combinatorial structure of unneighborly polytopes is not well-understood. For example, not even the minimum number of vertices that an unneighborly polytope can have is known exactly; see Part III.

Part II

Perles’ Skeleton Theorem

Chapter 6

Skeleta of Polytopes with Few Vertices

Perles’ Skeleton Theorem, a beautiful result by Micha A. Perles in the theory of polytopes with few vertices, is discussed in this chapter and proved in the following two. It is from an unpublished manuscript written by Perles around 1970 [89], reported on and reproved by Kalai [62]. We will apply this theorem to problems by Marcus [71] and Bienia & Las Vergnas [23, Exercise 9.35(iv)*]; see Problems 10.4.1 and 11.0.1 in Chap- ters 10 and 11, respectively. Why are three entire chapters of this thesis devoted to this theorem? There are a number of reasons: (i) Perles’ original account on this theorem seems to be lost. I asked him about it during a visit at the Hebrew University of Jerusalem in the spring of 2007. He said that he did not have it, or rather, dass er es nicht habe, as he speaks perfect German [88]. (ii) Kalai’s proof in [62] establishes an interesting connection to the Erd˝os- Rado Sunflower Lemma [40]. Unfortunately, the exposition is rather sketchy and contains some inaccuraries. It also skips some key observations. (For example, it is not proved that a polytope on few vertices does not have sunflowers of empty pyramids of “large” size. The proof of this fact requires that the core of a sunflower is a face, which is not quite as obvious as in the simplicial case.) Most importantly, the bound on the number of vertices in missing pyramids given cannot be

101 6. Skeleta of Polytopes with Few Vertices

used as stated, since it depends on a parameter r that needs to be guessed from the context. This, however, is the result we need for our applications. (iii) While none of the proofs given in Chapters 7 and 8 reaches the ele- gance of Kalai’s “sunflower proof,” our methods yield better bounds on the number of missing pyramids than the bound implied by Kalai’s argument. (iv) The generalizations to strong spheres, graded relatively complemented lattices, and pyramidallyPL perfect lattices we prove in Chap- ter 8 were so far not substantiated in print. As stated by Kalai in [62, Section 2.4], his sunflower proof generalizes to the setting of graded relatively complemented lattices, but this does not seem to be straightforward and needs care in some touchy details. According to Kalai [62, Section 2.4], Perles’ original proof also applies to this much more general setting; but see (i). For one of our applications, namely Problem 11.0.1, we need at least the level of generality of strong spheres. PL I believe these reasons warrant the time and effort spent on reproving this theorem in the following two chapters. (Besides this, I had a lot of fun doing it.) In this chapter, we give an introduction to, and overview of, Perles’ Skeleton Theorem and its ramifications. This is the essence: Theorem 6.0.1 (Perles’ Skeleton Theorem [89] [62]). For parameters k and γ, the number of combinatorial types of k-skeleta of d-polytopes on d + γ + 1 vertices is bounded by a constant independent of the dimension d. The key step in proving this result is to bound the number of empty pyramids in the k-skeleton. While bounds on the number of empty simplices in simplicial polytopes were worked out exactly in a brilliant paper by Nagel [84], the general case has so far not been given much attention. As mentioned, Kalai gave a proof [62] that relies on the concept of sunflowers from the famous Erd˝os- Rado Sunflower Lemma [40]. We will discuss in Section 6.2 that, for polytopes with few vertices, (a) bounding the number of combinatorial types of k-skeleta independently of d, (b) bounding the number of empty pyramids in the k-skeleton independently of d, and (c) bounding the dimension in which k-skeleta can be realized “up to taking pyramids” independently of d,

102 6.1. Reconstruction of Skeleta are essentially equivalent ways of stating Perles’ original result. We will therefore refer to any such statement as Perles’ Skeleton Theorem. Chapters 7 and 8 are then entirely devoted to the problem of bounding the number of empty pyramids in varying generality and by varying means.

6.1 Reconstruction of Skeleta

In this chapter we need the two notions of “empty simplex” and “empty pyramid.” We generalize these concepts in Chapter 7.

Deﬁnition 6.1.1 (Vertex-induced, empty simplex, empty pyramid). Let be a polytopal complex, and let U ( ) be a subset of the verticesC of . ⊆ V C The inducedC subcomplex of on U, denoted [U], is the polytopal complex [U] induced by the vertexC set U. That is,C it consists precisely of all C the faces F of with (F ) U. If for some subcomplex ′ of we have C V ⊆ C C ′ = [U] for some U, we say that ′ is vertex-induced. C ForC k 1, an empty k-simplex Cof is a vertex-induced subcomplex of that is isomorphic≥ to the boundary ofC the k-simplex. Similarly, an empty kC-pyramid of is a vertex-induced subcomplex of that is isomorphic to the boundaryC of a pyramidal k-polytope. C In the special case k = 1 we also speak of missing edges.

The combinatorial type of a simplicial polytope (or more generally, a simplicial complex) is uniquely determined by its empty simplices: A set of vertices forms a face if and only if it does not contain the vertices of an empty simplex. For general polytopal complexes, for example, boundary complexes of nonsimplicial polytopes, knowing the empty simplices is not enough, as the following example shows.

Example 6.1.2. Consider the 2-skeleta of the following two polytopes:

P := ∆ ¤, and 1 7 ∗ P := bipyr(∆ ) bipyr(∆ ). 2 3 ∗ 3

Both have f0 = 12 and the only empty simplices are two disjoint missing edges: in P1, these are the diagonals of the quadrilateral two face; in P2, these are formed by the apex vertices of the bipyramids. In terms of empty simplices the 2-skeleta are indistinguishable. But P1 has a quadrilateral 2-face, and P2 does not. The 2-skeleta are therefore not isomorphic.

103 6. Skeleta of Polytopes with Few Vertices

In the former example, we could have remedied the situation by reconstructing the 2-skeleta from the complexes induced by the vertices in empty simplices. The following example shows that this is not suﬃcient in general. Example 6.1.3. Let Q := pyr(¤), the pyramid over a quadrilateral. Con- sider the 3-skeleta of

P := bipyr(Q) bipyr(∆ ), and 1 ∗ 4 P := Q bipyr2(∆ ) 2 ∗ 4

They both have the same number of vertices. In P1, the boundary of Q forms an empty 3-pyramid. In P2, Q is a 3-face and there are no empty 3-pyramids. Thus, the 3-skeleta of P1 and P2 are not isomorphic. Let v be the apex of Q. This does not lie in an empty simplex, neither of P nor of P . However, the 3-skeleta of P and P induced on (P ) v 1 2 1 2 V 1 \ { } and (P2) v , respectively, are isomorphic. Consequently, we cannot distinguishV \ the { } 3-skeleta from the complexes induced by vertices in empty simplices. This example shows that we also need to consider vertices in empty pyramids. Indeed, this is enough, as we establish in the following. Definition 6.1.4 (Kernel). Let k 1 and let be a polytopal complex of dimension at least k. Let U be the set≥ of verticesC u of that are contained in some empty ℓ-pyramid with ℓ k. C ≤ We define the k-kernel Kerk( ) of to be the k-skeleton of [U]. If a polytopal complex is a k-kernelC of a polytopeC P with γ(P )= γCwe call a (k, γ)-kernel. C C The following lemma is an important step towards Perles’ Skeleton The- orem. Although its proof is straightforward, the lemma illustrates the im- portance of the concept of empty pyramids. Lemma 6.1.5 (Kalai [62]). The k-skeleton of a polytopal complex can be reconstructed, up to isomorphism, from Ker ( ) and the number fC( ). k C 0 C Proof. Let V be the set of vertices of and A = V (Kerk( )). Every vertex of A forms the apex of a pyramidC over every\V face of dimensionC at most k 1. Thus the set of faces of skel ( ) is precisely − k C F B : F Ker ( ), B A, dim(F )+ B k . { ∗ ∈ k C ⊆ | | ≤ } To reconstruct the complex up to isomorphism it clearly suffices to know Ker ( ) and the number of vertices. k C 104 6.2. Perles’ Skeleton Theorem

This lemma can also be stated as follows: The k-skeleton of “looks” like the k-skeleton of the join of Ker ( ) with a simplex of dimensionC k C f ( ) (Ker ( )) 1. 0 C − | V k C | − Thus, one important idea in connection with Perles’ Skeleton Theorem is to distinguish skeleta of polytopes only “up to taking pyramids in the k-skeleton.”

Deﬁnition 6.1.6 (Pyramidally equivalent). Let 1 and 2 be polytopal complexes in Rd. We say that and are pyramidallyC k-equivalentC if they C1 C2 have isomorphic k-kernels. If two complexes and are pyramidally k-equivalent, then there are C1 C2 nonnegative integers n1 and n2 such that skel (pyrn1 ( )) skel (pyrn2 ( )). k C1 ≃ k C2 Without loss of generality we can even assume that n1 =0 or n2 = 0. We give complete lists of pyramidally inequivalent skeleta of polytopes for small parameters in Section 7.4.

6.2 Perles’ Skeleton Theorem

In this section we state Perles’ Skeleton Theorem in diﬀerent guises. We write down three statements, Perles’ Skeleton Theorem I, II, and III. We justify calling all of them Perles’ Skeleton Theorem by showing that they are essentially equivalent: Each one of them can easily be deduced from any one of the others. The ideas are from Kalai’s paper [62].

6.2.1 The Number of Empty Pyramids We start with the number of empty pyramids. This is the statement we prove in Chapter 7 for polytopes and in Chapter 8 for more general objects. Theorem 6.2.1 (Perles’ Skeleton Theorem I). For any d-polytope P on d + γ + 1 vertices, the total number of empty ℓ-pyramids with ℓ k is bounded by a function of k and γ. ≤ If k 1 and is a polytopal complex, we denote by M(k, ) the total ≥ C C number of empty ℓ-pyramids of with ℓ k. Furthermore, we deﬁne C ≤ γ M(k, γ) := max M(k, P ) : P d d 0 ≥ { ∈P } that is, M(k, γ) is the maximum number of empty ℓ-pyramids with ℓ k of any polytope P with γ(P )= γ. This is well-deﬁned by Theorem 6.2.1.≤

105 6. Skeleta of Polytopes with Few Vertices

6.2.2 The Number of Combinatorial Types of Skeleta Lemma 6.2.2. Let P be a d-polytope on d+γ +1 vertices and k 1. Every empty k-pyramid of P has at most k + γ vertices. ≥ Proof. Clearly, a (k 1)-face G of P can have at most k + γ vertices, with equality if and only if−P isa(d k +1)-fold pyramid over G. The statement follows. − For ﬁxed γ, let N be the maximum number of vertices in the k-kernel of a polytope with γ(P ) = γ. Since the number of empty pyramids of is bounded by a function of k and γ, also the size of N is bounded byC a function of k and γ. Indeed, by Lemma 6.2.2 we have N (k + γ)M(k, γ). The number of k-kernels of d-polytopes on d + γ +≤ 1 vertices is then bounded by the number of k-dimensional polytopal complexes on N vertices such that each k-face has at most k+γ vertices. And this number is bounded by

k+γ k+γ N (k + γ)M(k, γ) exp exp 2 j ≤ 2 j j=1 j=1 ³X µ ¶´ ³X µ ¶´ exp (exp ((k + γ)M(k, γ))), ≤ 2 2 which is a function of k and γ. By Lemma 6.1.5, the k-skeleton can be reconstructed from the k-kernel. Thus, Theorem 6.2.1 implies the following theorem. Theorem 6.2.3 (Perles’ Skeleton Theorem II). The number of combinatorial types of k-skeleta of d-polytopes on d + γ + 1 vertices is bounded by a function of k and γ. For integers γ 0 and k 1 let C(d, k, γ) be the number of combinatorial types of k-skeleta≥ of d-polytopes≥ on d + γ + 1 vertices. Because of Theorem 6.2.3, it makes sense to deﬁne

C(k, γ) := max C(d, k, γ) . d 0 ≥ { } The function C(k, γ) counts the number of combinatorial types of k-skeleta of polytopes P with γ(P )= γ.

6.2.3 The Realizability Dimension of Kernels Let D(k, γ) be the smallest dimension such that C(k, γ)= C(D(k, γ), k, γ). For every e D(k, γ) we thus have C(e, k, γ)= C(k, γ). ≥ 106 6.2. Perles’ Skeleton Theorem

γ By taking pyramids, we can realize any k-kernel of a polytope in d as γ Pγ the k-kernel of a polytope in Pe . Thus the k-skeleta of polytopes in e are γ P pyramidally k-equivalent to k-skeleta of polytopes in Pd . We therefore call D(k, γ) the realizability dimension of (k, γ)-kernels. These observations show that Theorem 6.2.3 implies the following statement.

Theorem 6.2.4 (Perles’ Skeleton Theorem III). For integers k 1 and γ 0 there is d 0 such that for every e d the k-skeleton of≥ an e-polytope≥ on e + γ + 1≥vertices is pyramidally k-equivalent≥ to the k-skeleton of some d-polytope on d + γ + 1 vertices.

To bring this section to a close, we show that Theorem 6.2.4 implies Theorem 6.2.1: The number of vertices in the k-kernel of any polytope with γ(P )= γ is bounded by D(k, γ)+ γ + 1, which is a function of k and γ. Thus, also the number of empty pyramids is bounded by a function of k and γ.

Remark 6.2.5. We have also shown the following quantitative relations between the functions M(k, γ), C(k, γ), and D(k, γ):

k+γ (k + γ)M(k, γ) C(k, γ) exp , and ≤ 2 j j=1 ³X µ ¶´ k+γ D(k, γ)+ γ + 1 M(k, γ) . ≤ j j=1 X µ ¶ In particular, if we bound M(k, γ) we also have a bound for C(k, γ). There seems to be no hope of ﬁnding an explicit bound for the function D(k, γ). To do so, one would have to ﬁnd a way to realize a given k-kernel of a d-polytope on d + γ + 1 vertices in some dimension d′(k, γ) D(k, γ). Indeed, a simplicial polytopal complex can always be realized in a≥ polytope whose dimension is bounded by a function of the size of the complex; see [63, Proposition 20.2.3]. However, with the known techniques one does not have control over the size of γ.

Remark 6.2.6. By deﬁnition, the functions M(k, γ), C(k, γ), and D(k, γ) are weakly monotone in k. We list a few more properties: The function M(k, γ) is strictly monotone in γ. Take a polytope • that has M(k, γ) empty pyramids of dimension at most k and stack

107 6. Skeleta of Polytopes with Few Vertices

a facet of this polytope. Then γ goes up by one, and M(k, γ + 1) goes up at least by one. It is not, however, strictly monotone in k. Every d-polytope on d + 2 vertices has at most 2 emtpy pyramids of dimension at most k, and this bound is attainable for every k 1; see Section 7.4. ≥ The function C(k, γ) is strictly monotone both in k and in γ. By • doing a vertex split at the vertex of an empty k-pyramid, we obtain that C(k + 1, γ) > C(k, γ). To see that C(k, γ + 1) > C(k, γ), we take a polytope P with γ(P )= γ with a kernel of maximum size and stack a facet. Monotonicity properties of the function D(k, γ) do not seem to be • easy to prove. We do however have a lower bound on the size of D(k, γ). Lemma 5.4.2 and Theorem 5.4.14 imply that a polytope P with γ(P ) = γ with the maximum number of 2γ disjoint empty k-pyramids is necessarily of the form

P = (∆ ∆ ) (∆ ∆ ) . k ⊕ k ∗ · · · ∗ k ⊕ k γ factors in the join

Thus D(k, γ) dim(P|)= γ(2k + 1){z 1. } ≥ −

108 Chapter 7

Perles’ Skeleton Theorem for Polytopes

In this chapter we prove Perles’ Skeleton Theorem for polytopes by deriving a bound on the number of empty pyramids of bounded dimension. An empty pyramid has the important geometric property of being necessarily flat, in terminology by Richter-Gebert [94, p. 31]: It is flat, that is, if we take the convex hull, we obtain a polytope of the same dimension, and it is flat in every geometric realization—it is necessarily flat. A general empty face, which is defined similarly to empty pyramids, except that the combinatorial type may be arbitrary, may not be flat. For example, induced cycles in the graph of a polytope are empty 2-faces, and if such a cycle has length at least 4, it obviously might have an affine dimension that is larger than 2; see Figure 7.1. Why is the property of being flat important? It is important for our inductive approach to proving Perles’ Skeleton Theorem. We want to count the total number of empty pyramids, and we do this by taking vertex figures and counting inductively. A general empty face might “disappear” when we pass to a vertex figure; for an example consider the vertex figure at v of the empty 2-face in Figure 7.1. This is not true for flat empty faces: A flat empty face yields a flat empty face of one dimension less in a vertex figure, except for trivial cases. We prove this in Section 7.1. Indeed, counting flat empty faces seems to be the “right thing to do.” Although knowing the empty pyramids is sufficient for reconstructing the k-skeleton, they complicate the induction: An empty pyramid is not neces-

109 7. Perles’ Skeleton Theorem for Polytopes

Figure 7.1: The empty 2-face indicated by the bold lines is “invisible” in the vertex ﬁgure at v.

sarily an empty pyramid in a vertex figure. To see this, consider a vertex figure at the apex of an empty pyramid. The empty face in this vertex figure is isomorphic to the boundary of the base of the empty pyramid. But this might be of arbitrary combinatorial type. One can get around this problem and argue that we can still count all empty pyramids by considering them in the right quotients, but the result is slightly clumsy; see Section 8.4. It is more natural to simply count flat empty faces, and this is what we do. Here is a rough outline of the proof: In Section 7.1, we establish basic properties of flat empty faces: Flat • empty faces yield flat empty faces in quotients, and two distinct flat empty faces yield distinct flat empty faces in quotients. These two properties make sure that we do not miscount the total number of flat empty faces. In Section 7.2, we give the main parts of the proof: We bound the • number of missing edges, and then by induction the number of flat empty k-faces. To accomplish this, we have to select a small number of vertices such that we count all flat empty faces by counting flat empty faces in vertex figures at these vertices. We apply Grünbaums’ theorem (Theorem 2.1.2) on simplex refinements to accomplish this. The rest of the chapter consists of Section 7.4, where we look at pyramidally inequivalent complexes of polytopes for small parameters, and a survey of results related to the simplicial version of Perles’ theorem; see

110 7.1. Empty Faces in Polytopes

Section 7.5.

7.1 Empty Faces in Polytopes

We generalize the concept of empty simplices and empty pyramids here.

Definition 7.1.1 (Empty face, empty pyramid, empty simplex). Let Rd be a polytopal complex and k 1. C ⊂An empty k-face M of is a vertex-induced≥ subcomplex of that is combinatorially isomorphicC to the boundary complex of a k-polytopeC P . If P is a pyramid or a simplex, the empty face is called empty k-pyramidor empty k-simplex,respectively, as defined before. A flat empty k-face M is an empty k-face of that equals as a polytopal complex the boundary of the polytope conv( (CM)). If M is an empty k-face, flat empty k-face,V empty k-pyramid, or empty k-simplex of for some k, we say that M is an empty face, flat empty face, empty pyramidC , or empty simplex, respectively.

Deﬁnition 7.1.2 (Quotient of an empty face). Let P Rd be a d- ⊂ polytope, M an empty face of P , and let F be a face of M. Let G1,...,Gm be the faces of M that contain F , and consider a realization of the quotient polytope P/F . The polytopal subcomplex of P/F given by the set of faces G1/F, G2/F, . . . , Gm/F is called the quotient of the empty face M at F and{ will be denoted by M/F} .

7.1.1 Flat Empty Faces in Quotients Lemma 7.1.3. Let d k 1, let P be a d-polytope, and let M be a ﬂat empty k-face of P . Then≥ M≥is not contained in a k-face of P .

Proof. This is “geometrically clear”: We can assume that k = d, otherwise we intersect P with the k-ﬂat aﬀ(M). But P cannot have an empty d- face.

Lemma 7.1.4. Let P be a d-polytope and let 2 k , k d 1. For ≤ 1 2 ≤ − i = 1, 2, let Mi be a ﬂat empty ki-face. Suppose that M1 and M2 are distinct but intersect in a vertex v. Then there is a realization of P/v such that the following hold:

(i) For i = 1, 2, M /v is a ﬂat empty (k 1)-face of P/v. i i −

(ii) The ﬂat empty faces M1/v and M2/v are distinct.

111 7. Perles’ Skeleton Theorem for Polytopes

Proof. Let H be a hyperplane such that v is strictly on one side of H and (P ) v is strictly on the other side of H. Then the intersection P H V \ { } Rd 1 ∩ is a realization of P/v in H ∼= − . In the following we will denote this realization simply by P/v. Let i 1, 2 . All the proper faces of conv(M H) are faces of P/v and, ∈ { } i ∩ since ki 2, there is at least one such face of dimension at least zero. We first show≥ that conv(M H) is not a face of P/v. Let K := aff(M ) Rd i ∩ i i ⊂ be the k-dimensional affine span of Mi. Because v lies in Mi, the affine flat K intersects H in an affine (k 1)-flat K′ := H K . Assume that i − i ∩ i conv(Mi H) = Ki′ P/v is a (k 1)-face of P/v. The polytope P is contained∩ in the cone∩ with apex v −that is spanned by P/v. The k-flat K contains v and the (k 1)-face conv(M H) of P/v. Therefore, K i − i ∩ i intersects P in a k-face F . However, Mi is contained in Ki and therefore in F , a contradiction to Lemma 7.1.3. Clearly, if k = k , then M /v and M /v are distinct. So suppose k = 1 6 2 1 2 1 k2. If M1/v and M2/v are equal, then also K1 = K2. Consequently, both M1 and M2 lie in the k-flat K := K1 = K2. By convexity M1 M2 and, by symmetry, M M , so M = M , in contradiction to the hypothesis.⊂ 2 ⊂ 1 1 2 Corollary 7.1.5. Let P be a d-polytope and let 1 k , k d 1. For ≤ 1 2 ≤ − i = 1, 2, let Mi be a flat empty ki-face and suppose that M1 and M2 are distinct. Let F be a face of M1 and M2 of dimension at most min k1, k2 2. Then there is a realization of P/F such that the following hold:{ }− (i) For i = 1, 2, M /F is a flat empty (k dim(F ) 1)-face of P/F . i i − − (ii) The flat empty faces M1/F and M2/F are distinct.

Proof. If min k1, k2 = 1, the statement is trivial, as then F = and P/F = P . Otherwise,{ } apply Lemma 7.1.4 inductively. ∅

7.2 A Bound on the Number of Flat Empty Faces

The proof of the following lemma is similar to a proof by Perles of a bound on the number of missing edges [88]. Theorem 7.2.1. Let P be a d-polytope on d + γ + 1 vertices. Then the number of missing edges is bounded by γ(γ + 1). Proof. The statement holds for 2-dimensional polytopes, as the number of missing edges of a 2-polytope on 3 + γ vertices is 3+ γ γ(γ + 3) (3 + γ)= γ(γ + 1), 2 − 2 ≤ µ ¶ 112 7.2. A Bound on the Number of Flat Empty Faces with equality only for γ = 0 and γ = 1. Let P be a d-polytope on d + γ + 1 vertices with d 3. Choose a vertex ≥ v (P ), and let P/v be a vertex ﬁgure of P at v. Write γ′ := γ(P/v). Let the∈ setV D consist of the neighbors of v and v itself, and let B = (P ) D. V \ Then B = γ γ′. By| induction| − on the dimension, the number of missing edges in P/v is at most γ′(γ′ + 1). Suppose that e′ is an edge of P/v whose endpoints, under radial projection from v, map to a missing edge of P . Then e′ corresponds to a 2-face of P with at least 4 vertices. Thus, there is at least one vertex of this 2-face that belongs to B. No two such 2-faces share vertices in B, and so the number of such edges is at most B = γ γ′. Every other missing edge of P| is| incident− to at least one vertex of B. Since every vertex in a d-polytopes has degree at least d, the total number of missing edges is then at most

γ′(γ′ + 1) + γ γ′ + γ(γ γ′) γ(γ + 1), − − ≤ missing edges edges missing missing edges in P/v because of 2-faces incident to B | {z } | {z } | {z } as claimed. Remark 7.2.2. The bound of γ(γ + 1) is best possible. It is attained by P = I ∆γ, the prism over a γ-simplex. This is a simple polytope and any vertex× in 0 ∆ has exactly one neighbor in 1 ∆ . Thus any { } × γ { } × γ vertex of the bottom face of P lies in exactly γ missing edges. The total d γ 1 number of missing edges then is γ(γ + 1). Take pyr − − (P ) to obtain a d-polytope for which the bound on the number of missing edges is tight for any d γ + 1. ≥ Lemma 7.2.3. Let P be a d-polytope on d+γ+1 vertices and let φ : ∆d P be a refinement homeomorphism with affinely independent principal vertices→ D (P ). ⊆Then V there are at most γ flat empty faces induced by vertices in D and they are all empty simplices. Proof. If M is a flat empty ℓ-face induced by vertices in D, that is, (M) V ⊆ D, then the vertices of M are affinely independent. Clearly, M is an empty ℓ-simplex on ℓ + 1 vertices. 1 If τ ∆d is the ℓ-simplex with (τ)= φ− ( (M)), then φ(τ) contains a vertex⊆ besides the vertices of M.V This must beV a vertex in B := V D. Since φ is a bijection, there can be at most B = γ flat empty faces, which\ are necessarily all empty simplices. | |

113 7. Perles’ Skeleton Theorem for Polytopes

The example of the reﬁnement homeomorphism from the 3-simplex to the regular 3-crosspolytope in which the four equatorial vertices are principal, compare Section 2.1, shows that the previous lemma fails for more general reﬁnement homeomorphisms.

Theorem 7.2.4. Let P be a d-polytope on d + γ + 1 vertices and 1 k d 1. Then the number of ﬂat empty ℓ-faces with ℓ k is bounded≤ by≤ − ≤ γ(γ + 1)k.

Proof. We prove the statement by induction on k. Let v (P ) be a vertex of P . Let φ : ∆ P be a refinement homeomorphism∈ rooted V at v d → with affinely independent principal vertices D = φ( (∆d)). Furthermore, let B = (P ) D. The size of B is B = γ. V TheV case k\ = 1 is Theorem 7.2.1,| | so we have that the number of flat empty 1-faces is bounded by γ(γ + 1). Let k 2. The number of flat empty faces induced by vertices in D is bounded by≥ γ, according to Lemma 7.2.3. Every other flat empty face contains at least one vertex in B. For w B ∈ consider the vertex figure P/w and let γ′ = γ(P/w). By induction, the k 1 number of flat empty ℓ-faces with ℓ k 1 of P/w is at most γ′(γ′ + 1) − . By Lemma 7.1.4, this number bounds≤ the− number of flat empty ℓ-faces of P with 2 ℓ k that are incident to w, as any two such flat empty faces ≤ ≤ yield distinct flat empty faces in P/w. The number of missing edges incident to w B is bounded by γ γ′. k 1 ∈ k 1 − Therefore, we count at most γ′(γ′ + 1) − + γ γ′ γ(γ + 1) − flat empty ℓ-faces with ℓ k at w. This number bounds− all≤ flat empty ℓ-faces of P with ℓ k that≤ contain w. The≤ sum over all vertices in B, together with the number of flat empty faces induced by vertices in D, yields a total bound of

2 k 1 k γ + γ (γ + 1) − = γ(γ + 1) , as claimed.

As every empty pyramid is a ﬂat empty face we have also shown the following.

Corollary 7.2.5 (Perles’ Skeleton Theorem I). Let k 1 and γ 0. Then the total number of empty ℓ-pyramids with ℓ k ≥is bounded by≥ γ(γ + 1)k. ≤

114 7.3. Disjoint Empty Faces

7.3 Disjoint Empty Faces

In contrast to Theorem 7.2.4, we derive here a bound on the number of disjoint empty faces in a polytope. This is much simpler, we get a tight bound, and we even have a characterization of polytopes that attain equality.

Lemma 7.3.1. Let P be a d-polytope on d+γ +1 vertices and d 2. Then P has at most 2γ vertex-disjoint empty faces. ≥

Proof. By Lemma 5.4.2, the polytope P has a max 1, (d γ) -dimensional simplex face S. Any empty face of P has a vertex{ outside− S}. Therefore, there can be at most d + γ + 2 (d γ + 2) = 2γ disjoint empty faces. − − The bound of 2γ is tight. It is attained by polytopes of the form

∆ (∆ ∆ ) (∆ ∆ ) . . . (∆ m ∆ m ), n ∗ j1 ⊕ k1 ∗ j2 ⊕ k2 ∗ ∗ j ⊕ k where k ...,k , j ,...,j 1, d = n + j + k + . . . + j + k + m and 1 m 1 m ≥ 1 1 m m f0 = d + γ +1 = n +1+ j1 + k1 + . . . + jm + km + 2m, that is, γ = m. There are exactly 2 disjoint empty faces (in this case empty simplices) in each factor of the join. Thus altogether there are 2m = 2γ disjoint empty faces. In Theorem 5.4.14 we have shown that these are the only polytopes that attain the bound of 2γ. Lemma 7.3.1 is an important ingredient of Kalai’s proof [62] of Per- les’ Skeleton Theorem, which uses a variant of the Erd˝os-Rado Sunflower Lemma [40]. Let X = A1,...,Am be a collection of sets. The collection X is called { } m a sunflower of size m if G := k=1Ak = Ai Aj and Ai G = for all i, j 1,...,m ,i = j. The set∩ G is called the∩ core of the− sunflower6 ∅ and ∈ { } 6 the sets Ai G the petals of the sunflower. Using Lemmas− 7.3.1 and 8.2.3, and Corollary 8.2.10, one can show that a d-polytope on d + γ + 1 vertices does not have a “sunflower of empty pyramids” of size 2γ + 1 (consider the vertex sets of empty pyramids as a set system). A slight variation of the proof of the Erd˝os-Rado Sunflower Lemma [40] yields that every collection of nonempty sets, each of which does not exceed a given size, that does not have a “large” sunflower is “small”; compare Kalai [62]. This gives another proof of Perles’ Skeleton Theorem. From this argument one can derive a bound of (k + γ)!(2γ)k+γ on the number of vertices in the k-kernel. (This proof does not generalize in an obvious way to flat empty faces.)

115 7. Perles’ Skeleton Theorem for Polytopes

7.4 Pyramidally Inequivalent Complexes

We give complete lists of (k, 1)-kernels and of (1, 2)-kernels to illustrate Perles’ theorem.

7.4.1 Kernels of Polytopes on d + 2 Vertices The classiﬁcation of d-polytopes on d + 2 vertices [51, pp. 97–101], see also Theorem 5.4.14, tells us that every such polytope P is given by parameters k , k ,d N with k + k d as 1 2 ∈ 1 2 ≤ d (k1+k2+2) P = pyr − (∆ ∆ ). k1 ⊕ k2

This polytope has exactly two missing faces: one missing k1-simplex and one missing k2-simplex. Thus for every k only the following three types of kernels appear: (i) There is no empty face of dimension at most k, so the kernel is empty. In this case, k1, k2 k + 1. (ii) There is exactly one≥ empty face of dimension at most k. This corresponds to the case where, up to symmetry, k1 k and k2 k + 1. (iii) There are two disjoint empty faces of dimensions≤ k k and≥ k k. 1 ≤ 2 ≤ 7.4.2 Kernels of Polytopes on d + 3 Vertices Polytopes on d+3 vertices correspond to 2-dimensional Gale diagrams. We give a full list of kernels that appear for k = 1. We need the following basic building blocks of graphs and the operation of disjoint union to combine them: Deﬁnition 7.4.1 (Elementary graphs, disjoint union). Let k 1. We denote by Ek the graph on 2k vertices with k edges (all of them disjoint),≥ by P k the path on k edges, and by Ck the simple cycle on k edges. We write G G for the disjoint union of graphs G and G under the 1 · 2 1 2 assumption that∪ (G ) (G )= . V 1 ∩ V 2 ∅ We have the following conditions on the number of missing edges: There are at most 4 disjoint missing edges, by Lemma 7.3.1. If there • are exactly 4 missing edges, then the combinatorial type of the polytope is determined, by Theorem 5.4.14. There are at most 6 missing edges in total, by Theorem 7.2.4. • Furthermore, we have the following structural conditions imposed by the geometry of the Gale diagram:

116 7.4. Pyramidally Inequivalent Complexes

(a) (b) (c)

2 2

2 (d) (e)

Figure 7.2: Gale diagrams of 4-polytopes on 7 vertices that realize 1-kernels of polytopes on d + 3 vertices.

The complement graph of a d-polytope on d + 3 vertices does not • contain a 3-cycle or a 4-cycle. If the complement graph contains a 5- or 6-cycle, then there are no • further missing edges. The graph E1 P 4 is not the complement graph of a d-polytope on · • d + 3 vertices. ∪ (The proof of these conditions is left to the reader.) The list in the following theorem is a complete list of graphs that do not violate any of the above conditions. Thus, to proof the theorem we only need to ﬁnd a realization for each of the given kernels.

Theorem 7.4.2. The following is a complete list of (1, 2)-kernels of polytopes:

E1,E2,E3,E4, • P 2, P 3, P 4, • E1 P 2, E2 P 2, E1 P 3. • ∪· ∪· ∪· 117 7. Perles’ Skeleton Theorem for Polytopes

C5, C6. •

Furthermore, the realizability dimension D(1, 2) of (1, 2)-kernels is 5, that is, every (1, 2)-kernel is the 1-kernel of some 5-polytope on 8 vertices.

Proof. We go through the list and give a realization for each of the graphs as a (1, 2)-kernel: Graph E1: This graph is not realizable as the 1-kernel of a 3-polytope • on 6 vertices, as K5 is not planar. However, it is realized by a 4- polytope on 7 vertices, given by the Gale diagram in Figure 7.2(a). Graph E2: A graph on 6 vertices with E2 as its complement contains • a K3,3-subgraph, so this graph is not realizable as the 1-kernel of a 3-polytope on 6 vertices. It is realized by a 4-polytope on 7 vertices with the Gale diagram in Figure 7.2(b). Graph E3: A 3-polytope on 6 vertices that realizes this graph as its • 1-kernel is the octahedron. Graph E4: By Lemma 7.3.1, every d-polytope on d + 3 vertices with • this graph as its complement is an iterated pyramid over the join of two quadrilaterals. The one with the smallest dimension clearly is the join of two squares, that is, a 5-polytope on 8 vertices. Graph P 2: The graph K P 2 has a K -subgraph, so P 2 is not the • 6 \ 3,3 1-kernel of a 3-polytope on 6-vertices. It is realized by a 4-polytope given by the Gale diagram in Figure 7.2(c). Graph E1 P 2: Because of a K -subgraph, this is not the 1-kernel of · 3,3 • a 3-polytope∪ on 6 vertices. It is realized by a 4-polytope on 7 vertices given by the Gale diagram in Figure 7.2(d). Graph E2 P 2: This is realized as the 1-kernel of a 4-polytope on 7 · • vertices, given∪ by the Gale diagram in Figure 7.2(e). This is realized by a 3-polytope on 6 vertices, a vertex split of a 5-gon. • Graph E1 P 3: This is realized by a 3-polytope P on 6 vertices. Let Q · • be a quadrilateral∪ in R2 and a one of its edges. Let P :=(Q, a) (e,e) for a line segment e. ⊕ Graph P 4: Let Q be a quadrilateral and take R := pyr(Q). Construct • a polytope by stacking onto one of the triangular faces of R. The resulting 3-polytope on 6 vertices realizes P4 as a 1-kernel. Graph C5: This is realized by a 5-gon. • Graph C6: This is realized by a prism over a triangle. • The largest dimension we needed was d = 5, and thus D(1, 2) = 5.

118 7.5. Empty Simplices in Simplicial Polytopes

7.5 Empty Simplices in Simplicial Polytopes

There is a wealth of results related to the simplicial version of Perles’ Skele- ton Theorem by Kalai [62] and Nagel [84]. A short survey of these results is given here. Let P be a simplicial d-polytope. An empty pyramid of P is an empty simplex, so M(k, P ) counts the number of empty ℓ-simplices of P with ℓ k. Let ≤ d MS(k, γ) := max M(k, P ) : P γ . d 0 ≥ { ∈ S } Kalai [62] gave a proof of Perles’ Skeleton Theorem for simplicial polytopes that uses the Erd˝os-Rado Sunﬂower Lemma [40]. His proof implies a bound k+1 of (γ + 1) (k + 1)! for MS(k, γ). In the same paper he stated the following conjecture by Kalai, Klein- schmidt & Lee.

Conjecture 7.5.1 (Kalai, Kleinschmidt & Lee [62, Conjecture 2]). Let (h ,...,h ) be the h-vector of a simplicial d-polytope, and let 1 k 0 d ≤ ≤ d 1. For all simplicial d-polytopes with h-vector (h0,...,hd), the number of− empty k-simplices is maximized by the Billera-Lee polytopes constructed in the proof of the suﬃciency part of the g-Theorem [17] [18].

This conjecture was proven by Nagel [84, Theorem 2.3] following ideas by Kalai [63, Theorem 20.5.35], who observed that Conjecture 7.5.1 essentially follows from a result by Migliore & Nagel [81]. It was shown by Kalai [62, Theorem 3.8] that, for 0 j k and ﬁxed ≤ ≤ α, the number of empty k-simplices of a simplicial d-polytope with gj α is bounded by a constant that is independent of d. Another proof of≤ this statement was given by Nagel [84, Theorem 4.20]. It is open whether such a statement holds for general polytopes. With respect to the number of empty simplices in simplicial polytopes, Nagel proved the following optimal bounds for MS(k, γ).

Theorem 7.5.2 (Nagel [84, Theorem 4.15]). Let P be a simplicial d- polytope on d + γ + 1 vertices, γ 1 and k 1. Then ≥ ≥ γ+k γ 1 , if 1 k

The bound for d/2 k is attained by the cyclic polytopes, that is, the polytopes that arise≤ as the convex hull of points on the moment curve, according to the following result by Terai & Hibi [112].

Theorem 7.5.3 (Terai, Hibi [112]). Let Cd(n) be the cyclic d-polytope on n = d + γ + 1 vertices. Then

γ + d/2 γ + d/2 1 M(k,C (n)) = ⌊ ⌋ + ⌊ ⌋ − , d γ 1 γ 1 µ − ¶ µ − ¶ for every k d/2. ≥ The following problem on empty simplices of simplicial polytopes was posed by Kalai [62].

Problem 7.5.4 ([62, Problem 1]). Let be a simplicial complex. Let m ( ) denote the number of empty k-simplicesC of . k C C Characterize the vectors (m1,...,md 1) that arise from simplicial d- − polytopes.

120 Chapter 8

Generalizations of Perles’ Skeleton Theorem

In this chapter we prove three generalizations of Perles’ Skeleton Theorem. We prove a bound on the number of empty pyramids in strong PL spheres parallel to the geometric proof given in Chapter 7. Finding equivalent topological statements for the geometric statements in Chapter 7 is a large part of the work. In the end, we arrive at the same bound for the number of empty pyramids. We then pass to purely combinatorial objects. It was remarked by Kalai, see [62, Section 2.4], that Perles had shown his skeleton theorem for a class of lattices called pyramidally perfect [89]. As there is no written account of this generalization available we establish this result in this chapter. In fact, we consider two classes of lattices: the graded relatively complemented ones, and the pyramidally perfect ones. The objects in this chapter are related to polytopes by the following chain of inclusions:

polytopal face lattices face lattices of strong spheres { } ⊂ { PL } graded relatively complemented lattices ⊂ { } pyramidally perfect lattices . ⊂ { } For d 4 there are many nonpolytopal strong (d 1)-spheres; see for example≥ the Br¨uckner sphere [33], or the ones constructedPL − by Pfeiﬂe & Ziegler [91], and Kalai [60].

121 8. Generalizations of Perles’ Skeleton Theorem

The class of graded relatively complemented lattices is important, because it contains, besides the face lattices of polytopes, the class of geometric lattices: the lattices of flats of matroids [85]. Recall that a finite graded lattice is geometric if it is relatively complemented and if it is semimodular, that is, the rank function rk satisfies for any two elements x and y the inequality rk(x)+rk(y) rk(x y)+rk(x y). It is easy to see that a polytopal face lattice is geometric≥ ∨ if and only∧ it is boolean. (Show that, whenever a simplicial polytope is not isomorphic to ∆d, there is a ridge and a vertex whose join is the polytope itself. The statement then follows by induction.) The class of graded relatively complemented lattices is thus a much larger class than the class of face lattices of polytopes. The same arguments apply for strong spheres. We shall also see that the class of pyramidally perfectPL lattices is a larger class than the class of graded relatively complemented ones. Thus the three inclusions above are really proper inclusions. Since we move from more concrete to more abstract objects, the reader may be worried that we reprove things again and again. This is only true to a small extent. In most cases the proofs for the more concrete objects yield more information or better bounds. For example, the bounds on the size of the kernel we prove mimic the above hierarchy. Although I do not know if better bounds can be proved for the more general objects, the proofs given definitely do not generalize. The two main reasons are that (a) there is no equivalent to Grünbaums’s theorem on simplex refinements [50] (Theo- rem 2.1.2) for the two lattice classes under consideration, and that (b) the class of pyramidally perfect lattices is not closed under taking upper intervals.

8.1 Strong Spheres PL Deﬁnitions and notation are taken from [23, Chapter 4.7]. Basic references for topology are Hudson [55], Zeeman [117], and Rourke & Sander- sonPL [97].

8.1.1 Basic Notions

A geometric simplicial complex is a ﬁnite collection of simplices in some Re such that K (i) if F then also all faces of F are in , (ii) if F,G∈ K then F G is a face both ofKF and G. ∈ K ∩ 122 8.1. Strong Spheres PL

e Let be a geometric simplicial complex in R with vertices v1,...,vn. We expressK every point x of the underlying space of , that is, the e || K || K set of points in R that lie in some simplex of , by barycentric coordinates. nK n These are real numbers λi such that x = i=1 λivi, i=1 λi = 1, and such that the set vi : λi > 0 is the set of vertices of a simplex of . A map {Rd } nP Pn K f : is linear if f(x) = f( i=1 λivi) = i=1 λif(vi) for every || Kn || → x = i=1 λivi . It is piecewise linear if it is linear on some simplicial subdivision of ∈. || K || P P TwoP geometricK simplicial complexes Re and Rd are homeomorphic if there is a piecewise linear mapK ⊂f : C ⊂ PLRd that is a homeomophism. || K || → || C || ⊂

Deﬁnition 8.1.1 (Simplicial PL ball, PL sphere). The underlying space of a geometric simplicial complex is a d-ball (respectively, a (d 1)-sphere) if it admits a triangulation thatPL is homeomorphic toPL the standard− d-simplex (respectively, boundary of thePL standard d-simplex).

Recall from Section 4.1 the definition of regular cell complexes and the basic terminology connected with them. Every regular d-dimensional cell complex P can be embedded with a flat embedding of its order complex, which is isomorphic to the barycentric subdivision. (A flat embedding or a geometric realization of an abstract simplicial complex is a geometric simplicial complex that is combinatorially isomorphic to the abstract simplicial complex.)

Deﬁnition 8.1.2 (PL ball, PL sphere; compare [23, Lemma 4.7.25]). A regular cell complex P is a cellular d-ball (respectively, a cellular (d 1)-sphere) if one (and thus every)PL geometric realization of its orderPL complex− is a simplicial d-ball (respectively, a simplicial (d 1)- sphere). PL PL −

Clearly, this deﬁnition is equivalent to requiring the following: Every geometric realization of the order complex of P admits a triangulation that has a simplicial subdivision that is combinatorially isomorphic to some simplicial subdivision of the d-simplex (respectively, boundary of the d-simplex).

Deﬁnition 8.1.3 (Strong PL sphere). A strongly regular cellular d- sphere that is a d-sphere is called a strong d-sphere. PL PL The structures deﬁned in this section are related by inclusion as shown in Figure 8.1. If P is a strongly regular cellular (d 1)-sphere, we set γ(P ) := f0 d 1. It is easy to see that a strongly regular− cellular (d 1)-sphere has at− least− − 123 8. Generalizations of Perles’ Skeleton Theorem

regular cell complexes

cellular spheres

geometric strongly regular simplicial cellular sphere complex

cellular ball ball PL PL cellular sphere sphere PL PL strong sphere PL

Figure 8.1: Inclusion relations among the structures deﬁned in this section. A dotted line indicates the relation via barycentric subdivision. d + 1 vertices. Thus γ(P ) 0 for every strongly regular cellular (d 1)- sphere. ≥ − The augmented face poset ( ) of a regular cell complex is the face poset (C), which has a 0ˆ correspondingF C to the empty setC by Deﬁni- F tion 4.1.1, together with an elementb 1ˆ such that F < 1ˆ for every F ( ). ∈F C

8.1.2 Basic Constructions and Properties Let be a regular cell complex. Then I, where I = [0, 1] denotes the unitC interval, is again a regular cell complex.C × The cells of I are either of type F 0 , F 1 , or they are of type F I, where CF ×is a cell of . The cell complex× { } ×I { is} called the prism over ×. C The pyramid pyr(C × ) over , also called the coneC over , is obtained from C C C I by identifying all points in 0 . The pyramid pyr( ) is again a Cregular × cell complex. The vertex thatC ×{ arises} from identifyingC the points in 0 is the apex of pyr( ). The complex is called the base of . If aC ×{ complex} is a pyramidC over a complex Cwith apex v, we alsoC write C K = pyrv( ). C We needK a number of properties of spheres that are collected in the PL 124 8.2. Empty Faces in Strong Spheres PL following theorem. Theorem 8.1.4 (cf. [23, Theorem 4.7.21] [23, Proposition 4.7.26]). (i) (Newman’s theorem) The closure of the complement of a d-ball embedded in a d-sphere is itself a d-ball. PL PL PL (ii) The cone over a d-sphere is a (d + 1)-ball. PL PL (iii) If P is a sphere, then every closed cell σ P is a ball. PL ∈ PL (iv) If P is a sphere, then there exists a regular cell decomposition PL PL P ∆ of the d-sphere with anti-isomorphic augmented face poset:

(P ∆) = (P )op. F ∼ F Theorem 8.1.4 (iii) and (iv)b are crucialb for extending many of the arguments that work for polytopes to strong spheres. They allow us to define vertex figures, or more general, face figuresPL of spheres. Let F and G be two cells of P with F G. Then by the twoPL latter properties of the above theorem, the interval⊆ [F,G] in the face poset of P is the augmented face poset of a strong sphere, denoted by G/F . Such a strong sphere is called a quotientPLof P . The same is true for intervals [F, 1]ˆ inPL the augmented poset of P , in which case we denote the resulting sphere by P/F and call it a face figure of P . In the special case when FPLis a vertex, we call P/F a vertex figure. For our comfort in the combinatorial arguments that follow it is also good to know that the augmented face poset of a strong sphere is a graded lattice with diamond property (actually, this is truePL as well for regular cell decompositions of spheres with intersection property; see [23, Corollary 4.7.12]).

8.2 Empty Faces in Strong Spheres PL In this section, we deﬁne empty faces in strong spheres and prove a number of properties of them. PL Let U ( ) be a subset of the vertices of a regular cell complex . The ⊆ V C C subcomplex of induced by U, denoted by [U], is the regular cell complex induced by theC vertex set U. That is, [U]C consists precisely of all the cells F of with (F ) U. If a subcomplexC is an induced subcomplex for some set ofC vertices,V we⊆ call it vertex-induced. For k 1, an empty k-face of is a vertex-induced subcomplex that is a regular≥ cell decomposition of theC (k 1)-sphere. − 125 8. Generalizations of Perles’ Skeleton Theorem

If this cell decomposition is combinatorially isomorphic to the boundary of a pyramid or a simplex, the empty face is called empty k-pyramid or empty k-simplex, respectively. If the dimension does not matter, we also call the cell decomposition simply an empty face, empty pyramid or empty simplex, respectively. Empty 1-faces will also be called missing edges. If satisfies the intersection property, then clearly also [U] satisfies the intersectionC property. C Let P be a strong sphere, M an empty face of P , and let F be a PL cell of M. Let G1,...,Gm be the cells of M that contain F , and let P/F be a face figure of P at F . The strongly regular cell subcomplex of P/F given by the set of cells G /F, G /F, . . . , G /F , where G /F is the cell { 1 2 m } i of P/F that corresponds to the cell Gi in P , is called the quotient of the empty face M at F . We will denote this complex by M/F .

8.2.1 Empty Pyramids in Quotients The following lemma is the topological equivalent to Lemma 7.1.3.

Lemma 8.2.1 ([23, Lemma 4.7.3]). Let Sk be a k-dimensional sphere and A Sk a proper subset of Sk. Then A is not homeomorphic to a k-sphere.⊂

Proof. This follows, for example, from Alexander duality [82].

One of the key features of empty pyramids is that they behave nicely with respect to intersection with cells and with other empty pyramids.

Lemma 8.2.2. Let P be a strongly regular cellular (d 1)-sphere, let F P be a k-cell of P , and let M be an empty k-pyramid of−P . Then the strongly⊂ regular cell complex F M is a cell of F and hence of P . ∩ Proof. Since M is an empty k-pyramid, there is a (k 1)-cell G M and a vertex v (M), such that M = ∂(pyr (G)). By the− intersection⊂ property, ∈ V v the intersection F ′ := F G is a face of P . ∩ If F ′ = G, then the vertex v is not a vertex of F . Otherwise, every proper face of M is also a face of F . However, both ∂F and M are (k 1)- spheres, so ∂F = M, by Lemma 8.2.1. This contradicts the facts that−F is a k-face and that M is an empty k-face. In the other case F ′ = G. Either the vertex v lies in F , in which case 6 the intersection F M equals pyr (F ′), or v does not lie in F . In the latter ∩ v case we have F M = F ′. ∩ 126 8.2. Empty Faces in Strong Spheres PL

(a) The two geometric 2-faces indicated (b) The empty 2-face indicated by the by the bold lines intersect in two compo- bold lines intersects the empty 2-face from nents. Figure 8.2(a) in two edges.

Figure 8.2: Empty faces in 3-polytopes.

The following lemma holds in much more generality for pyramidally perfect lattices; see Lemma 8.9.5. We therefore postpone the proof until later. Lemma 8.2.3. Let P be a strongly regular cellular (d 1)-sphere, let k 1, − ≥ and let M1 and M2 be distinct empty k-pyramids of P . Then the regular cell complex M M is a cell of P . 1 ∩ 2 Remark 8.2.4. It is not true that the intersection of two geometric empty faces in a polytope is a single face. Consider the geometric empty faces of the regular 3-dimensional crosspolytope indicated by the bold lines in Figure 8.2(a). General empty faces can also intersect locally in a complex that is larger than a single face. Consider, for example, the intersection of the empty 2- face indicated by the bold lines in Figure 8.2(b) with the empty 2-face in Figure 7.1 at the beginning of the previous chapter. Lemma 8.2.5. Let k,d N with 1 k d 1. Let P be a strong ∈ ≤ ≤ − (d 1)-sphere, and let M = ∂ pyrv(G) be an empty k-pyramid, where vPL (P−) and G P is a (k 1)-cell of P . ∈Let V F be a cell⊂ of M of− dimension at most k 2. Then M/F is an empty (k dim(F ) 1)-face in P/F with M/F = −∂(pyr (G)/F ). − − v Proof. Let G1,...,Gm be the cells of M that contain the cell F . Denote by P/F a face ﬁgure of P at F , that is, a strong sphere with face poset that PL 127 8. Generalizations of Perles’ Skeleton Theorem is isomorphic to the interval [F, 1]ˆ in the augmented face poset of P . For i = 1,...,m, let Gi′ := Gi/F be the cells of M/F in P/F that correspond to the faces G1,...,Gm. We claim that M/F is an empty cell. Suppose there is a (k dim F 1)- cell in P/F that has M/F as its boundary. Then this cell corresponds− − to a k-cell F ′ in P that intersects M in G1,...Gm. By Lemma 8.2.2, the set of cells G1,...,Gm then is exactly the set of cells of a cell H P of dimension{ at most k} 1. Since M is an empty k-pyramid and F⊂has at most dimension k 2− there are at least two cells of dimension k 1 that contain F and that− are cells of M. Consequently, they are containd− in H. This is clearly a contradiction. Since the set G ,...,G contains at least one cell of dimension k 1, { 1 m} − the set of cells G′ ,...,G′ in P/F contains at least one ((k 1) dim(F ) { 1 m} − − − 1)-cell. Thus M/F is an empty (k dim(F ) 1)-cell. − − Corollary 8.2.6. Let P be a strong (d 1)-sphere and v (P ). Let 2 k d 1 and let M be an empty k-pyramid− with v M.∈ Then V M/v is an≤ empty≤ (−k 1)-pyramid of P/v if and only if at least one∈ of the following two cases applies:−

(i) There is w (P ) v and a (k 1)-cell G with M = ∂ pyr (G), or ∈ V \ { } − w (ii) there is a (k 1)-pyramid G P with M = ∂ pyr (G). − ⊂ v Proof. We have that M/v = ∂ pyr (G)/v for v (P ), G is a (k 1)-cell, w ∈ V − and that pyrw(G)/v is a pyramid if and only if v G or if G is a pyramid. The statement then follows immediately from Lemma∈ 8.2.5.

The following lemma is the equivalent of Corollary 7.1.5.

Lemma 8.2.7. Let P be a strong (d 1)-sphere and let 1 k , k d 1. − ≤ 1 2 ≤ − For i = 1, 2, let Mi be an empty ki-pyramid and suppose that M1 and M2 are distinct. Let F be a cell of M1 and of M2 of dimension at most min k , k 2. Then M /F and M /F are distinct empty cells of P/F . { 1 2} − 1 2

Proof. Let k′ := k dim F 1 for i = 1, 2. By Lemma 8.2.5, M /F is an i i − − i empty k′-cell of P/F for i = 1, 2. Clearly, if we have k = k , the empty i 1 6 2 cells M1/F and M2/F are distinct. So assume k1 = k2 and let k := k1 = k2. Consider the intersection F ′ := M M . According to Lemma 8.2.3, F ′ is a cell of P of dimension 1 ∩ 2 at most k 1. We also have F F ′, since F was assumed to be a cell − ⊆ of M1 and of M2. Since Mi, i = 1, 2, is an empty k-pyramid there is at least one (k 1)-cell G of M distinct from F ′ that contains F , which was − i i 128 8.3. Simplex Refinements of Spheres PL assumed to be of dimension at most k 2. Clearly Gi is not a cell of Mj for i, j = 1, 2 , so G /F is not a cell− of M /F . Thus M /F and M /F { } { } i j 1 2 are distinct empty cells. Deﬁnition 8.2.8. Let P be a strong (d 1)-sphere and let F a k-cell PL − of P . Denote by βF (P ) the number of empty (k + 1)-pyramids that contain F . Lemma 8.2.9 (cf. [62, Lemma 2.1]). Let P be a strong (d 1)-sphere on d + γ + 1 vertices and F a cell of P . Then PL − β (P ) γ(P ) γ(P/F ) γ(F ). F ≤ − − Proof. Suppose that dim(F )= k. Then the quotient P/F has dim(P/F )= d k 1 and m := f (P/F ) is the number of (k + 1)-cells in P that contain − − 0 the cell F .

Let F1,...,Fm be the (k + 1)-cells that contain F , and M1,...,MβF (P ) the empty (k + 1)-pyramids that contain F . For every 1 i m, let vi be a vertex in (Fi) (F ). Then vi = vj, whenever i = j,≤ since≤F F = F . Likewise, forV every\V 1 i β (P ) let6 w 6 i ∩ j ≤ ≤ F i be a vertex in (Hi) (F ). Then also wi = wj whenever i = j, since by Lemma 8.2.3 weV have\VM M = F . Lemma6 8.2.2 implies that6 F M = F i ∩ j i ∩ j for all 1 i m and 1 j βF (P ), so we also have vi = wj. This≤ shows≤ that ≤ ≤ 6 f (P/F )+ β (P ) f (P ) f (F ). 0 F ≤ 0 − 0 If we substitute using the equations f (P/F ) = d k + γ(P/F ), f (P ) = 0 − 0 d + γ(P )+1, and f0(F )= k + γ(F ) + 1, we get d k + γ(P/F )+ β (P ) d + γ(P ) + 1 (k + γ(F )+1), − F ≤ − which, if simplified, is the statement we wanted to prove. Corollary 8.2.10. Let P be a strong (d 1)-sphere on d+γ+1 vertices, and let F be a cell of P . Then β (P )PLγ(P−) γ(P/F ). F ≤ − 8.3 Simplex Refinements of Spheres PL In this section we generalize Grünbaum’s theorem on simplex refinements of polytopes [50], Theorem 2.1.2, to strong spheres. We prove the generalization in two partsPL by (a) proving that the closed star of a vertex is a refinement of the pyramid over the boundary of a vertex figure at that vertex, and (b) using this property as a replacement for the radial projection in the proof of Grünbaum’s theorem.

129 8. Generalizations of Perles’ Skeleton Theorem

ast(v) v

star(v)

v (a) The star of the vertex . (b) The antistar of the vertex v.

Figure 8.3: The star and the antistar of the vertex of a 2-sphere (only partly drawn). Their intersection is the link.

8.3.1 Stars, Antistars, and Links We need to decompose a sphere into distinguished subcomplexes, the (closed) star, the (closed) antistar, and the link. They are introduced in the following deﬁnition; compare with Deﬁnition 1.3.6 for polytopal complexes.

Deﬁnition 8.3.1 (Star, antistar, link). Let be a regular cell comples, C and let F be a cell of . We deﬁne C the star at F by • star (F ) := G : there is H such that F H and G H , C { ∈ C ∈ C ⊆ ⊆ }

the antistar at F by ast (F ) := G : F G , • C { ∈ C 6⊆ } and the link at F by link (F ) := ast (F ) star (F ); • C C ∩ C see Figure 8.3.

A basic relation of these subcomplexes is given by the following lemma that we need below.

Lemma 8.3.2. Let be a regular cell complex and let F be a cell of . Then C C

link (F )= ast∂H (F ). C H ,F H ∈[ C ⊆

130 8.3. Simplex Refinements of Spheres PL

Proof. We have

link (F ) = ast (F ) star (F ) C C ∩ C = G : there is H : G H,F H,F G { ∈ C ∈ C ⊆ ⊆ 6⊆ } = G : G H,F G { ∈ C ⊂ 6⊆ } H ,F H ∈C[⊆ = ast∂H (F ), H ,F H ∈C[⊆ so the statement holds. Lemma 8.3.3. Let P be a strong (d 1)-sphere and v (P ). Then PL − ∈ V starP (v) and astP (v) are strong (d 1)-balls, and linkP (v) is a strong (d 2)-sphere. PL − PL − Proof. We prove the statement by induction on d. For dimensions d 3, every strong (d 1)-sphere is isomorphic to the boundary of a d-polytope,≥ so the statementPL − holds. (For d = 3, this is Steinitz’ theorem [108] [109]; see [118, Theorem 4.1].) So let P be a strong (d 1)-sphere, where d 4, and let v (P ). Form the strongly regularPL cell− complex P˜ by replacing≥ the star∈ of VP at v be the pyramid over the link of P at v with apexv ˜. Then P and P˜ are homeomorphic. Indeed, by induction, the antistar of every facet of PPLthat contains v is a ball, so the pyramid over it is also a ball. Thus, every facet F ofPLP that contains v is homeomorphic toPL the pyramid over the antistar of ∂F atv ˜. These homeomorphismsPL can be made to coincide on intersections of facets (the technical statement one needs here is that two simplicial subdivisions of the same underlying space have a common refinement [117, Chapter 1]). Thus starP (v) and and starP˜(˜v) are homeomorphic, and so are P and P˜. PL Since starP˜(˜v) is a cellular (d 1)-ball (see [97, Chapter 2]), also star (v) is a cellular (d 1)-ball.PL Newman’s− theorem implies that also P PL − astP (v) is a cellular (d 1)-ball. Since the link is the boundary of the star, it is a (d PL2)-sphere.− PL − 8.3.2 A Replacement for Radial Projection Definition 8.3.4 (Refinement homeomorphism). Let and be regular cell complexes. A refinement homeomorphism is a homeomorphismC K φ : such that for every F there is a subcomplex CF withkKk→kCkφ( F )= C . We then say that∈ Kis a refinement of . ⊆ C k k k F k C K 131 8. Generalizations of Perles’ Skeleton Theorem

The principal vertices of the refinement are the vertices φ( ( )), that is, the images of the vertices of . V K K The following lemma replaces the radial projection in the proof of The- orem 2.1.2, Grünbaum’s theorem on simplex refinements of polytopes. Lemma 8.3.5. Let P be a strong (d 1)-sphere, and let v (P ). Furthermore, let P/v be a vertex figurePL of P− at v. ∈ V Then the star starP (v) is a refinement of the pyramid over P/v such that the principal vertices of the refinement are U v , where U is a subset of the neighbors N(v) in the graph of P . ∪ { } Proof. We denote by F/v the cell of P/v that correponds to the cell F of P .

We inductively construct maps φi : skeli(pyrv(P/v)) skeli(starP (v)) for i 0, 1,...,d 1 that satisfy the following two properties:→ ∈ { − } (i) The map φi is a reﬁnement homeomorphism onto its image. (ii) For every j-cell F of P that contains v with j min i + 1,d 1 we ≤ { − } have that φi(∂ pyrv(F/v)) = ∂F and φi(F/v) = astF (v).

Clearly, the map φ0 exists, so suppose that 1 1 d 2. Suppose that G/v is an (i 1)-cell of P/v that≤ corresponds≤ − to the i-cell G − of P . We define φi on pyrv(G/v) as follows. By (ii), the map φi 1 is defined − on the boundary of pyrv(G/v). We define φi identically on the boundary and extend it on the interior to a homeomorphism between pyrv(G/v) and the cell G. (It is easy to prove that this can be done; see for example [23, Lemma 4.7.2].) Next, suppose that G/v is an i-cell of P/v that corresponds to the (i+1)- cell G of P . By (ii) and by Lemma 8.3.2, we have that φi 1(∂G/v) = − link∂G(F ). Define φi identically to φi 1 on ∂G/v. Then extend this to − a homeomorphism between G/v and ast (v), which is a i-ball by ∂G PL Lemma 8.3.3.

The map φi is a refinement homeomorphism from skeli(pyrv(P/v)) to skeli(starP (v)), because P has the intersection property. By construction it is clear that the map satisfies (ii). By Lemma 8.3.2 and by (ii) we have that the map φd 1 restricted to the − (d 2)-skeleton of P/v is a refinement homeomorphism between P/v and link− (v), as both are (d 2)-spheres and consequently P − skeld 2(P/v)= P/v and skeld 2(linkP (v)) = linkP (v). − − Thus φd 1 is a refinement homeomorphism between pyrv(P/v) and starP (v). − By (ii) for j = 1 we have that the principal vertices are U v , where U is a subset of the neighbors of v. ∪ { }

132 8.4. Perles’ Skeleton Theorem for Spheres PL

8.3.3 Proof of Grünbaum’s Theorem for Strong Spheres PL Definition 8.3.6 (Rooted refinement). Let P be a strong (d 1)- sphere and v a vertex of P . We say that a refinement homeomorphismPL − with principal vertices U v is rooted at v if U is a subset of the neighbors of v in G(P ). ∪ { }

Theorem 8.3.7. Let P be a strong (d 1)-sphere, and let v be a vertex of P . PL − Then there is a reﬁnement homeomorphism from ∂∆d to P that is rooted at v.

Proof. We prove the statement by induction on d. For d 1, every (d 1)- ≤ − sphere is isomorphic to the boundary of ∆d. Let d 2 and let P/v be a vertex ﬁgure of P at v. Let v be a vertex ≥ 0 of ∆d. By induction there is a reﬁnement homeomorphism

φv : link∂∆d (v0)= ∂∆d 1 P/v, − → so clearly there is a refinement homeomorphism φv∗ from pyrv0 (link∂∆d (v0)) = star∂∆d (v0) to pyrv(P/v). By Lemma 8.3.5, starP (v) is a refinement of pyrv(P/v). Concatenating the refinement φv∗ with this refinement, we have a refinement homeomorphism from star∂∆d (v0) to starP (v). Extending this homeomorphism onto ast∂∆d (v0) yields the desired refinement homeomorphism. The statement about the principal vertices follows from the corresponding statement in Lemma 8.3.5. Lemma 8.3.5 yields in principle the same as radial projection. In particular, with a more detailed analysis one could prove the strengthenings that we proved in Theorem 2.1.2 for polytopes analogously for spheres. As we have no use for them, we contain ourselves with the versionPL at hand.

Corollary 8.3.8 (cf. Corollary 2.1.5). Let P be a strong (d 1)- sphere, and let v (P ) a vertex of P . Then G(P ) contains aPL subdivision− ∈ V of Kd+1 rooted at v. Corollary 8.3.8 was proved in much more generality by Barnette [8] [10]; but see also the discussion in Section 4.3.

8.4 Perles’ Skeleton Theorem for Spheres PL We replace the geometric proof of the bound on the number of missing edges by an argument that involves the subdivision of a complete graph.

133 8. Generalizations of Perles’ Skeleton Theorem

Theorem 8.4.1. Let P be a strong (d 1)-sphere on d+γ +1 vertices. Then the number of missing edges isPL bounded− by γ(γ + 1).

Proof. Let v be a vertex of P and let G be a reﬁnement of Kd+1 rooted at v. Let D denote the set of principal vertices of G and B = (P ) D. V \ If for vertices v1,v2 D the edge v1v2 is a missing edge of P , then at least one of the vertices∈ in B subdivides this edge. Therefore there can be at most γ missing edges between vertices in D. Additionally, every vertex in B has degree at least d in P , thus there can be at most γ2 empty edges incident to B. This totals to at most γ(γ + 1) empty edges.

The following two lemmas yield the generalization of Lemma 7.2.3. We had proven this lemma for polytopes with the aid of a reﬁnement homeomorphism with aﬃnely independent principal vertices. The following statement is the combinatorial analogon to this.

Lemma 8.4.2. Let P be a strong (d 1)-sphere and let φ : ∂∆d P be a refinement homeomorphism withPL principal− vertices D (P ).→ Let 1 k d and let U D with U = k. ⊆ V ≤If U≤ is the vertex-set⊆ of a| face| F , that is, U = (F ), then F is a (k 1)-simplex. V − Proof. Let j := dim F . By the definition of a refinement homeomorphism, there is a j-simplex τ ∂∆ such that F is a face of the strongly regular ⊆ d cell complex φ(τ). 1 1 Let τ ′ ∆ be the (k 1)-simplex with vertices φ− (U)= φ− ( (F )). ⊆ d − V Since j k 1 we have that τ ′ and τ intersect in the set of vertices 1 ≤ − φ− ( (F )). Thus we have τ ′ = τ and it follows that dim F = k 1. So F is a (Vk 1)-simplex. − − The previous lemma yields that every empty face induced by a subset of principal vertices is simplicial. Of course, this implies that an empty pyramid induced by a subset of principal vertices is an empty simplex.

Lemma 8.4.3. Let P be a strong (d 1)-sphere on d+γ+1 vertices and PL − let φ : ∂∆d P be a reﬁnement homeomorphism with principal vertices D (P ). → ⊆Then V there are at most γ empty pyramids induced by vertices in D and they are all empty simplices.

Proof. If M is an empty ℓ-pyramid induced by vertices in D, that is, (M) D, then by Lemma 8.4.2 every proper face of M is an (ℓ 1)- simplex.V ⊆ Thus M is an empty ℓ-simplex on ℓ + 1 vertices. −

134 8.4. Perles’ Skeleton Theorem for Spheres PL

1 Let τ ∂∆d be the ℓ-simplex with (τ) = φ− ( (M)). The strongly regular complex⊆ φ(τ) P is a pure ℓ-dimensionalV V complex. Therefore, ⊆ φ(τ) contains a vertex besides the vertices of M. Since φ is a bijection, we then have an injection from the set of empty pyramids induced on D into B := (P ) D. WeV conclude\ that D can induce at most B = γ empty pyramids, which are necessarily all empty simplices. | | We now come to the proof of the main theorem of this section, the generalization of Theorem 7.2.4. The argument is slightly more intricate as empty pyramids are not necessarily empty pyramids in vertex figures. We have to make sure that we nevertheless count all of them. Theorem 8.4.4. Let P be a strong (d 1)-sphere on d + γ + 1 vertices and 1 k d 1. Then the total numberPL − of empty ℓ-pyramids with ℓ k is bounded≤ ≤ by γ−(γ + 1)k. ≤ Proof. We prove the statement by induction on k. Let us fix some vertex v (P ), and consider a refinement homeomorphism φ : ∂∆ P rooted ∈ V d → at v. Let the principal vertices be D = φ( (∆d)), and B = (P ) D. Then the size of B is B = γ. V V \ | | Theorem 8.4.1 yields that the number of missing edges is bounded by γ(γ + 1). So, let k 2. Lemma 8.4.3 implies that the number of empty pyramids induced by vertices≥ in D is bounded by γ. The remaining empty ℓ-pyramids, ℓ k, of P contain each at least one vertex in B. Let M be such an empty ℓ≤-pyramid with ℓ 2. ≥ Claim. There is w (P ) and an (ℓ 1)-cell F with M = ∂ pyr (F ) and ∈ V − w (F ) B = . V ∩ 6 ∅ Proof of the claim. Suppose otherwise that for all possible choices of w and

F with M = ∂ pyrw(F ) we have (F ) B = . Fix one such choice w and F . Since all vertices of FV lie in∩ D we∅ get by Lemma 8.4.3 that F is an (ℓ 1)-simplex. Let F ′ F be a facet of F andw ˜ − ˜ ⊆ ˜∈ (F ) (F ′). Then for F := pyrw(F ′) andw ˜ we have M = pyrw˜(F ) andV \V(F˜) B = , since w (F˜) B. This proves the claim. V ∩ 6 ∅ ∈ V ∩ Corollary 8.2.6 now implies that for every empty ℓ-pyramid with 2 ℓ k that is not induced by vertices in D there is a vertex b B such that≤ ≤ ∈ M/b is an empty (ℓ 1)-pyramid of P/b. Let −

Γ(b) := M ′ : there is an empty ℓ-pyramid, ℓ k, in P with M/b = M ′ , { ≤ } 135 8. Generalizations of Perles’ Skeleton Theorem and Γ := Γ(b) b B [∈ By Lemma 8.2.7 we have a surjection from Γ to the set of empty ℓ-pyramids, 2 ℓ k, of P that contain a vertex of B. This shows that the number of ≤ ≤ empty ℓ-pyramids with 2 ℓ k of P is bounded by γ + Γ . For b B, let P/b be≤ a vertex≤ ﬁgure. The boundary of| P/b| is a strong ∈ (d 2)-sphere on (d 1) + γ′ +1 = d + γ′ vertices, where γ′ = γ PL − − k 1 − βw(P ). By the induction hypothesis, there are at most γ′(γ′ + 1) − empty ℓ-pyramids in P/b with ℓ k 1. Additionally, there are at most βb(P ) ≤ − k 1 many missing edges at b. Therefore, we count at most γ′(γ′+1) − +βb(P ) k 1 ≤ γ(γ + 1) − empty pyramids of P at b. Thus, the total number of empty ℓ-pyramids with ℓ k is bounded by ≤ 2 k 1 k γ + γ (γ + 1) − γ(γ + 1) , ≤ as claimed.

8.5 Pyramidally Perfect Lattices

According to Kalai [62], Perles’ original proof of his theorem in [89] applies to pyramidally perfect lattices. Here is the deﬁnition of this class of lattices.

Deﬁnition 8.5.1 (Pyramidally perfect lattice; see [62]). Let L be a graded atomic lattice, a (L), and x L with a x. Then a is pyramidal over x if (x a)= (x∈A) a . ∈ 6≤ The latticeA ∨L is pyramidallyA ∪ { } perfect if every atom that is pyramidal over x L is also pyramidal over every y x. ∈ ≤ A complete list of pyramidally perfect lattices with at most 8 elements is given in Figure 8.4. The following proposition is immediate from the deﬁnitions of graded, atomic, and pyramidally perfect. It is however important to note (a) the usefulness of this proposition, as it allows us to do induction by taking lower intervals, and (b) that the condition that x equals 0ˆ in (ii) and (iii) cannot be dropped. For example, the rightmost lattice in Figure 8.4(b) is the smallest example of a pyramidally perfect lattice L that has nonatomic upper intervals.

Proposition 8.5.2. Let L be a lattice, x, y L, and let L′ := [x, y]. ∈

(i) If L is graded, then L′ is graded.

136 8.6. Boolean Intervals

a b

(a) All pyramidally perfect lattices on at most 7 elements. (b) All pyramidally perfect lattices on 8 elements. The rightmost lattice has upper intervals [a, 1]ˆ and [b, 1]ˆ that are not atomic. Figure 8.4: A complete list of pyramidally perfect lattices on at most 8 elements.

(ii) If L is atomic and x = 0ˆ, then L′ is atomic.

(iii) If L is pyramidally perfect and x = 0ˆ, then L′ is pyramidally perfect.

8.6 Boolean Intervals

Definition 8.6.1 (Direct product [107]). Let P1 and P2 be two posets. The product of P1 and P2 is the poset on the set P1 P2 in which (x1, x2) (y , y ) if and only if x y and x y . × ≤ 1 2 1 ≤ 1 2 ≤ 2 If L1 and L2 are lattices then L1 L2 is a lattice. The direct product of two polytopal lattices is a lattice that× is isomorphic to the face lattice of the join of the two corresponding polytopes. Definition 8.6.2 (Product with a finite set). Let L be a lattice, and let A be a finite set with A L = . We define the product of L with A by ∩ ∅ L A := L B(A). ∗ × If for x L and A (L) (x) there is y L such that x y, ∈ ⊆ A \A ∈ ≤ [0ˆ, y] = [0ˆ, x] A, and (y)= (x) A, we also write y = x A. ∼ ∗ A A ∪ ∗ If A = a we write pyra(L) := L A and call pyra(L) a pyramid over base L with{ }apex a. If for x L and∗ a (L) (x) there is y L ∈ ∈ A \A ∈ such that x y, [0ˆ, y] = pyr ([0ˆ, x]), and (y)= (x) a , we also write ≤ ∼ a A A ∪ { } y = pyra(x).

137 8. Generalizations of Perles’ Skeleton Theorem

If L is a graded lattice of rank r and n := A , then L A has rank r +n. | | ∗ Deﬁnition 8.6.3 (γ). For a graded atomic lattice of rank r on n atoms we deﬁne γ(L) := n r. − The parameter γ behaves as for polytopes, as the following lemma shows.

Lemma 8.6.4. Let L be a graded atomic lattice.

(i) For every lower interval of L we have γ(L) γ(L′). In particular, γ(L) 0. ≥ ≥

(ii) If γ(L)= γ(L′) for a lower interval such that rk(L′) = rk(L) 1, then − a (L) (L′) is pyramidal over L′. ∈A \A Proof. Let L be a graded atomic lattice of rank r := rk(L). We prove the statement by induction on r, where the case r = 0 is trivial. Suppose that r 1 and let n := (L) . Let x L be of rank r 1. ≥ |A | ∈ − Let L′ := [0ˆ, x] and n′ := (L′) . |A | By induction, γ(L′) = γ(L′′) for every lower interval L′′ in L′. Since L is atomic there is at least one atom in (L) with a x, as otherwise A 6≤ 1ˆ would not be the join of atoms. Thus n n′ + 1 and it follows that ≥ n r n′ + 1 r = n′ (r 1) = γ(L′). Thus we have γ(L) γ(L′) for − ≥ − − − ≥ every lower interval L′ in L. If γ(L) = γ(L′) for L′ with rk(L′) = r 1, there is exactly one atom − a (L) (L′), that is, a is pyramidal over x. ∈A \A For pyramidally perfect lattices this lemma immediately implies the following corollary; compare [23, Exercise 4.4(b)].

Corollary 8.6.5. Let L be a pyramidally perfect lattice. If γ(L) = γ(L′) for a lower interval L′ with rk(L′) = rk(L) 1 then L is a pyramid over L′. In particular, if γ(L) = 0, then the lattice −L is boolean.

We now formulate and prove the analog of Lemma 5.4.2. This lemma stated that a d-polytope on d + γ + 1 vertices has a (d γ)-face that is a simplex. −

Lemma 8.6.6. Let L be a pyramidally perfect lattice of rank r on r + γ atoms and x L with rk(x) r γ. Then there∈ is y L of≤ rank−rk(y) = r γ and a set of atoms A (L) with A (x)∈ = , such that x A −= y. The set A is given by⊆ A = (y) (∩Ax). ∅ ∗ A \A

138 8.7. Reconstruction of Skeleta

Proof. We prove the statement by induction on r. The case r = 0 is trivial, so suppose that the statement is true for every pyramidally perfect lattice of rank r 1, and let x be an element of L. If rk(x−)= r γ, take y := x and A := and the statement holds. − ∅ If rk(x) < r γ we have in particular that x = 1.ˆ Let x′ be a coatom − 6 of L with x L′ := [0ˆ, x′]. Such a coatom exists, since L is graded. Let ∈ γ′ = γ(L′). We have rk(L′) = r 1 and 0 γ′ γ by Lemma 8.6.4. By the − ≤ ≤ induction hypothesis, there is y′ L′ of rank r 1 γ′ and a set of atoms ∈ − − A′ = (x) (y′), such that x A′ = y′. A \A ∗ If γ′ γ 1, then we set y := y′ and A := A′ and the statement follows. ≤ − Otherwise, L is a pyramid over L′ by Corollary 8.6.5. Let a (L) (L′). ∈A \A Then there is y L with y = pyr (y′). With A := A′ a we then get the ∈ a ∪ { } statement. Corollary 8.6.7. Every atom of a pyramidally perfect lattice of rank r on r + γ atoms is contained in a boolean interval of rank r γ. − 8.7 Reconstruction of Skeleta

Definition 8.7.1 (Induced sets). Let L be a pyramidally perfect lattice and let A (L). We denote by L[A] the poset on the set ⊆A L[A] := x L : (x) A , { ∈ A ⊆ } with the order induced by L. We call L[A] the subset of L induced by A. If L′ L is induced by some set of atoms, we say that L′ is an atom-induced subset⊆ of L. An atom-induced subset of a lattice is a finite meet-semilattice with 0.ˆ If L is graded, then L′ is graded, too. Definition 8.7.2 (k-skeleton). The k-skeleton of a graded poset L is the set of all elements of rank at most k. We denote the k-skeleton by skelk(L). Definition 8.7.3 (Kernel). Let L be a pyramidally perfect lattice and let k be an integer with k 1. Let ≥ A := a (L) : there is x L such that rk(x) k 1 { ∈A ∈ ≤ − and a is not pyramidal over x . }

We deﬁne the k-kernel Kerk(L) of L to be the k-skeleton of the subset of L induced by A, that is, Kerk(L) := L[A].

139 8. Generalizations of Perles’ Skeleton Theorem

On the face poset level, the Definition 6.1.4 given for the kernel of polytopes coincides with the above definition of the kernel. To show this, we have to show that the set A defined in Definition 8.7.3, considered as a set of vertices of a polytope P , is exactly the set of vertices in empty pyramids of P of dimension at most k, that is, A = (Kerk(P )). If a A, let j be the smallest dimension such that a is notV pyramidal over every∈ face of dimension j. By the definition of A, there is at least one face of dimension at most k 1 such that a is not pyramidal over this face. This implies that 0 j k− 1. Let F be a j-face, over which a is not pyramidal. Then, by≤ the≤ choice− of j, the vertex a is the apex of an empty pyramid with base F , and thus lies in an empty pyramid of dimension at most k. If a (Ker (P )), then either a is an apex of an empty ℓ-pyramid with ∈ V k ℓ k, and so it is not pyramidal over some (ℓ 1)-face, or it is the vertex of≤ a base F of an empty pyramid M. In this case,− let G F be a facet of F that does not contain a, and let v (P ) such that ⊂M = ∂(pyr (F )). ∈ V v Then pyrv(G)isa(k 1)-face over which a is not pyramidal. This equivalence of− the definitions also follows from Corollary 8.9.4 that we prove below. We now prove the generalization of Lemma 6.1.5, which stated that the k-skeleton of a polytope is reconstructable from the k-kernel.

Lemma 8.7.4. Let L be a pyramidally perfect lattice. Then the k-skeleton of L can be reconstructed, up to isomorphism, from Kerk(L) and the total number of atoms.

Proof. Given Kerk(L) and A, we deﬁne a poset on the set

K := (x, B) : x Ker (L), B A and rk(x)+ B k , { ∈ k ⊆ | | ≤ } by the following order: If y1 =(x1, B1) and y2 =(x2, B2) are two elements of K with x1, x2 Kerk(L) and B1, B2 A, then y1 K y2 if and only if x x and B ∈B . ⊆ ≤ 1 ≤ 2 1 ⊆ 2 Then there is an order-preserving injective map from K to skelk(L). Let y K with y = (x, B). As B (Kerk(L)) = , every atom in B is pyramidal∈ over every element of rank∩A at most k 1.∅ But then there is a − unique element y′ with (y′)= (x) B and y′ = x B. Clearly, the map A A ∪ ∗ π : K skel (L) deﬁned by π(y)= y′ is order-preserving. → k But there is also an order-preserving injective map from skelk(L) to K. Let y skelk(L). Let A′ = (y) A and B = (y) A, that is, A′, B is a partition∈ of the atoms of y intoA elements\ that lieA in∩ the k-kernel and those that do not.

140 8.8. Relatively Complemented Lattices

Let x := a′. a′ A′ ∈ _ If we can show that (x)= A′, then it follows that y′ :=(x, B) is in K and A that the map π∗ : skel (L) K deﬁned by π∗(y)= y′ is injective. k → Assume to the contrary that there is b B with b (x). Let A′′ A′ ∈ ∈A ⊆ be the smallest subset of A′ such that b (x′′), where ∈A

x′′ := a′′. a′′ A′′ ∈ _ Clearly, A′′ = . Let a′′ A′′, let Z = A′′ a′′ , and denote by z the join of the atoms6 in∅ Z. Then∈ clearly \ { }

(z) = A′′, b / (z) and z b = x′′. A 6 ∈A ∨ But then b is not pyramidal over z, which is of rank rk(z) k 1. This is a contradiction to b B A. ≤ − ∈ ⊆

8.8 Relatively Complemented Lattices

Recall that a lattice L is complemented if every element x L has a complementary element y L, that is, an element y such that∈x y = 1ˆ and x y = 0,ˆ and that it∈ is relatively complemented if every interval∨ is complemented.∧

Lemma 8.8.1. Let L be a graded relatively complemented lattice. Then L is pyramidally perfect.

Proof. The lattice L is atomic and graded by Theorem 1.2.2, so it only remains to show that whenever an atom is pyramidal over some element, it is also pyramidal over every element below that element. We prove this by induction on the rank of L, where the base case is trivial. Suppose that rk(L) 1. Let a (L) and suppose that a is pyramidal over x for some x with≥a x.∈A We can assume that x is a coatom of 6≤ ˆ L, otherwise we apply the induction hypothesis on the lattice [0, x a] ∼= ˆ ∨ pyra([0, x]) and get that a is pyramidal over every y x. Furthermore, we only need to show that a is pyramidal over every element≤ covered by x. By induction the statement then follows. Let y be one of the elements in L that are covered by x. Then rk(y)= r 2. By Theorem 1.2.2, the interval [y, 1]ˆ contains at least 4 elements, that− is, it contains at least one element of rank r 1 besides x. − 141 8. Generalizations of Perles’ Skeleton Theorem

Claim. There is exactly one element in [y, 1]ˆ of rank r 1 besides x, and this element is y a. Furthermore, (y a)= (y−) a . ∨ A ∨ A ∪ { } Proof of the claim. Since a is pyramidal over x, we have (L)= (x) a . A A ∪{· } As L is atomic, an element z of [y, 1]ˆ that is not x has to satisfy (z) = (y) a . Otherwise, (z) contains a proper superset of Aatoms ofAy whose∪ { join} is x. But thenA

a′ = 1ˆ, a′ (z) ∈A _ in contradiction to rk(z) = r 1. Clearly, we then have z = y a, and the claim is proved. − ∨

Thus, we have shown that (y a)= (y) a , so a is pyramidal over y. A ∨ A ∪{ } The list of pyramidally perfect lattices in Figure 8.4 shows that every pyramidally perfect lattice of rank at most 2 is a graded relatively complemented lattice. However, for rank larger than 2 this is false, as the rightmost example in Figure 8.4(b) shows.

8.9 Empty Pyramids in Lattices

We generalize the notion of “empty pyramid” to pyramidally perfect lattices. We also prove some of the lemmas of Chapter 7 in this generality. At some point, when we start to take upper intervals, we have to restrict the proofs to graded relatively complemented lattices. Indeed, the largest subclass of pyramidally perfect lattices that is closed under taking upper intervals is the class of graded relatively complemented lattices. We must be careful when generalizing empty pyramids to pyramidally perfect lattices, as we still want to be able to reconstruct the k-skeleton from the set of vertices in empty pyramids. One could say that the lattice in Figure 8.5(c), which is the lattice of flats of the geometry in Figure 8.5(a), does not have any “missing edges,” as any two atoms do have an element of rank 2 that lies above both of them. Yet we need to be able to distinguish the 2-skeleton of this lattice from the 2-skeleton of the lattice in Figure 8.5(d), which is the lattice of flats of the geometry in Figure 8.5(b). As we show in Corollary 8.9.4, the following definition of empty pyramids satisfies this requirement.

142 8.9. Empty Pyramids in Lattices

(a) A geometry on 4 points . . . (b) Another geometry on 4 points . . .

Figure 8.5: Two geometries of rank 3 on 4 points with nonisomorphic 2-skeleta.

Deﬁnition 8.9.1 (Empty pyramid). Let L be a pyramidally perfect lattice, M L an atom-induced subset of L, and ⊆ k := max rk(x) : x M + 1. { ∈ } Suppose there is an element y M and an atom a (M) with ∈ ∈A (i) rk(y)= k 1, − (ii) a = (M) (y), { } A \A (iii) a is pyramidal over every x < y, and

(iv) a is not pyramidal (in L) over y.

Then we call M an empty k-pyramid. We call y a base and a an apex of M, respectively.

Lemma 8.9.2. Let L be a pyramidally perfect lattice and k 2. Let M be an empty k-pyramid, and let y M and a (M) be a base≥ and apex of ∈ ∈ A M, respectively. Then every element of M of rank k 1 is either y or it can be written as z a with z ⋖ y. − ∨ 143 8. Generalizations of Perles’ Skeleton Theorem

Proof. Lety ˜ = y be of rank k 1. Then a (˜y), otherwise (˜y) (y) andy ˜ = y. Let6 − ∈A A ⊆A z := b. b (˜y) a ∈A \{ } Then z < y, since (˜y) a _ (y). Then a is pyramidal over z and a z =y ˜. This impliesA that\ {z}⊆A⋖ y. ∨ It follows that, if M is an empty k-pyramid, the poset Mˆ := M 1ˆ , where 1ˆ is an additional element with x< 1ˆ for all x M, is isomorphic∪ { to} ˆ ∈ pyra(y) and thus that M is a pyramidally perfect lattice. Lemma 8.9.3. Let L be a pyramidally perfect lattice and k 2. Let M be an empty k-pyramid and a (M). Then there is an empty≥ ℓ-pyramid ∈A M ′ M with ℓ k such that a is an apex of M ′. ⊆ ≤ Proof. Suppose there is no y in M with rk(y) k 2 such that a is pyramidal over every x < y but not over y, that≤ is, there− is no empty ℓ- pyramid M ′ M with ℓ k 1. Since a is pyramidal over 0,ˆ we get, by induction, that⊆ a is pyramidal≤ − over every element of rank k 2. Since Mˆ = M 1ˆ is atomic, there is y M of rank k− 1 such that a / (y). It remains∪ { only} to show that a =∈ (M) (y). − ∈A ˆ { } A \A Let z M and b M with M = pyrb(z). Lety ˜ be covered by y and b / (˜y) (such∈ ay ˜ exists∈ since [0ˆ, y] is atomic). Now a is pyramidal overy ˜, that∈A is, (˜y a)= (˜y) a b. A ∨ A ∪ { } 6∋ By Lemma 8.9.2 we must havey ˜ a = z. But then ∨ (˜y a)= (z)= (M) b A ∨ A A \ { } and we have (˜y) = (M) a, b andy ˜ b = y, that is, a = (M) (y). A A \ { } ∨ { } A \ A This implies that the set of atoms in the k-kernel of L equals the set of atoms in empty ℓ-pyramids with ℓ k. ≤ Corollary 8.9.4. Let L be a pyramidally perfect lattice and k 1. Let ≥ A = (Kerk(L)) and B be the set of atoms in empty ℓ-pyramids with ℓ k. ThenAA = B. ≤ Proof. This follows at once from Lemma 8.9.3, as the case k = 1 is trivial.

The following lemma is the generalization of Lemma 8.2.3 to pyramidally perfect lattices.

144 8.10. Empty Pyramids in Upper Intervals

Lemma 8.9.5. Let L be a pyramidally perfect lattice. Let M1 and M2 be empty k-pyramids of L such that M = M . Then M M is an interval 1 6 2 1 ∩ 2 in L, that is, there is y L with [0ˆ, y]= M M . ∈ 1 ∩ 2

Proof. Since, for i = 1, 2, Mi is an empty k-pyramid, there are ai (Mi) and x M with rk(x )= k 1, such that M 1ˆ = pyr (x ). ∈A i ∈ i i − i ∪ { } ai i Lety ˜ := x x and M˜ = M M L. We distinguish three cases. 1 ∧ 2 1 ∩ 2 ⊂ Case (i). Suppose that a1 / M2 and a2 / M1. Then M˜ = [0ˆ, x1] [0ˆ, x2], and the statement holds for y∈:=y ˜. ∈ ∩ Case (ii). Suppose that, up to symmetry, a1 M2 and a2 / M1. If x =y ˜ = x x , then x = x , because∈ rk(x ) = rk(∈ x ). Thus 1 1 ∧ 2 1 2 1 2 (M1) = (x1) a1 = (M2) a2 = (x2). This is not possible, as MA must haveA at∪ least { } oneA element\ { of rank} Ak 1 besides x = x . Hence, 1 − 1 2 me must have x1 = x1 x2. In this case, a1 is pyramidal over x1 x2. If we set y := a (x 6 x ),∧ then clearly M˜ = [0ˆ, y]. ∧ 1 ∨ 1 ∧ 2 Case (iii). In the ﬁnal case we have a1 M2 and a2 M1. If x = x x = x , then clearly a =∈a and thus ∈M = M . Also, if 1 1 ∧ 2 2 1 2 1 2 a1 = a2 and x1 = x2 then a1 = a2 is pyramidal over (x1 x2), and we set y := a (x x6 ). ∧ 1 ∨ 1 ∧ 2 So, we can assume that x1 = x2 and a1 = a2. We have a1 (x1 x2) < x2 or a (x x ) < x , since otherwise6 M 6 M and M ∨ M ∧. Because 2 ∨ 1 ∧ 2 1 1 ⊆ 2 2 ⊆ 1 both a1 and a2 are pyramidal over x1 x2 the elements a1 (x1 x2) and a (x x ) have the same rank. Thus∧a (x x ) < x and∨a ∧(x x ) < 2 ∨ 1 ∧ 2 1 ∨ 1 ∧ 2 2 2 ∨ 1 ∧ 2 x1. Then a2 is pyramidal over a1 (x1 x2) and we get the statement with y := a a (x x ). ∨ ∧ 2 ∨ 1 ∨ 1 ∧ 2

8.10 Empty Pyramids in Upper Intervals

In this section, the proofs and statements are merely valid for graded relatively complemented lattices, as the upper intervals in a pyramidally perfect lattice are not necessarily pyramidally perfect.

Lemma 8.10.1. Let L be a graded relatively complemented lattice, let M1 be an empty k1-pyramid and M2 be an empty k2-pyramid distinct from M1, and x M M . Suppose that the sets M ′ := y : y x and y M ∈ 1 ∩ 2 1 { ≥ ∈ 1} and M ′ := y : y x and y M are empty pyramids in [x, 1]ˆ . Then 2 { ≥ ∈ 2} M ′ = M ′ . 1 6 2

Proof. The statement is clearly true if k1 = k2. If k1 = k2 we have by Lemma 8.9.5 that there is y L with [0ˆ, y]=6 M M , since M = M . ∈ 1 ∩ 2 1 6 2 145 8. Generalizations of Perles’ Skeleton Theorem

ab c

(a) (b)

Figure 8.6: Two types of empty pyramids that do not occur in polytopal face lattices.

ˆ Since x M1 M2 we have x y and thus [x, y]= M1′ M2′ [x, 1]. We ∈ ∩ ≤ ∩ˆ ⊆ had assumed that M1′ and M2′ are empty pyramids in [x, 1]. Accordingly, we have M ′ = M ′ M ′ and M ′ = M ′ M ′ . This implies M ′ = M ′ . 1 6 1 ∩ 2 2 6 1 ∩ 2 1 6 2 The next lemma is probably the most technical statement we need for the proof of Perles’ Skeleton Theorem for graded relatively complemented lattices. It generalizes Lemma 8.2.9 for polytopes, as it bounds the number of empty pyramids that “disappear” in upper intervals. The proof is considerably more complicated than for polytopes. This does not only have technical reasons. The situation for graded relatively complemented lattices is really more complicated than for polytopes. Here are two examples of situations we have to cope with that do not appear in polytopal face lattices. The bold part of Figure 8.6(a) is an empty 2-pyramid, because (a b) = a, b, c . However, the element a b has rank 2. In particular,A any∨ two atoms{ in} a, b, c form an empty 2-pyramid.∨ This is the reason why an atom of a graded{ relatively} complemented lattice may lie in 2γ empty 2-pyramids; see the bound in the following lemma. In a polytope a vertex may lie on at most γ missing edges. The bold part of Figure 8.6(b) also is an empty pyramid, as the only element above the bold part has three additional atoms. (It is easy to check that the lattice is graded and relatively complemented.) The reason why these types of empty pyramids exist is that a statement similar to Lemma 8.2.1 fails for graded relatively complemented lattices.

Lemma 8.10.2. Let L be a graded relatively complemented lattice of rank r on r + γ atoms, a (L), L′ := [a, 1]ˆ , and γ′ = γ(L′). Let = M ,...,M∈A be the set of empty pyramids M that have a M { 1 n} i base x M and an apex a (L) such that i ∈ i i ∈A a x , and • ≤ i 146 8.10. Empty Pyramids in Upper Intervals

M ′ := y M : y a is not an empty pyramid in L′ := [a, 1]ˆ . • i { ∈ i ≥ } Then n 2(γ γ′). ≤ − Proof. The lattice L′ has r 1+ γ′ atoms. Each of these atoms is a rank 2 element in L that is above−a and above at least one other atom in L. Since L′ has r 1+γ′ atoms, we can choose a set B (L) a of size − ⊆A \{ } r 1+ γ′ such that every atom of L′ can be written as a b with b B. Every− atom in C := (L) ( a B) either (a) does not lie∨ below a∈ rank 2 element that coversAa, or\ (b){ }∪ lies below a rank 2 element that covers a and that covers at least two other atoms. In the latter case, at most one of the atoms lies in the set B. Therefore, if m denotes the number of atoms in C that do not lie below a rank 2 element that covers a, we have at most e + 2(γ γ′ e)=2(γ γ′) e empty 2-pyramids in the set . Let I− −1,...,n be− the− set of indices of empty pyrmaidsM in that are empty⊆k {-pyramids} for some k 3. Then x = a for every i MI. Let ≥ i 6 ∈ Ai := (Mi a) ( (xi) a ). Since Mi is an empty pyramid, the atom a is notA pyramidal∨ \ A over x∪and { } we have B = . i i i 6 ∅ Since Mi is an empty pyramid and xi >a, the atom ai is pyramidal over a, that is, (a a ) = a,a . In particular, the element a′ := a a is an A ∨ i { i} i ∨ i atom of L′. The atom a′ is pyramidal over every element in [a, x ] L′, as i i ⊆ H′ is not empty in L′. Hence, for every b B the element b a is not in i ∈ i ∨ (L′) and we have I m. A We now show that| | ≤ whenever i = j, for i, j I, then B B = . Let 6 ∈ i ∩ j ∅ b B and b B . i ∈ i j ∈ j We have b (x a ), so a b x a = x a′ . Since b x and i ∈ A i ∨ i ∨ i ≤ i ∨ i i ∨ i i 6≤ i (xi ai′ )= (xi) ai′ , we have ai′ a bi. The same holds for j, so we haveA ∨ the followingA inequalities∪ { } ≤ ∨

a′ a b x a′ , (8.1) i ≤ ∨ i ≤ i ∨ i a′ a b x a′ . (8.2) j ≤ ∨ j ≤ j ∨ j We distinguish two cases: a′ = a′ and a′ = a′ . i 6 j i j Case (i). If a′ = a′ , then [a′ , (x x ) a′ ] [a′ , (x x ) a′ ]= , since i 6 j i i ∧ j ∨ i ∩ j i ∧ j ∨ j ∅ ai′ is pyramidal over xi and aj′ is pyramidal over xj. Clearly, this implies that b = b . i 6 j Case (ii). If ai′ = aj′ , then ai = aj and both ai and aj are pyramidal over a. Then obviously x = x . i 6 j Suppose a b = a b L′. Then 8.1 and 8.2 imply that a b ∨ i ∨ j ∈ ∨ i ∈ [ai′ , (xi xj) ai′ ], because ai′ is pyramidal over xi and over xj. But∧ then∨ there is w [a, x x ] such that w a = a b and thus a is ∈ i ∧ j ∨ i ∨ i i not pyramidal over w < xi. This contradicts the fact that Mi is an empty pyramid.

147 8. Generalizations of Perles’ Skeleton Theorem

Thus, a b = a b and this implies b = b . ∨ i 6 ∨ j i 6 j In total, we thus have that n 2(γ γ′) m + m = 2(γ γ′). ≤ − − − 8.11 Proof for Relatively Complemented Lattices

Theorem 8.11.1. Let L be a graded relatively complemented lattice of rank r on r + γ atoms. Let k 1 and let Γ be the collection of all empty ℓ-pyramids with ℓ k. ≥ Then ≤ Γ 2kγk. | | ≤ In particular, the number of empty ℓ-pyramids with ℓ k is bounded by a function independent of r. ≤ Proof. We prove the statement by induction on k. Every atom is pyramidal over 0.ˆ Thus if k = 1 we have Γ = 0. | | Let Br γ L be a boolean interval of rank r γ, which exists by − ⊆ − Corollary 8.6.7. Let B = (Br γ) and D = (L) B. Then B = r γ − and D = 2γ. A A \ | | − For| | every atom a (L) the lattice L = [a, 1]ˆ is graded, and relatively ∈A a complemented. Let Γa be the collection of all empty ℓ-pyramids in La k 1 k 1 with ℓ k 1. By induction, Γa 2 − γ(La) − . Additionally, by Lemma≤ 8.10.2,− there are at most 2(| γ | ≤γ(L )) many empty pyramids that − a contain a in one of their bases, but that do not appear as an empty pyramid k 1 k 1 k 1 k 1 in La. Therefore, there are at most 2 − γ(La) − + 2(γ γ′) 2 − γ − empty ℓ-pyramids with ℓ k that have a base that contains− a.≤ We claim that every empty≤ ℓ-pyramid with ℓ k is counted at some a D. ≤ ∈Let M be an empty ℓ-pyramid of L. Then there are b (L) and x L such that x is a base of M and b an apex. If (x) ∈D A= and ∈ A ∩ 6 ∅ a′ (x) D, then M is counted at a′. ∈AOtherwise∩ (x) B. Then [0ˆ, x] is boolean, and any other element of M of rank ℓA 1 is⊆ a base of M. Since M is an empty ℓ-pyramid with − ℓ r γ and Br γ is boolean of rank r γ, there has to be z M of rank ≤ − − − ∈ ℓ 1 such that (z) D = 0, −Thus, the totalA number∩ 6 of empty ℓ-pyramids of L with ℓ k satisﬁes ≤ k 1 k 1 k k Γ D 2 − γ − = 2 γ , | | ≤ | | as claimed. Lemma 8.11.2. Let L be a pyramidally perfect lattice of rank r on r + γ atoms. Every empty k-pyramid of L has at most k + γ 1 atoms. − 148 8.12. Proof for Pyramidally Perfect Lattices

Proof. The proof is the same as the one for polytopes; compare Lemma 6.2.2. Indeed, an element x of rank (k 1) has at most k + γ atoms, with equality − if and only if L arises from x by taking a pyramid r k times. − Corollary 8.11.3 (Perles’ Skeleton Theorem for graded relatively complemented lattices). For fixed k 1 and γ 0, the number of combinatorial types of k-skeleta of graded≥ relatively complemented≥ lattices of rank r on r + γ atoms is bounded. Proof. Since, for fixed k and fixed γ, there is only a finite number of graded relatively complemented lattices of rank at most r k + γ on r + γ atoms, we can assume that k r γ. ≤ By Theorem 8.11.1,≤ the− number of empty ℓ-pyramids with ℓ k is ≤ bounded by a function of k and γ. This implies that the size Kerk(L) is bounded by a function of k and γ, as an empty k-pyramid has at most k + γ 1 atoms by Lemma 8.11.2. Thus,− also the number of combinatorial types of k-skeleta is bounded by a function of k and γ, as the k-skeleton can by reconstructed, up to isomorphism, from the k-kernel and the total number of atoms, by Lemma 8.7.4.

8.12 Proof for Pyramidally Perfect Lattices

Definition 8.12.1 (Tetration). Let n 0. We define a function f(n) by ≥ ...2 f(n) := 22 , n f(n 1) that is, recursively f(n) is defined by f|(0){z } = 1 and f(n) = 2 − . For k, γ 0 we write f(k, γ)= f(2k + γ 1). ≥ − The following theorem can be seen as a generalization of Lemma 8.6.6, and it follows the idea that the k-skeleton looks like the (direct) product of a “small” part of the k-skeleton with a “large” boolean lattice; compare Chapter 6. Theorem 8.12.2. Let L be a pyramidally perfect lattice of rank r on r + γ atoms and k an integer with 1 k r γ. ≤ ≤ − Then there is a set of atoms Ak such that for every x skelk 1(L) there − is y L with x A = y, and such that the size of (L) ∈A is bounded by ∈ ∗ k A \ k ...2 f(k, γ) = 22 . 2k+γ 1 − | {z } 149 8. Generalizations of Perles’ Skeleton Theorem

Proof. We construct the set Ak inductively. If k = 1, then according to Lemma 8.6.6 there is y of rank r γ and A = (y) with 0ˆ A = y. As − 1 A ∗ 1 A r γ, we have (L) A 2γ f(1, γ). | 1| ≥ − A \ 0 ≤ ≤ Let k 1 and let Ak 1 be given with the desired properties. ≥ − Claim. Whenever some element x L of rank k does not have y L ∈ ∈ with y = x (Ak 1 (x)), then this x is a join of atoms in W := ∗ − \A (L) Ak 1. A \ − Proof of the claim. Let x L be of rank k and set W ′ := W (x). Let ∈ ∩A z := w′. w′ W ′ ∈ If rk(z)= k, then x = z and x is_ a join of elements in W . Otherwise, we have rk(z) < k and by the deﬁnition of Ak 1 there is y with − y = z (Ak 1 (z)). Clearly, x = z A′ for A′ = (x) Ak 1 and ∗ − \A ∗ A ∩ − consequently y = x (Ak 1 (x)). ∗ − \A Let x ,...,x be the elements of rank k such that no y L exists with 1 n ∈ y = xi (Ak 1 (xi)). Every such element is the join of atoms in the − set W , and∗ the\A size of W is bounded by f(k 1, γ). Since the lattice L is f(k 1,γ)− atomic, the number n is bounded by 2 − . By Lemma 8.6.6, there are elements y1,...,yn of rank r γ and sets of atoms B ,...,B (L) with − 1 n ⊆A y = x B , y = x B , ..., y = x B . 1 1 ∗ 1 2 2 ∗ 2 n n ∗ n Let B = B1 B2 Bn. The size of every Bi is bounded from below by B (r∩ γ)∩···∩(k + γ) = r (k + 2γ), since an element of rank k | i| ≥ − − − has at most k + γ atoms below it. Hence, for every Bi, i = 1,...,n, in the intersection we “lose” at most (k + 3γ) of the atoms of L. Then the size of B is bounded from below by B r + γ n(k + 3γ). | | ≥ − We set Ak := Ak 1 B. Then for every element x of rank k there is − y L with y = x A .∩ The size of A is bounded from below by ∈ ∗ k k Ak Ak 1 n(k + 3γ). | | ≥ | − | − Thus, the size of (L) A is bounded from above by A \ k r + γ Ak r + γ Ak 1 + n(k + 3γ) − | | ≤ − | − | f(k 1,γ) f(k 1, γ) + 2 − (k + 3γ) ≤ − f(k, γ), ≤ as claimed.

150 8.12. Proof for Pyramidally Perfect Lattices

Corollary 8.12.3 (Perles’ Skeleton Theorem for pyramidally perfect lattices). For ﬁxed k 1 and γ 0, the number of combinatorial ≥ ≥ types of k-skeleta of pyramidally perfect lattices of rank r on r + γ atoms is bounded.

Proof. Since, for fixed k and fixed γ, there is only a finite number of pyramidally perfect lattices of rank at most r k + γ on r + γ atoms, we can assume that k r γ. ≤ By Theorem≤ 8.12.2,− there is a set of atoms A such that for every x ∈ skelk(L) there is y L with x A = y, and such that the size of (L) A is bounded by f(k,∈ γ). Clearly∗ no element of A lies in the k-kernelA and\ so the size of Kerk(L) is as well bounded by f(k, γ). Thus, also the number of combinatorial types of k-skeleta is bounded by a function of k and γ, as the k-skeleton can by reconstructed, up to isomorphism, from the k-kernel and the total number of atoms, by Lemma 8.7.4.

151

Part III

Unneighborly Polytopes

Chapter 9

Nonsimplicial Mani Polytopes

The theory of illumination of convex bodies, which has its roots in papers by Soltan [104] and Gr¨unbaum [49], is the main source for the material in this chapter. In particular, the result presented here relates to a problem on illuminated polytopes (that is, polytopes in which every vertex lies on an inner diagonal) that was posed by Hadwiger in 1972 [52]. He describes the problem as follows:

Vermutlich gilt die folgende Aussage: Hat ein Polytop, also ein kompaktes konvexes Polyeder P , des n-dimensionalen euklidischen Raumes die Eigen- schaft, dass sich zu jeder seiner Seitenflächen noch wenigstens eine andere mit ihr disjunkte Seitenfläche aufweisen lässt, so gilt fur¨ die Anzahl f der Seitenflächen von P die Ungleichung f 2n. Offensichtlich gilt Gleich- ≥ heit beim Hyperwurfel,¨ also beim 2n-Zell. Demnach ist also 2n die kleinste mögliche Seitenflächenzahl fur¨ Polytope der oben genannten Eigenschaft.1

He goes on with an argument that his conjecture is true for dimensions n 3. Only the four combinatorial types that are displayed in Figure 9.1 are≤ candidates for counterexamples—none of them is one, as none of them satisﬁes the “disjoint facets condition.” He then acknowledges the diﬃculty of the general problem:

1 The following statement is probably true: If a polytope, that is, a compact convex polyhedron P , in n-dimensional Euclidean space has the property that for every facet one can ﬁnd a disjoint facet, then the number f of facets of P satisﬁes f 2n. Obviously, we have equality for the hypercube, that is, for the 2n-cell. Accordingly,≥ 2n is the smallest possible number of facets of polytopes with this property.

155 9. Nonsimplicial Mani Polytopes

(a) A 2-simplex.

(b) A 3-simplex.

Figure 9.1: A complete list of polytopes in dimensions n = 2 and n = 3 with less than 2n facets.

Seltsamerweise scheint es, dass die Abkl¨arung, ob unsere Vermutung fur¨ alle Dimensionen n richtig ist oder nicht, viel schwieriger ist, als ein Kon- vexgeometer bei erster Konfrontation mit der Frage anzunehmen geneigt ist. Bereits einige haben sich vergeblich bemuht.¨ 2

He concludes by mentioning a letter by Gr¨unbaum, in which the latter conﬁrms the conjecure for n = 4, and by asking, whether the conjecture is true for all dimensions. (We now switch back from n to d to denote dimension.)

2 Strangely, it seems that the clarification, whether our conjecture is true for all dimensions n or not, is much more difficult than a convex-geometer would be inclined to believe upon first confrontation with this problem. Already several have tried in vain.

156 9.1. Illuminated Polytopes

Hadwiger’s question asks, in its polar-dual formulation, whether an illuminated d-polytope has at least 2d vertices. The obvious examples with equality are the d-crosspolytopes. Mani settled this question in an extraordinary paper from 1974 [70]. Not only did he give a negative answer to Hadwiger’s question, by constructing simplicial illuminated d-polytopes on fewer than 2d vertices. He also determined the extremal function s(d) for the number of vertices of such a polytope precisely. As it turns out, this function is roughly given by d + 2√d, that is, for large d there are illuminated polytopes on much fewer vertices than anticipated; see Theorem 9.1.4 below. Mani’s construction of illuminated d-polytopes on s(d) vertices is easy to understand—we will review it below. Proving the tight lower bound for the function s(d) is much harder, even with a simpliﬁcation found by Rosenfeld [96]. The argument consists to a large part of a series of delicate geometric “repositionings”: Given an illuminated polytope on the minimum number of vertices, one can carefully move the vertices to obtain a simplicial illuminated polytope of speciﬁc combinatorial type that is similar to those constructed by Mani. For these it is easy to show that they have at least s(d) vertices. In this chapter we are concerned with the set of illuminated polytopes on s(d) vertices and a question by Mani related to this set. His extremal examples constructed in [70] are simplicial and he asked whether nonsimplicial such polytopes exist. We give a precise answer to Mani’s question. While for d 5 the only ≤ extremal illuminated polytopes are the d-crosspolytopes, we show that for every d 6 there exists a nonsimplicial one. ≥

9.1 Illuminated Polytopes

In this section we deﬁne illuminated polytopes and state Mani’s result on the extremal function of the number of vertices of these polytopes [70]. All polytopes in this chapter are assumed to be full-dimensional. Suppose that we have a polytope, and imagine that this polytope is hol- low. Think of each vertex of the polytope as an inﬁnitesimally small light source that casts light on those points of the boundary that one can “see” from the vertex. Thus, a vertex does not cast light on all the points of proper faces of the polytope that contain that vertex. In particular, it does not cast light on itself. If all vertices are illuminated by the light cast from other vertices, then also the whole boundary is illuminated. Indeed, convex combinations of illuminated points are again illuminated. This image

157 9. Nonsimplicial Mani Polytopes explains the choice of the word “illuminated” in the following deﬁnition.

Deﬁnition 9.1.1 (Inner diagonal, illuminated). Let P be a d-polytope in Rd. For u, v Rd, the line segment ∈ [u, v] := λu + (1 λ)v : 0 λ 1 { − ≤ ≤ } is an inner diagonal of P if u and v are vertices of P and [u, v] int P = , that is, the segment [u, v] hits the interior of P . ∩ 6 ∅ The polytope P is called illuminated if every vertex of P lies on an inner diagonal.

A more general definition of diagonals that comprises the definition of inner diagonals can be given; see Section 10.2.3. Inner diagonals of polytopes were studied extensively by Bremner & Klee [31]. They obtained two upper bound theorems on the number of inner diagonals of general polytopes. They showed that, for fixed d and f0, the maximum number of inner diagonals is f d + 1 0 df + , 2 − 0 2 µ ¶ µ ¶ and that this is only attained by stacked polytopes [31, Theorem 3.9]. This is the same number as the maximum number of missing edges of a simplicial polytope, by the Lower Bound Theorem [7] [9]. In the proof of the upper bound for inner diagonals we can assume that the polytope is simplicial, as a perturbation of the vertices only increases the number of inner diagonals. This is similar to the first step in the proof of the Upper Bound Theorem by McMullen [76]. Bremner & Klee also showed that, for fixed d and fd 1, the maximum − number of inner diagonals is attained by certain simple polytopes [31, The- orem 3.10]. In addition, they analyzed with great care inner diagonals of 3-polytopes. We are interested in the minimal number of vertices of an illuminated polytope and the set of polytopes that attain this number.

Deﬁnition 9.1.2 (Mani polytopes). For d 1 we deﬁne the parameter ≥ s(d) := min f (P ) : P is an illuminated d-polytope , { 0 } that is, s(d) denotes the minimum number of vertices of an illuminated d- polytope. We call an illuminated polytope on s(d) vertices a Mani polytope.

158 9.1. Illuminated Polytopes

√4d+1 1 Deﬁnition 9.1.3. For d 1, we set p(d) := − . ≥ ⌈ 2 ⌉

The term p(d) shows up in the function of the minimum number of vertices of illuminated polytopes. The following theorem is the main result in Mani’s paper [70].

Theorem 9.1.4 (Mani [70]). For every d 1, ≥

s(d) = min 2d,d + p(d)+ d/p(d) + 1 . { ⌈ ⌉ }

For d 5, the function s(d) equals 2d, for d = 6, 7 it equals 2d = d + p(d)+ ≤d/p(d) + 1, and for d 8 it equals d + p(d)+ d/p(d) + 1 < 2d. The value⌈ of p⌉(d) is one possible≥ solution to ﬁnding a⌈ positive⌉ integer p such that p + d/p is minimized for ﬁxed d. The global minimum of the function f(x) =⌈ x +⌉ d/x in the continous variable x is given by √d. The integer solution is not unique in general: If d = 6, then both p = 2 and p = 3 attain the minimum of 5. A given p is a minimizer for the term p + d/p at least for every d in the range p(p 1) < d p(p + 1), that is, p⌈ + ⌉d/p q + d/q for − ≤ ⌈ ⌉ ≤ ⌈ ⌉ every positive integer q. The following lemma then implies that p(d) is a minimizer for p + d/p . ⌈ ⌉

√4d+1 1 Lemma 9.1.5. Let p := p(d)= 2 − . Then the following inequalities hold: ⌈ ⌉ p(p 1)

Proof. We prove the upper bound ﬁrst, as the calculation is shorter. We use that x + k = x + k for all x R and all integers k. ⌈ ⌉ ⌈ ⌉ ∈

√4d + 1 1 √4d +1+1 p(p +1) = − 2 2 » ¼ » ¼ √4d + 1 1 √4d +1+1 − ≥ 2 2 µ ¶ µ ¶ = d

For the strict lower bound, we ﬁrst calculate a (nonstrict) lower bound, and argue later why this implies a strict lower bound. We use that x x+1 ⌈ ⌉ ≤ ⌊ ⌋ 159 9. Nonsimplicial Mani Polytopes for all x R. ∈ √4d + 1 1 √4d + 1 3 p(p 1) = − − − 2 2 » ¼ » ¼ √4d +1+1 √4d + 1 1 − (9.1) ≤ 2 2 ¹ º ¹ º √4d +1+1 √4d + 1 1 − (9.2) ≤ 2 2 µ ¶ µ ¶ = d

√4d+1+1 Note that the bound in (9.1) is strict if and only if 2 is an integer, which is the only case in which (9.2) is not strict. That is, there is at most equality in (9.1) or in (9.2), but not both. Why does the minimum of p+ d/p show up in the function s(d)? This is partially answered by the construction⌈ ⌉ of the extremal polytopes, and we get back to this question after the discussion of the construction. Before we move on, I want to mention a related open problem on illuminated polytopes. Deﬁnition 9.1.6 (Primitively illuminated). Let P be a d-polytope. Then P is said to be primitively illuminated by its vertices if P is illuminated and no proper subset of (P ) illuminates P . That is, there is no proper subset U (P ) such thatV for every v (P ) there is u U such that ⊂ V ∈ V ∈ [u, v] is an inner diagonal of P . One can show that a d-polytope is primitively illuminated by its vertices if and only if the graph of inner diagonals, that is, the graph

G (P ) :=(V, (u, v) : [u, v] is an inner diagonal of P ) inn { } is a perfect matching on V . Question 9.1.7 (Boltiyanski, Martini & Soltan [29]). Is there a bound on the number of vertices of a primitively illuminated d-polytope? (The d- cube is primitively illuminated by its vertices. However, no example on more than 2d vertices is known.)

9.2 Mani’s Simplicial Illuminated Polytopes

For reference and comparison we review Mani’s construction of simplicial Mani polytopes [70].

160 9.2. Mani’s Simplicial Illuminated Polytopes

2 d Let Cd(n) := conv( (t,t ,...,t ) : t [n] ) be the cyclic d-polytope on n vertices v ,...,v .{ ∈ } { 1 n} We need Gale’s evenness criterion, which gives a combinatorial description of the cyclic polytopes.

Theorem 9.2.1 ([118, Theorem 0.7]). The cyclic d-polytope Cd(n) on n vertices has the following properties:

(i) It is a simplicial polytope.

(ii) (Gale’s evenness criterion) A set S [n] with S = d is the index set ⊆ | | of the vertices of a facet of Cd(n) if and only if

for all i

(i) For every j 1,...,q + 1 the set [d + p] Cj is the index set of a facet, by Gale’s∈ { evenness criterion;} see Theorem\ 9.2.1. (ii) The sets C cover the vertices of Q, that is, q C = (Q). j j=1 j V Denote by F the facet with vertex set (Q) v : i C , and let (P ) j V \{ Si ∈ j} Id be the polytope obtained from Q by stacking a vertex onto each of the facets F1,...,Fq+1. The polytope d(p) is illuminated “by construction,” as a stacked vertex lies on inner diagonalsI with all vertices that lie in the complement of the stacked facet. If we set p := p(d), the polytope := Id d(p) is a Mani polytope by Theorem 9.1.4 for d 5. Since Cd(p) is simplicial,I so is . ≥ Id 161 9. Nonsimplicial Mani Polytopes

(b) The polytope 3. (a) The polytope 2. I I Figure 9.2: The polytopes that result from Mani’s construction in dimensions 2 and 3.

Example 9.2.2. Let us look at examples in low dimensions. For d = 2, 3, the polytope is obtained from a simplex by stacking all Id facets. Thus, the number of vertices is 2(d + 1) and “we get nothing but pretty pictures”; see Figure 9.2. For d = 4, 5, the polytope d has one vertex more than the d-dimensional crosspolytope. I For d = 6, 7, the polytope d is an illuminated polytope on the same number of vertices as the d-crosspolytope,I which is 2d. It is not isomorphic to the crosspolytope, though: The vertices of the crosspolytope all have the d 1 same degree of 2 − >d and a stacked vertex of d only has degree d. For d 8, Mani’s polytopes have strictly fewerI vertices than the crosspolytopes,≥ so d = 8 is the ﬁrst interesting case. We have p(8) = 3, so we need to stack onto some of the facets of C8(11). The sets Cj, j = 1,..., 4, are schematically displayed in the following ﬁgure:

C4 1 2 3 4 5 6 7 8 9 10 11. C1 C2 C3 z }| { The four sets C1,C| 2{z,C3},C4|are{z facet} | complements{z } of C8(11). As their union covers all vertices of C8(11), the resulting polytope after stacking is illuminated. It has 11 + 4 = 15 < 16 = 2d vertices. With the above construction, we have proven the following theorem by Mani [70]. Theorem 9.2.3 (Mani [70]). Let d 2. Then there is a simplicial illuminated d-polytope on d + p(d)+ d/p≥ (d) + 1 vertices. Id ⌈ ⌉ 162 9.3. Nonsimplicial Mani Polytopes

As we have shown in Section 9.1, we cannot improve the construction by choosing a diﬀerent p. Of course, we know by Theorem 9.1.4 that the polytopes have the minimum number of vertices for d 5. A ﬁrst step Id ≥ towards a proof of that theorem is to show that the polytopes d have the minimum number of vertices among all illuminated polytopes Iconstructed in a similar way [70, Lemma 1]. We do not go further in that direction, as we have other plans.

9.3 Nonsimplicial Mani Polytopes

Mani’s extremal illuminated polytopes are simplicial, and he asked whether nonsimplicial ones exist. Question 9.3.1 (Mani [70, p. 66]). Are there nonsimplicial Mani polytopes? This question is also mentioned by Bremner & Klee [31], with an erroneous answer for low dimensions. They claim that it follows from Mani’s results in [70] that for d 7 the only Mani polytopes are the d-crosspolytopes. It is true in dimensions≤ d 7 that the set of Mani polytopes includes the d-crosspolytope. However,≤ as we have seen in Example 9.2.2 there are at least two combinatorial types of simplicial Mani polytopes in dimensions d = 6, 7. We show below that there are even nonsimplicial ones.

9.3.1 Unique Mani Polytopes For dimensions d = 3and d = 4 one can make complete lists of all polytopes on at most 2d vertices, which is the interesting range for the number of vertices in the case of illuminated polytopes. For d = 3 making such a list is an exercise for an introductory class on polytopes [119]. For d = 4 one can do the enumeration up to 7 vertices using Gale diagrams by hand [51, pp. 112–113]. The 4-polytopes on 8 vertices can be enumerated [2], but already because of their large number (according to Altshuler & Steinberg [2] there are 1294 diﬀerent combinatorial types) one certainly needs the help of a computer to accomplish this task; see also [103]. Checking these lists, one ﬁnds that the only illuminated polytopes in dimensions d = 3, 4 on 2d vertices are the d-crosspolytopes. We show here that in dimensions 1 d 5 Mani polytopes are combinatorially unique. Thus the only type that≤ ≤ appears is the d-crosspolytope. This can be inferred from results by Mani [70] and Rosenfeld [96].

163 9. Nonsimplicial Mani Polytopes

Deﬁnition 9.3.2 (Self illuminated, opposite sets). Let P be a d- polytope. A set of vertices U (P ) is said to illuminate itself if for ⊆ V every vertex v U there is a vertex u U such that [u, v] is an inner diagonal. ∈ ∈ A set W (P ) is said to lie opposite the vertex v (P ) if for every w W the⊆ V segment [v,w] is an inner diagonal and (P∈) V(W v ) illuminates∈ itself. V \ ∪ { } Let Γ(P ) := max W : W lies opposite some v (P ) . {| | ∈ V } The following is the main result from [96] with a slightly stronger statement that is easily extracted from Rosenfeld’s proof.

Theorem 9.3.3 (Rosenfeld [96]). Let P be an illuminated d-polytope. If Γ(P ) = 1, then f0 2d and there is a perfect matching on the inner diagonals. ≥

The next lemma is easily derived from results by Mani [70, Lemma 1, Proposition 2, and Proposition 3].

Lemma 9.3.4 (Mani [70]). Let d 3, let P be a Mani d-polytope, and assume that Γ(P ) 2. Then f (P ) ≥d + p(d)+ d/p(d) + 1. ≥ 0 ≥ ⌈ ⌉ Corollary 9.3.5. Let d 3, let P be a Mani d-polytope with Γ(P ) 2. ≥ ≥ Then d 6. ≥ Proof. By Lemma 9.3.4, the number of vertices is at least d + p(d) + d/p(d) + 1. But for 3 d 5 we have that d + p(d)+ d/p(d) + 1 > 2d. ⌈Since P⌉is a Mani polytope≤ ≤ we must have d 6. ⌈ ⌉ ≥ Theorem 9.3.6. There is exactly one combinatorial type among all Mani d-polytopes for 1 d 5. It is given by the d-crosspolytope. ≤ ≤ Proof. The cases d = 1, 2 are trivial, so assume d 3. Let P be a Mani polytope with 3 d 5. ≥ By Corollary≤ 9.3.5,≤ we have Γ(P ) = 1. Theorem 9.3.3 and existence of the crosspolytopes then imply that f0(P ) = 2d and that there is a perfect matching on the inner diagonals. Since any facet of P can contain only one vertex of any inner diagonal, the set of facets is a subset of the facets of the d-crosspolytope. This implies that P is the d-crosspolytope.

9.3.2 Nonsimplicial Mani Polytopes Nonsimplicial Mani polytopes can be constructed in diﬀerent ways.

164 9.3. Nonsimplicial Mani Polytopes

One method that works for some dimensions is the pseudo-stacking operation, described, for example, by Paffenholz & Werner [86]; see also [1]. In all dimensions except for d = p2 and d = p(p + 1), Mani’s construction leaves some room for local modifications. In these cases one can move one of the stacked vertices into a facet defining hyperplane, and this results in a nonsimplicial Mani polytope. We describe in this section a unified construction that settles the question for all dimensions. Recall that we want to cover the vertices by facet complements of maximal size, that is, we want to construct a d-polytope on d + p vertices that has “many” disjoint facet complements of size p. If we consider Gale diagrams, this problem becomes nearly trivial, as the positive circuits of a Gale diagram correspond exactly to the facet complements of the polytope. The construction in the proof of the following theorem follows this idea.

Theorem 9.3.7. There exists a nonsimplicial Mani d-polytope in every dimension d 6. ≥ Proof. For every d 6 we construct a nonsimplicial Mani d-polytope (recall that for d = 6, 7 we≥ have d+p(d)+ d/p(d) +1=2d). Let p 1, q := d/p , and choose a k with 1 k q 1.⌈ ⌉ ≥ ⌈ ⌉ We construct a nonsimplicial≤ ≤ − polytope Q that has q + 1 simplex facets, such that stacking onto these facets produces a nonsimplicial illuminated d- polytope. (What we describe here is in fact a whole family of such polytopes, indexed by the parameter k.) We describe Q in terms of a Gale diagram A. Let

B = e1,...,ep 1, 1 , { − − } where 1 denotes the vector in which all entries are 1. This is a positive p 1 basis of R − of cardinality p. The vectors in A are the following:

(1) Take k copies of B, and denote them by B1,...,Bk. (2) Take q k copies of B, and denote them by B˜1,..., B˜q k. − − − (3) Furthermore, take the vectors 1,e1,...,ed+p pq 1 one more time. − − − Then the number of vectors in A is d + p. By Theorem 1.3.3, every Bi, i = 1,...k, and every B˜j, j = 1,...,q k corresponds to facet complements of size p in Q, that is, to complements− of simplex facets . If we augment the set 1,e1,...,ed+p pq 1 to a positive {− − − } basis B′ by taking the last pq d vectors of B we have that − 1

B : i = 1,...,k B˜ : j = 1,...,q k B′ { i } ∪ { j − } ∪ { } 165 9. Nonsimplicial Mani Polytopes is a set of subconfigurations of A that correspond to complements of simplex facets of Q. These complements cover all vertices of Q. Stacking onto the corresponding facets we obtain an illuminated polytope P on d + p + q + 1 vertices. For p = p(d) we get that P is a Mani polytope. If 1 k q 1, then there is a set of two vectors in A that correponds to a facet≤ complement,≤ − so Q is nonsimplicial, unless p = 2. For d 7, we have p(d) 3 and we indeed get a nonsimplicial polytope. However,≥ for d = 6 we get≥ p = p(d)=2 and q = 3. In this case, we choose p = 3 instead of p(d). Then q = d/p = 2 and we have f0(P )=12= s(6), that is, P is a Mani polytope. ⌈ ⌉ In both cases, the polytope P is nonsimplicial, because Q is nonsimplicial and we only stack onto simplex facets. Example 9.3.8. We look at two examples that arise from the above description. For d = 6, we get a polytope Q as in the proof of Theorem 9.3.7 by constructing the Gale diagram in Figure 9.3(a) with p = 3, q = 2, and k = 2. We have seen this polytope before: It is the example of a 6-polytope Lockeberg constructed in [69] to show that in general simplex refinements cannot have two prescribed principal vertices; see Figure 2.1. The Gale diagram has three disjoint positive bases that cover all vectors: the bases B1 = B2 = e1,e2, 1 and the basis B˜1 = e1, e2, 1 . These bases correspond to complements{ − } of simplex facets of Q{−. Stacking− onto} these three facets produces a nonsimplicial illuminated 6-polytope on s(6) = 12 vertices. For d = 16, we display the result of the construction as an affine Gale diagram in Figure 9.3(b), where all points that “touch” represent different copies of a single point. (See [118, Chapter 6] for affine Gale diagrams and how they are related to ordinary Gale diagrams.) In this case, the polytope Q has f0 = 20 and five disjoint simplex facet complements of size four that cover all vertices. This yields a nonsimplicial illuminated 16-polytope on s(16) = 25 vertices. The following remarks are due to McMullen [79]. The construction is highly modifiable: Every vector configuration that contains at least two disjoint positive bases is a Gale diagram of a polytope. Thus, one may arbitrarily put the right number of positive bases of size p p 1 in R − (and double some vectors to get the right number of vectors). The polytopes constructed by Gale diagrams in the proof of Theo- rem 9.3.7 have a simple direct description. They can be obtained by applying a series of vertex splits on a prism over a simplex of suitable dimension.

166 9.3. Nonsimplicial Mani Polytopes

(a) Stacking onto the right set of facets a (b) Stacking onto the right set of facets of polytope with this Gale diagram yields a a polytope with this Gale diagram yields nonsimplicial Mani 6-polytope. a nonsimplicial Mani 16-polytope.

Figure 9.3: Gale diagrams of building blocks for nonsimplicial Mani polytopes.

Consider the following matrix of size 2p 2p: × 1 1 01 0 2 1 01 0  . . . .  ......   .  10 10     0 0 1 1   ··· ···  p + 1  1 1 11 1 1 1 1   ··· ···  p + 2  1 1 1 1  .  − −  .  1 1 1 1   . −. − . .   ......      2p  1 1 1 1   − −  The ﬁrst p + 1 rows contain as columns the vertices of the standard prism p+1 over a (p 1)-simplex lifted to the aﬃne hyperplane with xp+1 =1in R . The last− p 1 rows contain as columns the vectors of a Gale diagram − p 1 of this polytope. It consists of a positive basis in R − of size p and its negative. Doubling a vector of this Gale diagram corresponds to a vertex split at the corresponding vertex.

167

Chapter 10

Counterexamples to Marcus’ Conjecture

A minimal positive k-spanning vector configuration is a positively spanning vector configuration that is still positively spanning after the deletion of any (k 1) arbitrarily chosen vectors and is inclusion-minimal with respect to this− property. Marcus conjectured in [71] that the size of a minimal positive k-spanning vector configuration in Rr is bounded by 2kr. Why is this conjecture plausible? For k = 1 it is the classical Blumenthal–Robinson Theorem [26]; see also Theorem 10.2.1 below. Shephard gave an elegant Gale duality proof of this theorem [101]. The corresponding statement for linear k-spanning configurations, which was proven by Marcus [71], implies Marcus’ conjecture for centrally symmetric vector configurations. Furthermore, Dalmazzo [36] has shown a related result in graph theory on minimally k-edge-connected multidigraphs. A simpler proof was given by Berg & Jordán; see [14, Theorem 3]. Dalmazzo’s result implies Marcus’ conjecture for vector configurations that can be interpreted as the incidence matrix of such a multidigraph; we refer the reader to Marcus [71] for details. For k = 2, Marcus’ conjecture, in its Gale-dual formulation for polytopes, directly relates to Hadwiger’s problem on illuminated polytopes: A minimal positive 2-spanning configuration is a Gale diagram of an unneighborly polytope, a polytope in which every vertex lies on a missing edge. Thus

169 10. Counterexamples to Marcus’ Conjecture in particular, Gale diagrams of illuminated polytopes are minimal positive 2-spanning conﬁgurations. The conjectured bound of 4m on the size of a minimal positive 2-spanning conﬁguration translates to a lower bound of 4(d + 1)/3 on the number of vertices of an unneighborly polytope. ⌈ In a subsequent⌉ paper by Marcus [73], the following short note appeared at the end:

Note added in proof As this goes to press, the author has discovered an unneighborly polytope of dimension 36 having only 49 vertices.

(For d = 36, the conjectured bound of 4(d + 1)/3 equals 50.) Unfortunately, Marcus did not give⌈ any hint⌉ how he did obtain such a polytope. I tried to contact Marcus by writing to the California State Polytechnic University in Pomona in 2007. He had been on the faculty there for a number of years. Sadly, I received the information that he had died some years ago. Marcus’ counterexample is also mentioned in the “Mathematical Re- views” and “Zentralblatt” reviews of Marcus paper [73] by McMullen. How- ever, McMullen told me that he had never seen the counterexample him- self [79]. So it seems that Marcus’ counterexample is lost. However, as noted, illuminated polytopes are unneighborly, and as such polytopes exist on roughly d+2√d vertices, it is clear that for large enough d Marcus’ conjecture fails for k = 2. And indeed, Mani’s illuminated polytope in dimension 36 has exactly 49 vertices. So, it is likely that Marcus’ example was similar. In this chapter we work out the connection between Mani’s illuminated polytopes and minimal positive k-spanning configurations and prove the following two main results: Starting from Mani’s simplicial illuminated polytopes on s(d) vertices, • or from our nonsimplicial ones constructed in Chapter 9, we give counterexamples to Marcus’ conjecture for all k 2. Although nearly trivial for even k, the construction requires some≥ care for odd k. We prove an upper bound on the size of minimal positive k-spanning • r configurations in R , adding to a result by Marcus on the size of minimal positive 2-spanning configurations [71]. Mani’s illuminated polytopes imply that there is no bound on the size of minimal positive k-spanning configurations that is linear in r if k 2. Our bound is, for fixed k, polynomial in r, and we achieve this≥ with an application of Perles’ Skeleton Theorem.

170 10.1. Unneighborly Polytopes

We also look at classes of unneighborly polytopes for which the conjectured bound holds; see Section 10.2.3.

10.1 Unneighborly Polytopes

Davis’ paper [37] is a seminal piece for the theory of positive linear dependence with important classification results. As he remarks, the theory of pointed cones reduces to the study of polytopes, whereas we know now, with the advent of diagram techniques, that also the theory of solid cones, as he calls them, is essentially the theory of polytopes. In this section we work out the connection between minimal positive k- spanning configurations and a certain class of polytopes using Gale duality. r Let V = (v1,...,vn) be a vector configuration in R . Recall from Sec- tion 1.3.5 that V is said to be positively spanning for a vector space W Rr if nonnegative combinations of vectors of V span the space W . ⊆ We call V a positive spanning configuration if it is positively spanning for Rr. The configuration V is a minimal positive spanning configuration, also called a positive basis, if V is inclusion-minimal with respect to being positively spanning, that is, every proper subconfiguration of V is not positively spanning for Rr (it might be positively spanning for its linear span). Marcus [71] [73] generalized the notion of positively spanning in the same way as k-connectedness of graphs generalizes connectedness. Definition 10.1.1 (Positively k-spanning). For k 1, we call a vector configuration V positively k-spanning if for every U ≥ V of size at most k 1 the configuration V U is positively spanning. ⊆ −We call V a minimal positive\ k-spanning configuration if V is inclusion- minimal with respect to being positively k-spanning. The notion of positively k-spanning can be interpreted in terms of polytopes. To do so, we need the following definition of k-unneighborly polytopes. Definition 10.1.2 (k-unneighborly). Let P be a d-polytope with the following two properties: (i) It is (k 1)-neighborly, that is, every set of k 1 vertices is the set of vertices− of a simplex (k 2)-face. − − (ii) Every vertex is a vertex of an empty (k 1)-simplex. − Then P is called k-unneighborly. If k = 2, we call P unneighborly.

171 10. Counterexamples to Marcus’ Conjecture

If V is a vector configuration in Rr that is positively k-spanning for k 2, we can associate to V a polytope P of dimension n r 1 on n ≥ − − vertices that has V as a Gale diagram. This polytope is (k 1)-neighborly, according to the criterion in Theorem 1.3.3: Whenever we− remove at most k 1 vectors from V , the remaining vectors are still positively spanning. −If V is a minimal positive k-spanning vector configuration, then P is k-unneighborly: For every vector v V there are k 1 vectors U, such that the vectors in V (U v ) are not∈ positively spanning− for their linear span, as a minimal positive\ ∪ {spanning} configuration minus one vector is still linearly spanning for the whole space. By Theorem 1.3.3, the vertices that correspond to U v do not form a face of P . ∪ { }

10.2 The Sizes of Minimal Positive Spanning Conﬁg- urations

We are interested in the size of minimal positive k-spanning configurations. Throughout the rest of the chapter, we denote the size of the minimal positive k-spanning configuration V by n. A lower bound for n is trivially given by 2k + r 1: Take a hyperplane spanned by r 1 vectors of the configuration. Then− the positive half-space − and the negative half-space bounded by this hyperplane contain each at least k of the vectors of V , as V is positively k-spanning. It was shown by Marcus [73] that this lower bound is attainable. This result is better known as Gale’s Lemma [43]; see for example [75, pp. 64–66]. Harder to get are upper bounds on the size of V . We begin with a classical result for k = 1.

10.2.1 The Blumenthal-Robinson Theorem For k = 1 we have the following classical result, which was ﬁrst shown by Blumenthal & Robinson [26], according to Davis [37, p. 744]. We therefore call it the Blumenthal–Robinson Theorem.

Theorem 10.2.1 (Blumenthal & Robinson [26], Davis [37], Shep- hard [101]). If V is a positive basis for Rr of size n, then r + 1 n ≤ ≤ 2r. If n = 2r, then the conﬁguration is given by a basis (b1, . . . , br) and (λ1b1, . . . , λrbr), where λi < 0 for i = 1,...,r.

Shephard’s proof [101] of this theorem is the simplest and most concep- tual: He considers V as a Gale transform of a full-dimensional aﬃne point set P in (n r 1)-dimensional space. Since for any v V the conﬁguration − − ∈ 172 10.2. The Sizes of Minimal Positive Spanning Configurations

V v is not positively spanning, all points of P are double-points (one might\ { } consider these as empty vertices). Thus we have that n 2(n r), ≥ − that is, n 2r. If equality≤ holds in this equation, then P is an (n r 1)-simplex with doubled vertices and any Gale transform of P has the− desired− form. Thus, the Blumenthal–Robinson Theorem translates to the following nearly trivial statement about affine point sets: If every point of a d- dimensional affine point configuration lies on an empty vertex, that is, a double point of this configuration, then this point configuration has at least 2d + 2 vertices.

10.2.2 Marcus’ Conjecture and Unneighborly Polytopes Marcus conjectured [71] that the result by Blumenthal & Robinson [26] generalizes to minimal positive k-spanning vector configurations in the following way. Conjecture 10.2.2 (Marcus [71] [73]; disproved (Corollary 10.3.2)). Let V Rr be a minimal positive k-spanning vector configuration of size n. Then⊆ n 2kr. If n = 2kr, then V is contained in r lines through the origin. ≤ By taking k copies of a linear basis and k copies of its negative, we obtain a minimal positive k-spanning configuration on exactly 2kr vectors. Marcus [71] proved his conjecture for r = 2, and he proved it for minimal positive 2-spanning sets in Rr with r 4. For r 5, he gave a quadratic bound on the size of minimal positive≤ 2-spanning≥ sets. Theorem 10.2.3 (Marcus [71]). Let V Rr be a minimal positive 2- ⊂ spanning set and n := V . Then | | 4r, for r 4 n ≤ ≤ r(r + 1)/2 + 5, for r 5. ½ ≥ Marcus’ conjecture translates to the following conjecture on k-unneighborly polytopes; see also Marcus [71]. Conjecture 10.2.4 (Marcus [71] [73]; disproved (Corollary 10.3.3)). Let P be a k-unneighborly d-polytope, then 2k f (P ) (d + 1). 0 ≥ 2k 1 − 2k If f0 = 2k 1 (d + 1), then P is of type − (∆k 1 ∆k 1) (∆k 1 ∆k 1). − ⊕ − ∗ · · · ∗ − ⊕ −

173 10. Counterexamples to Marcus’ Conjecture

For unneighborly polytopes, this conjecture asserts that the number of vertices satisﬁes f 4(d + 1)/3 , 0 ≥ ⌈ ⌉ and the following gives a construction for polytopes that attain equality for every d 2. ≥ Indeed, what we describe here are whole classes of polytopes, as not even the combinatorial type is determined by the description in one case. However, all polytopes constructed are unneighborly, and the number of vertices is the same for all polytopes in the same class.

(1) For d = 3ℓ 1, we let d be the set of polytopes obtained by taking a join of ℓ-quadrilaterals.− M Because the graph of the join of two polytopes is the join of the graphs of the polytopes, every polytope in , for d = 3ℓ 1, is unneighborly. Md − (2) For dimensions d = 3ℓ, we take a polytope P in d 1, that is, a join M − of ℓ-quadrilaterals, and realize this polytope in the hyperplane xd = 0 in Rd. We then “attach” a quadrilateral along a common edge. That d 1 d 1 is, we take two vertices (v0, 0) R − R and (v1, 0) R − R of P that are connected by an edge,∈ and take× the convex hull∈ of P×and the two points (v0, 1) and (v1, 1). Denote the set of all polytopes obtained in this way by . Md As the points (v0, 0), (v1, 0), (v0, 1), (v0, 1) form a quadrilateral, the resulting polytope{ is unneighborly. The combinatorial} type of these polytopes depends on the realization of the join of quadrilaterals. For example, for d = 3 one may obtain a prism over a triangle, but also the polytope in which one square of this prism is “broken” into two triangles. (3) For dimensions d = 3ℓ+1, we again start with a join of ℓ-quadrilaterals, that is, with a polytope P in d 2, and deﬁne Q as the vertex sum of − this polytope with a quadrilateral,M that is, Q is of type (P,v ) (¤,v ), 0 ⊕ 1 where v0 is a vertex of P and v1 is a vertex of ¤. Denote by d the set of all such polytopes. M

Clearly, also every polytope in d for d = 3ℓ + 1 is unneighborly. A similar construction for polytopesM that attain equality was given by Marcus [71]. Indeed, one may construct polytopes in all dimensions such 2k that the integral bound of f0 2k 1 (d + 1) is tight for all d by mod- iﬁcations, similar to those above,≥ ⌈ on− joins of⌉ sums of simplices of higher dimensions. Comparing the functions s(d) and 4(d + 1)/3 , we ﬁnd that Marcus’ conjecture on 2-unneighborly polytopes⌈ fails for d⌉ = 36 (as claimed by

174 10.2. The Sizes of Minimal Positive Spanning Configurations

Marcus [73]) and for all d 39. For 4 d 29 and d = 31, 32 the polytopes provide unneighborly≥ polytopes≤ on≤ fewer vertices than Mani’s Md examples (the polytopes in 2 and 3 have the same number of vertices as the crosspolytopes). In allM otherM dimensions (d = 30, 33, 34, 35, 37, 38) both constructions give the same number of vertices. It is not known what the extremal function for the number of vertices of unneighborly polytopes is. By Theorem 10.2.3, a minimal positive 2- spanning configuration in Rr on n vectors satisfies n r(r + 1)/2 + 5. ≤ According to this result, an unneighborly polytope has at least f d + 2(d 4) 0 ≥ − vertices. Mani’s unneighborly polytopesp have roughly d + 2√d vertices (which is best possible for simplicial polytopes). So, for general polytopes the answer lies somewhere between the rough estimates d+√2d and d+2√d. It is not even known whether unneighborly polytopes on less than s(d) vertices exist for “large” d. McMullen observed [79] that unneighborly polytopes on s(d) vertices that are not illuminated exist at least in dimensions d p2 + 1,p2 + 2,p2 + 3,p(p + 1) + 1,p(p + 1) + 2,p(p + 1) + 3 . ∈ { } These can be obtained by “local” modifications of Mani’s illuminated polytopes, for example, by taking a join with a quadrilateral, a subdirect sum with an edge, or a subdirect sum with a quadrilateral, similar to the modifications described above. Before we construct counterexamples for all k 2, we look at classes of polytopes for which the polytopes are extremal≥ examples. Md 10.2.3 Special Classes that Satisfy Marcus’ Bound We consider unneighborly polytopes that are unneighborly because of low- dimensional diagonals. That is, we fix a constant m and look at those polytopes where every vertex lies on a diagonal through the relative interior of a q-dimensional face with 2 q m. ≤ ≤ The polytopes d that we constructed in the last section are of this type with m = 2 andM much of the (false) intuition why one would believe Conjecture 10.2.4 to be true is probably derived from these examples. Definition 10.2.5 (Dimension of diagonals [31]). Let P be a d-polytope. For 1 m d, an m-diagonal or a diagonal of dimension m of P is a segment≤ [v,w],≤ where v,w are vertices of P , such that the smallest face that contains [v,w] is of dimension m.

175 10. Counterexamples to Marcus’ Conjecture

The 1-diagonals of a polytope are the edges of the polytope and the d-diagonals are the inner diagonals. Similar to Marcus’ conjecture, one may ask for the minimum number of vertices of a polytope such that every vertex lies on a diagonal of dimension at most, or exactly, some ﬁxed constant. For inner diagonals this number is given by Mani’s result on illuminated polytopes [70]; see Theorem 9.1.4. Theorem 10.2.6. Let P be d-polytope and ﬁx 2 m d 2. Suppose that every vertex lies on a diagonal of dimension q with≤ 2≤ q− m. Then ≤ ≤ (m + 2)(d + 1) f (P ) . 0 ≥ m + 1 » ¼ In particular, if m = 2, then f (P ) 4(d + 1)/3 . 0 ≥ ⌈ ⌉ Proof. Because every vertex of P lies on a diagonal of dimension at most m, there are faces F1,...,Fℓ of P with the following properties: (i) We have m := dim F m. i i ≤ (ii) The face Fi carries an mi-diagonal of P , that is, the polytope Fi has an inner diagonal. This implies f (F ) m + 2. 0 i ≥ i (iii) The polytope P is the convex hull of the vertices of the faces F1,...Fℓ, that is, P = conv F ,...,F . { 1 ℓ} But then

ℓ ℓ d ℓ 1+ m ℓ 1+ m = ℓ(m + 1) 1, ≤ − i ≤ − − j=1 j=1 X X and we get that

ℓ (m + 2)(d + 1) f (P ) m + 2ℓ = d + ℓ + 1 . 0 ≥ i ≥ m + 1 j=1 X » ¼ For m = 2, this reads as f (P ) 4(d + 1)/3 . 0 ≥ ⌈ ⌉ For m = d, this yields the trivial lower bound of f0 d + 2. Another peculiar property of a join of quadrilaterals≥ is that it can be realized as a Lawrence polytope, that is, a polytope obtained from a series of Lawrence extensions; see Ziegler [118, Theorem and Definition 6.26]. These polytopes have the property that they have a centrally symmetric Gale diagram. It follows from Marcus’ result on linearly k-spanning configurations [71] that every Lawrence polytope also satisfies the conjectured bound of f 4(d + 1)/3 . 0 ≥ ⌈ ⌉ 176 10.3. Counterexamples to Marcus’ Conjecture

The joins of quadrilaterals are also the only polytopes with the maximum possible number of disjoint missing edges, by Theorem 5.4.14 and Lemma 7.3.1. Thus, the polytopes d are indeed extremal examples for a number of classes of polytopes, butM as we have already seen not for the class of unneighborly polytopes. Theorem 10.2.6 implies that unneighborly polytopes that have less vertices than Mani’s illuminated ones must have high-dimensional diagonals.

10.3 Counterexamples to Marcus’ Conjecture

We now construct counterexamples to Conjectures 10.2.2 and 10.2.4 for all k 2. ≥ Theorem 10.3.1. Let k 2 and r 1 and write ≥ ≥ k r r + 1 f(k, r) := + r + 1 . 2 2 2 ¹ º µ¹ º ¹ º ¶ Then there is a minimal positive k-spanning conﬁguration in Rr of size n, where f(k, r), if k is even, n = r+1 ( f(k, r)+ 2 + 1, if k is odd. Proof. The construction depends¥ on¦ the parity of k, so we distinguish two cases. Case k = 2ℓ. We choose d depending on whether r is even or odd. If r = 2p, we set d := p2. Then

4p2 + 1 1 4p2 + 1 1 4p2 p 1 < − − = p, − p 2 ≤ &p 2 ' ≤ &p2 ' that is, p = p(d). If r = 2p + 1, we set d := p(p + 1). Then

4p(p + 1) + 1 1 (2p + 1)2 1 p(d)= − = − = p. &p 2 ' &p 2 ' Written uniformly, we have d = r/2 (r + 1)/2 and p(d)= r/2 . Let A be a Gale diagram of ⌊ . Then⌋⌊ the size⌋ of A is ⌊ ⌋ Id d r r + 1 A = d + p(d)+ +1= + r + 1. | | p(d) 2 2 » ¼ ¹ º ¹ º 177 10. Counterexamples to Marcus’ Conjecture

Take every vector of this conﬁguration ℓ times and denote the resulting vector conﬁguration by Aℓ. (On the polytope side, we have just taken the wreath product of with an (ℓ 1)-simplex [57].) The size n of A is Id − ℓ k r r + 1 ℓ A = + r + 1 = f(k, r). | | 2 2 2 µ¹ º ¹ º ¶

It remains to show that Aℓ is a minimal positive k-spanning configuration. Let H be an open halfspace of Rr. Since H contains at least 2 vectors of A, it contains at least 2ℓ = k vectors of Aℓ. Thus Aℓ is positively k- spanning. Furthermore, if u Aℓ, then u is also contained in A. Now there is an open halfspace H(u∈), such that H(u) contains exactly 2 vectors, one of them being u. But then H(u) contains exactly 2ℓ vectors of Aℓ, one of them being u. Thus Aℓ is a minimal positive k-spanning configuration. Case k = 2ℓ + 1. We now construct configurations for k = 2ℓ + 1. Let A and Aℓ be constructed as in the first case, and let u1,...,uq+1 , where q = d/p , be those vectors of A that correspond to the{ stacked vertices} of ⌈ ⌉ . Id For every i = 1,...,q + 1 add one copy of ui to Aℓ, and denote the resulting vector configuration by A˜ℓ. Then the size n of A˜ℓ is

r + 1 r + 1 n = A + q +1= A + +1= f(k, r)+ + 1. | ℓ| | ℓ| 2 2 ¹ º ¹ º

It remains to show that A˜ℓ is a minimal positive k-spanning conﬁgura- tion for Rr. Every open halfspace contains at least 2 vectors of A, and at least one of them is an element of u ,...,u . Thus every open halfspace contains { 1 q+1} at least 2ℓ + 1 vectors of A˜ℓ. Consequently, A˜ℓ is positively k-spanning. If u is a vector of A u ,...u , then there is a vector u u ,...,u \{ 1 q+1} i ∈ { 1 q+1} and an open halfspace H(u, ui) such that H(u, ui) A = u, ui . Then H(u, u ) contains exactly 2ℓ+1 vectors of A˜ . If u is a vector∩ in{ u ,...,u} , i ℓ { 1 q+1} then there is a vector u A u1,...uq+1 and an open halfspace H(u, ui) such that H(u, u ) A =∈ u,\ u{ . Again,}H(u, u ) contains exactly 2ℓ + 1 i ∩ { i} i vectors of A˜ℓ. Thus, A˜ℓ is a minimal positive k-spanning conﬁguration.

Corollary 10.3.2. Conjecture 10.2.2 fails for all r 12 if k is even, and for all r 18 if k is odd. ≥ ≥ 178 10.4. Upper Bound on the Size

Proof. Simple calculations show that for r 12 and k 2, the strict inequality ≥ ≥ f(k, r) > 2kr, is satisﬁed, and for r 18 and k 3, the strict inequality ≥ ≥ r + 1 f(k, r)+ + 1 > 2kr 2 ¹ º is satisﬁed. The statement follows.

Corollary 10.3.3. Conjecture 10.2.4 fails for all d with

f(k, r) r 1, k even, r 12, d = − − ≥ ( f(k, r) (1 r)/2 , k odd, r 18. − ⌊ − ⌋ ≥ Based on the counterexamples one can also prove that Conjecture 10.2.4 indeed fails for all dimensions d d(r, k), for some function d(k, r). Let P be a counterexample,≥ that is, a polytope with a Gale diagram as constructed in the proof of Theorem 10.3.1 for large r. If we start the construction with , then P is a simplicial polytope. Id Take the join of P with several sums of (k 1)-simplices ∆k 1 ∆k 1 − − and denote the resulting polytope by Q. To get− the intermediate dimensions⊕ we can “attach” a sum of two (k 1)-simplices along an ℓ-simplex of P , − which is a face of Q, with 1 ℓ 2k 1, similar to the construction − ≤ ≤ − of the polytopes d. This is possible since P and ∆k 1 ∆k 1 are both − − simplicial polytopes,M so both really have an ℓ-simplex face.⊕ The resulting polytope may fail to be a counterexample only for “small” dimensions.

10.4 Upper Bound on the Size

As Marcus’ conjecture fails for all k 2, and there is no obvious substitute for it, we formulate a more modest goal≥ as the following problem; compare with Problem 11.0.1 in the next chapter.

Problem 10.4.1. Prove an upper bound on the size of minimal positive k-spanning conﬁgurations in Rr.

This was done by Marcus for the case k = 2 [71]; see Theorem 10.2.3 above. We add a bound on the size of minimal positive k-spanning conﬁg- urations by proving an upper bound for all k 2. ≥ 179 10. Counterexamples to Marcus’ Conjecture

Theorem 10.4.2. Let V Rr be a minimal positive k-spanning conﬁgura- tion for k 2, and let n :⊂= V . Then ≥ | | k 1 n kr(r + 1) − . ≤ Proof. Let P be a d-polytope that has V as a Gale diagram, where d = n r 1. Then P is k-unneighborly, that is, every vertex is contained in − − an empty (k 1)-simplex. By Theorem− 7.2.4, the number of empty (k 1)-simplices is bounded k 1 − by r(r + 1) − . Thus the number n of vertices of P is bounded by

k 1 k 1 n f0(∆k 1)r(r + 1) − = kr(r + 1) − , ≤ − as claimed. As we have seen in Theorem 10.3.1, there exist, for ﬁxed k, minimal positive k-spanning conﬁgurations of size Θ(r2). Thus there is still a large gap to the upper bound of O(rk). We formulate and prove the corresponding result for k-unneighborly polytopes.

Corollary 10.4.3. Let P be a k-unneighborly d-polytope. Then P has at least f d + k (d + 1)/k 0 ≥ vertices. p

Proof. Let P be k-unneighborly and write f0 = d + r + 1. Then

k 1 d + r + 1 kr(r + 1) − , ≤ by applying Theorem 10.4.2 to a Gale diagram of P . Thus we have (d + 1)/k (r + 1)k. ≤ This implies that r + 1 k (d + 1)/k and we get ≥ p k f = d + r + 1 d + (d + 1)/k, 0 ≥ as claimed. p

180 Chapter 11

Positive Spanning Sets in Oriented Matroids

In this chapter we generalize the result of Chapter 10 on the size of minimal positive k-spanning configurations to oriented matroids. Gale duality is at the core of the interrelation between minimal positive k-spanning configurations and k-unneighborly polytopes, as discussed in Chapter 10. The setup and questions generalize to oriented matroids: We speak of totally cyclic oriented matroids instead of positive spanning sets, of acyclic oriented matroids instead of affine point sets, and of oriented matroid duality instead of Gale duality. In this setting the following problem had been posed by Bienia & Las Vergnas [23, Exercise 9.35(iv)*]. Problem 11.0.1 (Bienia & Las Vergnas [23, Exercise 9.35(iv)*]). Let be a totally cyclic oriented matroid on a set E. AM subset A E such that conv (A)= E is called a positive spanning M set. Given an integer⊆ k 1, a positive k-spanning set is a subset A E ≥ ⊆ such that A S is a positive spanning set for all S A with S < k. Find a function\ f(k, r( )) such that for every⊆ positive k|-spanning| set A there is a positive k-spanningM set B A with B f(k, r( )). ⊆ | | ≤ M In this chapter we derive such a function from Perles’ Skeleton Theorem. Bienia and Las Vergnas [15] generalized the Blumenthal–Robinson The- orem (Theorem 10.2.1) to oriented matroids. Consequently, for k = 1 the function f(k, r( )) is given by 2r( ) and this bound cannot be improved. M M 181 11. Positive Spanning Sets in Oriented Matroids

We have at least two ways to obtain a function f(k, r( )) for k 2 by applying Perles’ Skeleton Theorem. M ≥ We may use the bound we proved in Theorem 8.11.1 for graded relatively complemented lattices. The face lattice (covector lattice) of an oriented matroid is a graded relatively complemented lattice [23, Theorem 4.1.14, Corollary 4.1.16]. To obtain the better bound proved for geometrically realizable totally cyclic oriented matroids in Theorem 10.4.2 we invoke the topological representation theorem for oriented matroids by Folkman & Lawrence [42], Edmonds & Mandel [39], and Lawrence [68]. This is a deep theorem in the theory of oriented matroids, which establishes the hard part of the following chain of inclusions,

polytopes matroid polytopes strong spheres , { } ⊆ { } ⊆ { PL } namely, that matroid polytopes can be represented by strong spheres. We then apply Theorem 8.4.4. PL

11.1 Oriented Matroids

We state in this section the basic definitions for oriented matroids. The theory of oriented matroids is a broad field, and many interesting things can be said about them. We keep this part brief and introduce just the right amount of notions to be able to state Bienia & Las Vergnas’ problem (Problem 11.0.1). Notation and definitions are taken from [23, Chapter 3], and we refer to [23] for more (indeed, much more) on oriented matroids. + A signed set X is a set X together with a partition (X , X−) into the + positive elements X and the negative elements X−. The set X is the support of X. A signed set is positive if X− = , and it is negative if X+ = . We write for the signed set ( , ). ∅ Let∅E be a finite∅ set. A signed subset ∅of∅E is a signed set with support in E. As usual, we identify a signed subset of E with an element in +, 0, E. In particular, if E = [n] we write a signed subset as a sign vector of{ length−}n. For an element e of a signed subset X E we define X(e) := 1, if + ⊆ e X , X(e) := 1, if e X−, and X(e) := 0, if e E X. ∈ − ∈ ∈ \ + The opposite of a signed set X is the signed set X with ( X) = X− + − − and ( X)− = X . If is a collection of signed sets we write for the collection− X : X C . − C We define{− oriented∈ C} matroids in terms of circuit systems. As for matroids there are many axiom systems for oriented matroids, all meticulously detailed in [23, Chapter 3].

182 11.2. Oriented Matroids, Polytopes, and Duality

Definition 11.1.1 (Oriented matroid). An oriented matroid is a pair (E, ) of a finite set E and a collection of signed subsets of EM, called the C C circuits of , that satisfies the following axioms: M (i) / , ∅ ∈ C (ii) = , C − C (iii) for all X,Y , if X Y , then X = Y or X = Y , ∈ C ⊆ − + (iv) for all X,Y , X = Y and e X Y − there is a Z such that ∈ C 6 − ∈ ∩ ∈ C

+ + + Z (X Y ) e and Z− (X− Y −) e . ⊆ ∪ \ { } ⊆ ∪ \ { } We deﬁne the composition X Y of signed sets X,Y by ◦ + + + + (X Y ) = X (Y X−) and (X Y )− = X− (Y − X ). ◦ ∪ \ ◦ ∪ \ A vector of an oriented matroid is any composition of circuits. The signed sets X,Y are said to be orthogonal if either X Y = or if there are e, f X Y such that X(e)Y (e)= X(f)Y (f). ∩ ∅ The following∈ proposition∩ establishes the− existence of a dual oriented matroid. Proposition 11.1.2 (Bland & Las Vergnas [24]; see [23, Proposition 3.4.1]). Let be an oriented matroid E with circuits . M C (i) There is a unique signature ∗ of the cocircuits of the underlying ma- C troid of such that for all X and Y ∗ we have that X and Y are orthogonal.M ∈ C ∈ C

(ii) The collection ∗ is the set of circuits of an oriented matroid on E, C called the dual of and denoted by ∗. M M (iii) We have ( ∗)∗ = , that is, the dual of the dual is the oriented matroid weM started with.M

The circuits and vectors of ∗ are called the cocircuits and covectors of , respectively. M M 11.2 Oriented Matroids, Polytopes, and Duality

We relate polytopes to oriented matroids and mention the relationship between Gale duality and oriented matroid duality. This establishes a connection to the material of the previous chapter.

183 11. Positive Spanning Sets in Oriented Matroids

11.2.1 From Polytopes to Oriented Matroids Given a d-polytope P , we define an oriented matroid := (P ) of rank r = d + 1 that encodes the combinatorial structure ofMP inM the following way; see [23, p. 164] and also [118, Chapter 6], where oriented matroids are discussed with the primary intention of applying them to polytope theory. d T Let P R have vertices (P ) = v1,...,vn , and let f(x) = c x + d be an affine⊂ function on Rd. V { } We record the signs of the values of f on the vertices of P as the sign vector sgn(f(v1),...,f(vn)). These sign vectors constitute the collection of covectors of an oriented matroid (P ). We get the covector lattice ofM (P ) if we order the sign vectors ac- M cording to the following partial order: Take the natural partial order on +, , 0 given by the two relations 0 < + and 0 < . The partial order on{ the− sign} vectors then is given by componentwise extension− of this order. For every proper nonempty face F of P we find a supporting hyperplane given by an affine function f(x)= cT x + d with f(v) = 0 for vertices v on F and f(v) > 0 for all vertices v of P not on F . The sign vector sgn(f(v1), f(v2),...,f(vn)) does not depend on the exact choice of f but only on the combinatorial structure of P . Obviously, it contains only positive or zero entries. That is, the covectors corresponding to faces of P are all contained in the interval = [0, +], where 0 = (0,..., 0) and + = (+,..., +). The top elementF in this interval corresponds to the F empty set, while the bottom element corresponds to P itself. Thus, the interval is the face lattice of the polar of P , or in other words op = (P ). TheF lattice op is called the Las Vergnas face lattice of the orientedF L matroid . F M

11.2.2 From Gale Diagrams to Oriented Matroids Let G be a Gale diagram of a d-polytope P . Then G lies in Rm, where m = f0(P ) d 1. The vector conﬁguration G gives rise to an oriented *− − * matroid = (P ) of rank r∗ = m in the following way: The sign M M vectors of linear dependences among g1,...,gn form the circuit system of an oriented matroid. In particular, positive dependences correspond to positive covectors in , that is, to faces of the polytope P . Most importantly, * is the oriented Mmatroid dual to . M According toM Ziegler [118, p. 183], the connection between oriented matroid duality and Gale duality was ﬁrst observed by McMullen [78]. It was worked out by Sturmfels [110].

184 11.3. Positive Spanning Sets in Oriented Matroids

11.3 Positive Spanning Sets in Oriented Matroids

When talking about polytopes and Gale diagrams of polytopes, we are talking about objects with the following properties: The vertices of a polytope form an aﬃne point set, and the vectors of a Gale diagram form a positive spanning vector conﬁguration. These two properties are translated into the world of oriented matroids by the notions of “acyclic” and “totally cyclic.”

Deﬁnition 11.3.1 (Acyclic, totally cyclic, positive spanning set). Let be an oriented matroid on a set E and denote by ( ) the circuits of M. C M MThe oriented matroid is acyclic if it does not contain a positive circuit. It is totally cyclic ifM every element is contained in a positive circuit. For A E, we deﬁne the convex hull of A in by ⊆ M conv (A) := A e E A : there is X (M) M ∪ { ∈ \ ∈ C + such that e X− and X A . ∈ ⊆ } If is totally cyclic, we call a subset A E with conv (A)= E a positive M spanningM set. ⊆

Bienia & Las Vergnas [15] have shown the generalization of the theorem by Blumenthal & Robinson, Theorem 10.2.1, in this setting.

Theorem 11.3.2 (Bienia & Las Vergnas [15]). Let be a totally cyclic M oriented matroid.

(i) A set A E is a positive spanning set if and only if r (A)= r( ) M and A is⊆ a union of positive circuits. M

(ii) For every positive spanning set A there is a positive spanning set B A such that B 2r( ). Furthermore, B = 2r if and only⊆ if r | | ≤ M | | B = i=1 ei,ei′ , where e1,...,er is a basis of and ei,ei′ is a positive circuit{ for} i = 1,...,r{ . } M { } S

Deﬁnition 11.3.3 (Positive k-spanning set). Given an integer k 1, a positive k-spanning set is a subset A E such that A S is a positive≥ spanning set for all S A with S < k.⊆ \ ⊆ | | Let be a totally cyclic oriented matroid of rank r. Problem 11.0.1 by BieniaM & Las Vergnas is to ﬁnd a function f(k, r) such that for every positive k-spanning set A E there is a positive k-spanning set B A with B f(k, r). ⊆ ⊆ | | ≤ 185 11. Positive Spanning Sets in Oriented Matroids

Given a positive k-spanning set A there is a minimal, that is, inclusion- minimal, positive k-spanning set B A. So, the above question can be ⊆ answered by ﬁnding a bound on the size of minimal positive k-spanning sets. Clearly, a positive k-spanning set is positively spanning, so B is a union of positive circuits, by Theorem 11.3.2 (i). By deﬁnition, (E B) then is totally cyclic. Thus we can assume that E is a minimal positiveM \ \k-spanning set for . For k 2 this implies that * is a matroid polytope, that is, all one-elementM subsets≥ of E are verticesM [23, Chapter 9].

11.4 The Topological Representation Theorem

The topological representation theorem for oriented matroids due to Folk- man & Lawrence [42], Edmonds & Mandel [39], and Lawrence [68] asserts that every oriented matroid arises as the oriented matroid of an arrangement of pseudospheres; see for example the exposition by Bj¨orner [21]. For us, the following is the relevant part of the topological representation theorem for oriented matroids: Theorem 11.4.1 (see [23, Theorem 4.3.5]). Let be an acyclic oriented matroid of rank r. The Las Vergnas lattice of Mis isomorphic to the M face lattice of a strong (r 2)-sphere. PL −

11.5 Upper Bound on Minimal Positive k-Spanning Sets

We now apply Perles’ Skeleton Theorem to Problem 11.0.1 by Bienia & Las Vergnas on positive k-spanning sets in oriented matroids. We deﬁne k-unneighborly for strong spheres in the same manner as for polytopes. PL Deﬁnition 11.5.1 (k-unneighborly). Let P be a strong sphere. Then P is called k-unneighborly , if (a) it is (k 1)-neighborly,PL that is, every set of k vertices is the set of vertices of a (k− 2)-cell, and (b) every vertex of P lies in an empty (k 1)-simplex. − − Lemma 11.5.2. Let be a totally cyclic oriented matroid on set E, and suppose that E is a minimalM positive k-spanning set for k 2. Denote by * the corresponding matroid polytope. ≥ M If P is a strong sphere isomorphic to the Las Vergnas face lattice of *, then P is k-unneighborly.PL M 186 11.5. Upper Bound on Minimal Positive k-Spanning Sets

Proof. Let F E be a set of size at most k 1. Then E F is positively spanning in ⊆, that is, it is the support of− a positive vector.\ Then F is M the zero-set of a positive covector of *, that is, of a face of *. For every v E, there is a set S ME with S = k and v MS, such that E S is not positive∈ spanning in ⊂. Then S |is| not a face of∈ *. \ M M We can now state and prove the upper bound on minimal positive k- spanning sets. This solves Problem 11.0.1. As in the realizable case there is no indication why the bound proved should by asymptotically optimal.

Theorem 11.5.3. Let be a totally cyclic oriented matroid of rank r and suppose that E is a minimalM positive k-spanning set of for k 2. Then M ≥ k 1 E kr(r + 1) − . | | ≤

Proof. The matroid polytope ∗ is isomorphic to a strong (d 1)- sphere P for d := E r M1 on d + r + 1 vertices. ThePL sphere−P is k-unneighborly by Lemma| | − 11.5.2.− Apply Theorem 8.4.4 to P . Then the number of vertices of P , which k 1 equals E as k 2, is bounded by kr(r + 1) − . | | ≥

187

Appendix

Appendix A

Poset of Structures

Figure A.1 summarizes the inclusion relations among structures that appear in this thesis; see also Figures 8.1 and 4.1. Structures that are central for this thesis are indicated by a bold font. Some of the inclusions are trivial by definition, as strong spheres and strongly regular cell complexes are of course regular cellPLcomplexes, and the face poset of any structure is a poset. The association of an oriented matroid to a polytope is discussed in Section 11.2. That matroid polytopes are isomorphic to strong spheres is a consequence of the Topological Representation Theorem foPLr oriented matroids; see Section 11.4. Strongly regular cell decompositions of manifolds are graph manifolds, that is, the collection of the graphs of the cells of such a cell complex form a graph manifold; see Chapter 4, in particular also the more detailed figure in Section 4.3. The face poset of a weak graph manifold is a graded relatively complemented lattice. Indeed, such a face poset satisfies the diamond property. The inclusion then follows from Björner’s criterion for relatively complementedness; see Theorem 1.2.2. Finally, graded relatively complemented lattices are pyramidally perfect. This was shown in Section 8.8.

191 A. Poset of Structures

posets

pyramidally perfect lattices (Chapter 8)

graded relatively complemented lattices (Chapter 8)

weak graph manifolds

graph manifolds regular cell complexes (Chapter 4)

strongly regular cell de- cellular spheres compositions of manifolds PL

strong spheres (ChaptersPL 8, 11)

matroid polytopes (Chapter 11)

polytopes (Chapters 2, 3, 5, 6, 7, 9, and 10) Figure A.1: Inclusion relations among the structures that appear in this thesis.

192 Bibliography

[1] A. Altshuler and I. Shemer, Construction theorems for polytopes, Israel J. Math., 47 (1984), pp. 99–110. (165) [2] A. Altshuler and L. Steinberg, The complete enumeration of the 4-polytopes and 3-spheres with eight vertices, Pacific J. Math., 117 (1985), pp. 1–16. (163) [3] C. A. Athanasiadis, On the graph-connectivity of skeleta of convex polytopes. Preprint, 10 pages, January 2008; arXiv: http://arxiv.org/abs/0801.0939. (1, 2, 3, 37, 42, 53, 55, 65, 73, 74) [4] , 2008. Connectivity and skeleta of convex polytopes, talk in the lecture series of the research training group Methods for Discrete Structures, Technische Universität Berlin, Berlin, June 16, 2008. (3, 38, 48) [5] M. L. Balinski, On the graph structure of convex polyhedra in n- space, Pacific J. Math. (1961), pp. 431–434. (1, 37, 41) [6] D. W. Barnette, A necessary condition for d-polyhedrality, Pacific J. Math. 23 (1967), pp. 435–440. (37) [7] , The minimum number of vertices of a simple polytope, Israel J. Math., 10 (1971), pp. 121–125. (158) [8] , Graph theorems for manifolds, Israel J. Math. 16 (1973), pp. 62– 72. (1, 37, 53, 55, 56, 57, 58, 59, 63, 65, 133) [9] , A proof of the lower bound conjecture for convex polytopes, Pa- cific J. Math., 46 (1973), pp. 349–354. (158)

193 Bibliography

[10] , Decompositions of homology manifolds and their graphs, Israel J. Math., 41 (1982), pp. 203–212. (37, 55, 58, 133)

[11] , A short proof of the d-connectedness of d-polytopes, Discrete Math. 137 (1995), pp. 351 – 352. (37)

[12] , Generating the centrally symmetric 3-polyhedral graphs, Graphs Combin., 12 (1996), pp. 139–147. (34)

[13] M. M. Bayer, Face numbers and subdivisions of convex polytopes, in Polytopes: abstract, convex and computational (Scarborough, ON, 1993), vol. 440 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 1994, pp. 155–171. (16)

[14] A. R. Berg and T. Jordan´ , Minimally k-edge-connected directed graphs of maximal size, Graphs Combin., 21 (2005), pp. 39–50. (169)

[15] W. Bienia and M. L. Vergnas, Positive dependence in oriented matroids. Preprint, 11 pages, 1990. (181, 185)

[16] L. J. Billera and A. Bjorner¨ , Face numbers of polytopes and complexes, in Handbook of Discrete and Computational Geometry, Second Edition, CRC Press, 2004, ch. 18, pp. 407–430. (20)

[17] L. J. Billera and C. W. Lee, Suﬃciency of McMullen’s conditions for f-vectors of simplicial polytopes, Bulletin Amer. Math. Soc., 2 (1980), pp. 181–185. (119)

[18] , A proof of the suﬃciency of McMullen’s conditions for f-vectors of simplicial polytopes, J. Combin. Theory Ser. A, 31 (1981), pp. 237– 255. (119)

[19] A. Bjorner¨ , The minimum number of faces of a simple polyhedron, European J. Combin., 1 (1980), pp. 27–31. (55)

[20] , On complements in lattices of ﬁnite length, Discrete Math. 36 (1981), pp. 325–326. (12)

[21] , Topological methods, in Handbook of Combinatorics, Vol. 2, Elsevier, Amsterdam, 1995, pp. 1819–1872. (54, 186)

[22] , Higher connectivity: graph-theoretical and topological, Oberwol- fach Report, 5 (2008), pp. 10–13. (37)

194 Bibliography

[23] A. Bjorner,¨ M. Las Vergnas, B. Sturmfels, N. White, and G. M. Ziegler, Oriented Matroids, vol. 46 of Encyclopedia of Math- ematics and its Applications, Cambridge University Press, second ed., 1993. (5, 54, 73, 101, 122, 123, 125, 126, 132, 138, 181, 182, 183, 184, 186)

[24] R. G. Bland and M. Las Vergnas, Orientability of matroids, J. Combin. Theory Ser. B., 24 (1978), pp. 94–123. (183)

[25] G. Blind and R. Blind, Convex polytopes without triangular faces, Israel J. Math., 71 (1990), pp. 129–134. (89)

[26] L. M. Blumenthal, Theory and applications of distance geometry, Oxford, 1953. (85, 169, 172, 173)

[27] A. I. Bobenko and B. A. Springborn, Variational principles for circle patterns and Koebe’s theorem, Trans. Amer. Math. Soc., 356 (2004), pp. 659–689. (50)

[28] B. Bollobas´ and A. Thomason, Highly linked graphs, Combina- torica, 16 (1996), pp. 313–320. (75)

[29] V. Boltyanski, H. Martini, and V. Soltan, On Gr¨unbaum’s conjecture about inner illumination of convex bodies, Discrete Com- put. Geom. 22 (1999), pp. 403–410. (160)

[30] J. A. Bondy and U. S. R. Murty, Graph Theory with Applica- tions, Elsevier, 1976. (38)

[31] D. Bremner and V. Klee, Inner diagonals of convex polytopes, J. Combin. Theory Ser. A, 87 (1999), pp. 175–197. (158, 163, 175)

[32] A. Brøndsted and G. Maxwell, A new proof of the d- connectedness of d-polytopes, Canad. Math. Bull. 32 (1989), pp. 252– 254. (37)

[33] J. M. Bruckner¨ , Uber¨ die Ableitung der allgemeinen Polytope und die nach Isomorphismus verschiedenen Typen der allgemeinen Achtzelle (Oktatope), Verhand. Konik. Akad. Wetenschap, 10 (1910). Erste Sectie. (121)

[34] S. S. Cairns, Introductory Topology, The Ronald Press Co., New York, 1961. (73)

195 Bibliography

[35] J. W. Cannon, Shrinking cell-like decompositions of manifolds. Codimension three, Ann. of Math. (2), 110 (1979), pp. 83–112. (55) [36] M. Dalmazzo, Nombre d’arcs dans les graphes k-arc-fortement connexes minimaux, C. R. Acad. Sci. Paris S´er. A-B, 285 (1977), pp. A341–A344. (169) [37] C. Davis, Theory of positive linear dependence, Amer. J. Math. 76 (1954), pp. 733–746. (171, 172) [38] R. Diestel, Graph Theory, vol. 173 of Graduate Texts in Mathe- matics, Springer-Verlag, Third ed., 2005. (7, 9, 78, 82) [39] J. Edmonds and A. Mandel, Topology of oriented matroids, PhD thesis, University of Waterloo, 1982. Ph.D. Thesis of A. Mandel, 333 pages. (6, 182, 186) [40] P. Erdos˝ and R. Rado, Intersection theorems for systems of sets, J. London Math. Soc., 35 (1960), pp. 85–90. (4, 101, 102, 115, 119) [41] G. Fløystad, Cohen-Macaulay cell complexes, in Algebraic and Ge- ometric Combinatorics, Proceedings of a Euroconference in Math- ematics, August 2005, Anogia, Crete, Greece, C. A. Athanasiadis, V. V. Batyrev, D. I. Dais, M. Henk, and F. Santos, eds., vol. 423 of Contemporary Mathematics, Providence, RI, 2007, American Math- ematical Society, pp. 205–220. (3) [42] J. Folkman and J. Lawrence, Oriented matroids, J. Com- bin. Theory Ser. B, 25 (1978), pp. 199–236. (6, 182, 186) [43] D. Gale, Neighboring vertices of a convex polyhedron, in Linear In- equalities and Related Systems, H. W. Kuhn and A. W. Tucker, eds., vol. 38 of Annals of Math. Studies, Princeton, 1956, Princeton Uni- versity Press, pp. 255–263. (172) [44] S. Gallivan, Properties of the one-skeleton of a convex body, PhD thesis, Univ. of London, 1974. (75, 76, 85) [45] , Disjoint edge paths between given vertices of a convex polytope, J. Combin. Theory Ser. A, (1985), pp. 112–115. (75, 76, 85, 87, 88, 96) [46] S. Gallivan, E. R. Lockeberg, and P. McMullen, Complete subgraphs of the graphs of convex polytopes, Discrete Math. 34 (1981), pp. 25–29. (1, 2, 24, 32, 33)

196 Bibliography

[47] M. Goresky and R. MacPherson, Intersection homology theory, Topology, 19 (1980), pp. 135–162. (55)

[48] P. Gruber and R. Schneider, Problems in geometric convexity, in Contributions to Geometry: Proceedings of the Geometry Symposium in Siegen 1978, J. T¨olke and J. Wills, eds., Birkh¨auser Verlag, Basel, 1979, pp. 255–278. (24)

[49] B. Grunbaum¨ , Fixing systems and inner illumination, Acta Math. Acad. Sci. Hung., 15 (1964), pp. 161–163. (155)

[50] , On the facial structure of convex polytopes, Bulletin Amer. Math. Soc., 71 (1965), pp. 559–560. (2, 4, 23, 27, 29, 122, 129)

[51] , Convex Polytopes, vol. 221 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2003. Second edition edited by V. Kaibel, V. Klee and G. M. Ziegler (original edition: Interscience, London 1967). (3, 7, 18, 19, 23, 26, 27, 32, 33, 34, 35, 73, 90, 92, 116, 163)

[52] H. Hadwiger, Ungel¨oste Probleme, Nr. 55, Elem. Math., 27 (1972), p. 57. (155)

[53] R. Halin, Zu einem Problem von B. Gr¨unbaum, Arch. Math. (Basel), 17 (1966), pp. 566–568. (35)

[54] F. Holt and V. Klee, A proof of the strict monotone 4-step conjecture, in Advances in Discrete and Computational Geometry (South Hadley, MA, 1996) (Providence RI), Contemporary Mathematics, vol. 223, B. Chazelle, J. E. Goodman, and R. Pollack, eds., Amer. Math. Soc., 1998, pp. pp. 201–216. (1, 37, 38, 39)

[55] J. F. P. Hudson, Piecewise Linear Topology, Benjamin, New York, 1969. (122)

[56] W. Jockusch and N. Prabhu, Cutting a polytope, Geometriae Dedicata, 54 (1995), pp. 307–312. (23, 24, 25)

[57] M. Joswig and F. H. Lutz, One-point suspensions and wreath products of polytopes and spheres, J. Combin. Theory Ser. A, 110 (2005), pp. 193–216. (178)

[58] H. A. Jung, Eine Verallgemeinerung des n-fachen Zusammenhangs f¨ur Graphen, Math. Ann., 187 (1970), pp. 95–103. (75, 81)

197 Bibliography

[59] G. Kalai, Rigidity and the lower bound theorem, I, Invent. Math, 88 (1987), pp. 125–151. (1, 55, 97)

[60] , Many triangulated spheres, Discrete Comput. Geom., 3 (1988), pp. 1–14. (121)

[61] , The number of faces of centrally symmetric polytopes (Research Problem), Graphs Combin., 5 (1989), pp. 389–391. (26, 34)

[62] , Some aspects of the combinatorial theory of convex polytopes, in Polytopes: abstract, convex and computational (Scarborough, ON, 1993), vol. 440 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 1994, pp. 205–229. (1, 3, 4, 16, 73, 85, 96, 101, 102, 104, 105, 115, 119, 120, 121, 129, 136, 200)

[63] , Polytope skeletons and paths, in Handbook of Discrete and Computational Geometry, Second Edition, CRC Press, 2004, ch. 20, pp. 455–476. Chapter 17 in the ﬁrst edition (1997). (1, 24, 51, 75, 107, 119)

[64] V. Klee, A property of d-polyhedral graphs, J. Math. Mech., 13 (1964), pp. 1039–1042. (37)

[65] A. V. Kostochka, The minimum Hadwiger number for graphs with a given mean degree of vertices (Russian), Metody Diskret. Analiz., (1982), pp. 37–58. (75)

[66] D. G. Larman and P. Mani, On the existence of certain conﬁgu- rations within graphs and the 1-skeletons of polytopes, Proc. London Math. Soc. 20 (1970), pp. 144–160. (3, 24, 33, 35, 75, 76, 78, 80, 81, 96)

[67] D. G. Larman and C. A. Rogers, Paths in the one-skeleton of a convex body, Mathematika, 17 (1970), pp. 293–314. (37)

[68] J. Lawrence, Shellability of oriented matroids complexes. Preprint, 8 pages., 1984. (6, 182, 186)

[69] E. R. Lockeberg, Reﬁnements in boundary complexes of polytopes, PhD thesis, University College London, London, 1977. (2, 23, 24, 26, 30, 32, 33, 166)

[70] P. Mani, Inner illumination of convex polytopes, Comm. Math. Helv., 49 (1974), pp. 65–73. (5, 157, 159, 160, 162, 163, 164, 176)

198 Bibliography

[71] D. A. Marcus, Minimal positive 2-spanning sets of vectors, Proc. Amer. Math. Soc. 82 (1981), pp. 165–172. (101, 169, 170, 171, 173, 174, 176, 179)

[72] , Simplicial faces and sections of a convex polytope, Mathematika, 29 (1982), pp. 119–127. (85)

[73] , Gale diagrams of convex polytopes and positive spanning sets of vectors, Discrete Appl. Math. 9 (1984), pp. 47–67. (170, 171, 172, 173, 175)

[74] J. Matouˇsek, Lectures on Discrete Geometry, vol. 212 of Graduate Texts in Mathematics, Springer-Verlag, 2002. (17)

[75] , Using the Borsuk-Ulam Theorem, Springer-Verlag, 2003. (172)

[76] P. McMullen, The maximum numbers of faces of a convex polytope, Mathematika, 17 (1970), pp. 179–184. (158)

[77] , Constructions for projectively unique polytopes, Discrete Math., 14 (1976), pp. 347–358. (18, 95)

[78] , Transforms, diagrams and representations, in Contributions to Geometry: Proceedings of the Geometry Symposium in Siegen 1978, J. T¨olke and J. Wills, eds., Birkh¨auser Verlag, Basel, 1979, pp. 92– 130. (17, 26, 76, 97, 184)

[79] , 2008. Personal communication. (166, 170, 175)

[80] K. Menger, Zur allgemeinen Kurventheorie, Fundamenta Math., 10 (1927), pp. 96–115. (9)

[81] J. Migliore and U. Nagel, Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers, Adv. Math., 180 (2003), pp. 1–63. (119)

[82] J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. (73, 126)

[83] M. Naatz, Acyclic orientations of mixed graphs, PhD thesis, Tech- nische Universit¨at Berlin, Berlin, 2001. (73)

[84] U. Nagel, Empty simplices of polytopes and graded Betti numbers, Discrete Comput. Geom., 39 (2008), pp. 389–410. (102, 119)

199 Bibliography

[85] J. G. Oxley, Matroid Theory, Oxford Science Publications, Oxford University Press, New York, 1992. (122)

[86] A. Paffenholz and A. Werner, Constructions for 4-polytopes and the cone of ﬂag vectors, in Algebraic and geometric combinatorics, vol. 423 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2006, pp. 283–303. (165)

[87] A. Paffenholz and G. M. Ziegler, The Et-construction for lattices, spheres and polytopes, Discrete Comput. Geom., 32 (2004), pp. 601–621. (38, 49, 50)

[88] M. A. Perles, 2007. Personal communication. (101, 112)

[89] , Truncation of atomic lattices. Unpublished manuscript, cited in [62], around 1970. (3, 96, 101, 102, 121, 136)

[90] M. A. Perles and N. Prabhu, A property of graphs of convex polytopes, J. Combin. Theory Ser. A, 62 (1993), pp. 155–157. (37, 38, 39)

[91] J. Pfeifle and G. M. Ziegler, Many triangulated 3-spheres, Math. Ann., 330 (2004), pp. 829–837. (121)

[92] H. Poincare´, Cinquième complément àl’analysis situs, Rend. Circ. Mat. Palermo, 18 (1904), pp. 45–110. (55)

[93] N. Prabhu, Problem 3.11, in DIMACS workshop on polytopes and convex sets, list of problems, DIMACS, Rutgers University, 1990. (24, 25)

[94] J. Richter-Gebert, Realization Spaces of Polytopes, vol. 1643 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1996. (109)

[95] N. Robertson and P. D. Seymour, Graph minors. XIII. The disjoint paths problem, J. Combin. Theory Ser. B, 63 (1995), pp. 65– 110. (3, 75, 81)

[96] M. Rosenfeld, Inner illumination of convex polytopes, Elem. Math., 30 (1974), pp. 27–28. (157, 163, 164)

[97] C. P. Rourke and B. J. Sanderson, Introduction to piecewise- linear topology, Springer-Verlag, New York, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69. (122, 131)

200 Bibliography

[98] G. T. Sallee, Incidence graphs of convex polytopes, J. Combin. The- ory, 2 (1967), pp. 466–506. (1, 2, 37, 41, 42, 43, 48)

[99] R. Sanyal, A. Werner, and G. M. Ziegler, On Kalai’s conjectures concerning centrally symmetric polytopes. 14 pages; arXiv: http://arxiv.org/abs/0708.3661, to appear in Discrete Com- put. Geom. (26, 34)

[100] P. D. Seymour, Disjoint paths in graphs, Discrete Math., 29 (1980), pp. 293–309. (75, 81)

[101] G. C. Shephard, Diagrams for positive bases, J. London Math. Soc., 4 (1971), pp. 165–175. (169, 172)

[102] Y. Shiloach, A polynomial solution to the undirected two paths problem, J. Assoc. Comp. Mach., 27 (1980), pp. 445–456. (75, 81)

[103] M. Sierke, Klassifizierung 3-dimensionaler Sphären mit 8 Ecken. Diplomarbeit, Technische Universität Berlin, Berlin, Germany, 1999. (163)

[104] P. S. Soltan, Illumination from within the boundary of a convex body, Math. Sbornik, 57 (1962), pp. 443–448. (155)

[105] D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions, Methuen, London, 1929. Reprint: Dover, New York, 1958. (18)

[106] R. P. Stanley, Interactions between Commutative Algebra and Combinatorics, Birkh¨auser, 1983. (55)

[107] , Enumerative Combinatorics, Volume I, Cambridge University Press, 1997. (7, 11, 137)

[108] E. Steinitz, Polyeder und Raumeinteilungen, Encyclop¨adie der mathematischen Wissenschaften, Band 3 (Geometrie), Teil 3AB12, (1922), pp. 1–139. (131)

[109] E. Steinitz and H. Rademacher, Vorlesungen ¨uber die Theorie der Polyeder unter Einschluss der Elemente der Topologie, Springer- Verlag, Berlin, 1976. Reprint der 1934 Auﬂage, Grundlehren der Mathematischen Wissenschaften, No. 41. (131)

[110] B. Sturmfels, Neighborly polytopes and oriented matroids, Euro- pean J. Combin., 9 (1988), pp. 537–546. (184)

201 Bibliography

[111] T.-S. Tay, Lower-bound theorems for pseudomanifolds, Discrete Comput. Geom., 13 (1995), pp. 203–216. (55)

[112] N. Terai and T. Hibi, Computation of Betti numbers of monomial ideals associated with cyclic polytopes, Discrete Comput. Geom., 15 (1996), pp. 287–295. (120)

[113] R. Thomas and P. Wollan, An improved linear edge bound for graph linkages, European J. Combin., 26 (2005), pp. 309–324. (75, 81, 97)

[114] A. Thomason, An extremal function for contractions of graphs, Math. Proc. Cambridge Philos. Soc., 95 (1984), pp. 261–265. (75)

[115] C. Thomassen, 2-linked graphs, European J. Combin., 1 (1980), pp. 371–378. (75, 81)

[116] A. Werner and R. F. Wotzlaw, Linkages in polytope graphs, Electronic Notes in Discrete Mathematics, 31 (2008), pp. 189 – 194. The International Conference on Topological and Geometric Graph Theory. (3)

[117] E.-C. Zeeman, Seminar on Combinatorial Topology. Mimeographed notes, Inst. des Hautes Etudes´ Sci. Paris, 1963. (122, 131)

[118] G. M. Ziegler, Lectures on Polytopes, vol. 152 of Graduate Texts in Mathematics, Springer-Verlag, 1995. (7, 13, 15, 17, 20, 24, 40, 50, 76, 131, 161, 166, 176, 184)

[119] , Classiﬁcation of 3-polytopes with 6 vertices. Handout in class Discrete Geometry: Polytopes and More, 2004. (163)

[120] , Convex polytopes: Extremal constructions and f-vector shapes, in Geometric combinatorics, vol. 13 of IAS/Park City Math. Ser., Amer. Math. Soc., Providence, RI, 2007, pp. 617–691. (50)

202 List of Symbols

(P1,F ) (P2,G) the subdirect sum of P1 and P2 with respect to the ⊕ faces F and G [n] the set of all natural numbers from 1 to n [x, y] the interval between elements x, y of a poset (L) the atoms of the lattice L A (x) the atoms in a lattice below the element x A a regular cell complex C [U] subset of the cells of the regular cell complex C induced by the set U C ast (F ) the antistar of the regular cell complex at the cell C F C star (F ) the star of the regular cell complex at the cell F C C ∆d the d-dimensional standard simplex the empty set ∅ (P ) the face lattice of the polytope P L ( ) face poset of the regular cell complex F C C ( ) the face poset of the polytopal complex F C C ( ) the augmented face poset of a regular cell complex F C γ(P ) the quantity f0 d 1 of the strong (d 1)- b sphere P − − PL −

203 List of Symbols

(n, j1, k1,...,jm, km) the polytope ∆n 1 (∆j1 ∆k1 ) (∆jm ∆km ) − G for parameters n, m∗ 0⊕ and j ,∗· k ·,...,j ·∗ ,⊕ k 1 ≥ 1 1 m m ≥ (n, m) the polytope ∆n 1 ¤ ¤ for n, m 0 G − ∗ ∗ · · · ∗ ≥ m times int P the set of interior points of the polytope P | {z } the join of two elements in a join-semilattice ∨ a geometric simplicial complex K κk,ℓ(d) the minimum connectivity of (k,ℓ)-incidence graphs in dimension d link (F ) the link of the regular cell complex at the cell F C C /F face ﬁgure of the graph manifold at F M M a face structure M the meet of two elements in a meet-semilattice ∧ 1ˆ the maximal element of a poset that has a 1ˆ ¤ a 2-dimensional cube σ◦ the interior of a subset σ of a topological space ( ) the vertices of the polytopal complex V C C ( ) the vertices of the face structure V M M (P ) the set of vertices of the polytope P V (P ) the set of vertices of the regular cell complex V C 0ˆ the minimal element of a poset that has a 0ˆ 0 the column vector in which every entry is zero Ck the simple cycle on k edges Ek the graph on 2k vertices with exactly k disjoint edges f ( ) the number of vertices of the regular cell complex 0 C C G( ) the graph of the regular cell complex C C G G the disjoint union of the graphs G and G (under 1 · 2 1 2 ∪ the assumption that V (G ) V (G )= ) 1 ∩ 2 ∅ G G the join of the graphs G and G 1 ∗ 2 1 2 Gc(P ) the graph of P directed according to the linear function c

204 List of Symbols

Gk( ) the(k, k + 1)-incidence graph of the regular cell C complex C G ( ) the(k, k+1)-incidence graph of the graph manifold k M M Gk(P ) the(k, k + 1)-incidence graph of the polytope P

Gcofacet(P ) the hypergraph of facet complements of the polytope P

Gk,ℓ the (k,ℓ)-incidence graph of the polytope P G ( ) the(k,ℓ)-incidence graph of the graph manifold k,ℓ M M k(d) the minimal linkedness of d-polytopes k(d, γ) the minimal linkedness of polytope in γ Pd K(d, γ) the maximal linkedness of a d-polytope on d+γ +1 vertices k(G) the largest integer k such that the graph G is k- linked k(P ) the minimal linkedness of the graph of the polytope P kS(d) the minimal linkedness of simlicial d-polytopes k (d, γ) the minimal linkedness of polytopes in γ S Sd M/F the quotient of the empty face M at the cell F P/F the face figure of the strong sphere P at the cell F PL P k the path on k edges P P the join of the polytopes P and P 1 ∗ 2 1 2 P P the direct sum of the polytopes P and P 1 ⊕ 2 1 2 P P the cartesian product of the polytopes P and P 1 × 2 1 2 T (P ) the (deep) truncation of the truncatable polytope P aff(V ) the affine hull of the set V Rd ⊆ ast (F ) the closed antistar of the face F in the polytopal C complex C B(A) the boolean lattice on the set of atoms A B(a,...,a ) the boolean lattice on atoms a ,...,a n { 1 n} 205 List of Symbols bipyr P the bipyramid over P (P ) the boundary complex of the polytope P B a polytopal complex C ∆ C3 the octahedron

Cd the d-dimensional standard cube ∆ Cd the d-dimensional standard crosspolytope cone(V ) the conical hull of the set V Rd ⊆ conv(V ) the convex hull of the set V Rd ⊆ dim(P ) the dimension of the polytope P E the edges of a graph E(G) theedgesofthegraph G F 3 the face of the polar polytope that corresponds to the face F f(P ) the f-vector of the polytope P G the complement of the graph G G a ﬁnite graph γ(P ) the quantity f d 1 0 − − h the h-vector of a simplicial polytope I the unit interval κ(G) the connectivity of the graph G

Km,n the complete bipartite graph on m + n vertices [n] Kn the complete graph G = ([n], 2 ) on n vertices link (F ) the link of the face F in the polytopal complex C ¡ ¢ C maxc(P ) the set of maxima of the polytope P with respect to the linear function c minc(P ) the set of minima of the polytope P with respect to the linear function c N the natural numbers N = 0, 1, 2,... { } n a natural number N(v) neighbors of the vertex v in a graph P a polytope γ the set of d-polytopes on d + γ + 1 vertices Pd 206 List of Symbols

P/F the face ﬁgure of P at the face F P ∆ the polar of the polytope P P/v the vertex ﬁgure of P at the vertex v pyrk(P ) the k-fold pyramid over P pyra(P ) the pyramid over P with apex a R the real numbers rk(L) therankofthegradedposet L rk(x) the rank of the element x in a graded poset γ the set of simplicial d-polytopes on d+γ+1 vertices Sd skel ( ) the k-skeleton of the polytopal complex k C C star (F ) the closed star of the face F in the polytopal com- C plex C V the vertices of a graph or polytope V (G) the vertices of the graph G

207

Index

acyclic...... 185 bipyramid...... 19 affine Gale diagrams ...... 166 Blumenthal–Robinson affine hull...... 13 Theorem . . . 169, 172, 173, affine point set...... 181 185 antistar classical ...... 172 face structure ...... 60 for oriented antisymmetry...... 10 matroids...... 181, 185 apex...... 19, 124 large simplex face ...... 85 apex of a pyramid...... 18 Shephard’s proof . . . 172 – 173 arrangement of boolean...... 12 pseudospheres...... 186 boolean intervals ...... 137 Athanasiadis’ conjecture . . vii, 53, boolean lattice...... 12 54 – 55 bound is not tight...... 81 atom-induced ...... 139 boundary complex...... 19 augmented face poset ...... 124 Brückner sphere...... 121 bad edge...... 68 cellular ball...... 123 bad vertex...... 43 cellular PL sphere...... 123 Balinski’s theorem...... 1 classificationPL of polytopes on d + 2 for graph manifolds...... 63 vertices ...... 3, 90, 92 for polytopes...... 38 combinatorially isomorphic poly- generalizations of ...... viii, 1 topal complexes ...... 19 is best possible ...... 41 combinatorially isomorphic poly- barycentric coordinates...... 123 topes...... 16 base...... 19, 124 complement graph...... 10 base of a pyramid...... 18 complemented...... 12 bipartite graph...... 8 complete graph...... 8

209 Index concatenation direct sum...... 19 of paths...... 8 disjoint empty faces ...... 115 of strong chains ...... 61 bound is tight...... 115 of strong walks...... 62 disjoint union of graphs ...... 116 of walks...... 62 d-polytopal...... 48 cone...... see conical hull d-simplex ...... 16 conical hull...... 13 d-simplicial graph ...... 80 conjecture by Athanasiadis...... see edges...... 7 Athanasiadis’ conjecture elementary graphs...... 116 by Kalai, empty face ...... 109, 126 Kleinschmidt & Lee . . 119 empty faces...... 111 by Marcus ...... see Marcus’ flat empty faces ...... 111 conjecture in regular cell complexes . 125 connected graph...... 9 quotient of...... 111 connectivity empty k-faces...... 111 and linkages ...... 75, 77, 81 flat empty k-faces ...... 111 of a graph...... 9 empty k-pyramids...... 111 of graph manifold skeleta . . 64 empty k-simplices ...... 111 of incidence graphs ...... vii empty pyramid...... 126 convex hull ...... 13, 185 empty pyramids...... 111 covector lattice ...... 182, 184 in lattices...... 142 cubical polytopes...... 89 in quotient ...... 126 – 128 cyclic polytopes ...... 120 in regular cell complexes . 126 number of empty empty simplex...... 126 simplices in ...... 120 empty simplices ...... 111, 186 bound on the number of. .119 d-graph complex ...... 57 in regular cell complexes . 126 d-graph manifold...... 56 empty vertices ...... 173 d-graph manifold with boundary endpoints ...... 7 57 Erd˝os-Rado Sunflower d-pseudo-graph manifold ...... 57 Lemma...... 115, 119 d-crosspolytope ...... 16 erroneous answer to Mani’s prob- d-cube...... 16 lem ...... 163 definitions of kernels coincide...... 140 face structure d-face structure ...... 56 strongly connected ...... 56 diamond property. . . . .12, 91, 125 face figure...... 15, 58 versus relatively of a strong sphere . . . . 125 PL complemented ...... 12 face lattice...... 15 dimension...... 19 face poset...... 19

210 Index face poset of a polytope ...... 14 bipartite...... 8 face structure...... 56 complement...... 10 antistar...... 60 complete...... 8 closed under taking faces . . 56 connected...... 9 edges...... 56 connectivity...... 9 faces...... 56 cycle...... 9 facets...... 56 edges...... 7 intersection property ...... 56 independent paths...... 8 link...... 60 induced subgraph...... 8 pure...... 56 inner vertices of a path . . . . . 8 ridges...... 56 isomorphic ...... 7 star...... 60 join...... 9 strong chain...... 56 k-connected...... 9 vertices...... 56 length of a cycle ...... 9 flat...... 109 length of a walk...... 8 flat embedding...... 123 minor...... 10 flat empty faces ...... 111 neighbors...... 8 bound on the number of. .112 path...... 8 in quotient...... 111 path concatenation ...... 8 flat empty k-faces ...... 111 path length...... 8 free vertex ...... 82 polytopal...... 48 f-vector...... 16 polytopality ...... 48 subgraph...... 8 Gale diagrams...... 17 subpath...... 8 example...... 17 vertices...... 7 geometry of...... 116 walk...... 8 main theorem of...... 17 graph complex...... 57 oriented matroid of...... 184 graph manifold...... 56 Gale duality ...... 181 (k,ℓ)-incidence graph...... 64 and oriented bad vertex ...... 68 matroid duality ...... 183 connecting walk ...... 66 Gallivan’s examples...... 76, 85 dual graph...... 64 γ ...... 16, 138 edge figure...... 58 generic...... 38 face figure...... 58 geometric lattices ...... 121 – 122 face numbers...... 60 geometric realization ...... 123 good edge...... 68 geometric simplicial complex . 122 good vertex...... 68 geometries of rank 3 good walk...... 66, 69 on 4 points...... 142 strong cycle...... 65 good vertex...... 43 strong walk...... 61 graph...... 7 vertex figure...... 58

211 Index graph manifold with boundary 57 of empty pyramids . . 127, 145 g-vector...... 16 of geometric empty faces . 127 intersection property Hadwiger’s problem on illuminated face structure ...... 56 polytopes...... 155 intersection property Mani’s counterexamples . . 157 polytopal complex ...... 19 Handbook of Discrete and Com- regular cell complex ...... 54 putational Geometry . . 75 isomorphic graphs...... 7 Hirsch conjecture...... 1 isomorphism of posets ...... 11 h-vector...... 16 join of two graphs...... 9 illuminate itself ...... 164 join of two polytopes ...... 18 illuminated...... 158 combinatorially...... 18 illuminated polytopes geometrically ...... 18 Hadwiger’s problem ...... 155 join-semilattice ...... 11 Mani polytope ...... 158 coatom...... 12 Mani’s counterexamples . 157, Kalai’s proof of Perles’ Skeleton 160 Theorem...... 115 Mani’s problem ...... 157 k-connected...... 9 Mani’s theorem ...... 159 kernels of polytopes minimum number on d + 2 vertices ...... 116 of vertices ...... 158, 159 on d + 3 vertices ...... 116 some examples ...... 162 k-ﬂat...... 13 incidence graph ...... 54 skew...... 13 inclusion relations among face struc- k-kernel...... 104 tures...... 57 (k,ℓ)-incidence graph...... 41 inclusion relations among regular k-linked...... 77, 77 cell complexes ...... 124 k-skeleton...... 20, 139 independent paths...... 8 k-unneighborly induced subcomplex . . . . . 103, 125 sphere ...... 186 induced subgraph ...... 8 polytope...... 171PL inner diagonal...... 158 in simple polytopes...... 158 large simplex face ...... 85 simpliciality ...... 158 Las Vergnas face lattice ...... 184 upper bound theorem by Brem- lattice...... 11 ner & Klee...... 158 atom-induced subset . . . . . 139 inner vertices of a path ...... 8 atomic...... 12 interdependence of chapters . . . . 2 coatomic...... 12 interior points...... 14 complemented...... 141 intersection empty k-pyramid...... 143 empty face with face . . . . . 126 empty pyramid...... 143

212 Index

apex...... 143 Menger’s theorem ...... 9 base...... 143 minimal linkedness ...... 77, 81 induced subset...... 139 3-polytopes...... 80 k-kernel...... 139 combinatorial types . . . . 77, 93 product...... 137 few vertex case ...... 84, 85 product with a finite set. .137 lower bound...... viii, 76, 81 pyramid...... 137 simplicial pyramid apex ...... 137 polytopes ...... 76, 77, 78 pyramid base...... 137 upper bound...... 83 relatively complemented . . 141 upper bound by Gallivan . . 88 lie opposite a vertex...... 164 minimal positive k-spanning linear...... 123 configuration . . . . 169, 171 link Marcus’ conjecture ...... 173 face structure ...... 60 minimal positive k-spanning versus vertex figure...... 20 set...... 186 linkages minimal positive spanning config- and connectivity . . . 75, 77, 81 uration...... 171 in polytopes...... vii minimally k-edge-connected . . 169 locality of...... 84 minor...... 81 Lockeberg’s examples...... viii minor of a graph...... 10 Lower Bound Theorem missing edges...... 126 for general polytopes ...... 97 bound is tight...... 113 for simplicial polytopes . . . 158 bound on the number . . . . 134 lower bound theorems ...... 55 bound on the number of. .112 lower intervals...... 11 more on oriented matroids . . . 182 ℓ-simplicial...... 16 more pyramidally perfect lattices than graded relatively com- main theorem for polytopes . . . 13 plemented ones ...... 142 Mani polytope...... 158 Mani’s problem on illuminated necessarily flat ...... 109 polytopes . . . . vii, 157, 163 neighborly ...... 186 Mani’s theorem on illuminated neighbors in a graph...... 8 polytopes ...... 159 Newman’s theorem ...... 125 Marcus’ conjecture . . vii, 169, 173 no “missing edges” ...... 142 lost counterexample ...... 170 nonpolytopal spheres. . . . .121 PL Marcus’ partial results . . . 173 nonsimplicial Mani on unneighborly polytopes...... 163, 164 polytopes ...... 173 nonsimplicial Mani polytopes matroid polytope...... 182, 186 some examples ...... 166 meet-semilattice...... 11 normal...... 57 atom...... 12 graph complex ...... 60

213 Index

graph manifold ...... 59 polar polytope ...... 15 normalization procedure...... 55 polarity...... 15 notation for subpaths ...... 78 polytopal...... 48 number of combinatorial types of polytopal complex...... 19 k-skeleta ...... 106 face...... 19 number of empty pyramids . . . 105 subcomplexes...... 19 – 20 number of empty simplices in sim- polytopal face lattices ...... 137 plicial polytopes ...... 119 polytopality...... 48 of incidence graphs ...... viii octahedron...... 14 polytope...... 13 oriented matroid dimension...... 14 acyclic ...... 181, 185 edge...... 14 circuit system ...... 184 face...... 14 convex hull...... 185 facet...... 14 covector lattice ...... 184 flag...... 14 dual...... 184 full-dimensional ...... 14 duality...... 181 oriented matroid of...... 184 Las Vergnas face lattice . . 184 proper face...... 14 positive spanning set . . . . . 185 ridge...... 14 realizable...... 182 trivial face ...... 14 totally cyclic ...... 181, 185 truncatable...... 49 partially ordered set . . . see poset vertex...... 14 path...... 8 polytopes on “few vertices”. . . .84 connects...... 8 poset...... 10 joins...... 8 chain...... 11 γ length...... 11 Pd ...... 17 Perles’ Skeleton Theorem . . . . . vii cover...... 11 equivalence of different versions graded...... 11 of...... 105 – 107 has a 1...... 11ˆ for simplicial polytopes . . . 119 has a 0...... 11ˆ generalization to gj ...... 119 interval...... 11 overview on bounds...... 5 isomorphism ...... 11 version I...... 105, 114 join...... 11 version II ...... 106 k-skeleton...... 139 version III...... 107 meet...... 11 piecewise linear ...... 123 partial order...... 10 ball...... 123 poset of structures ...... 192 PL homeomorphic...... 123 positive PL sphere...... 123 k-spanning set ...... 185 PL Poincaréhomology 3-sphere . . . 55 spanning configuration . . . 171

214 Index positive basis...... see rank minimal positive spanning of a poset...... 11 configuration of an element...... 11 positive spanning set ...... 185 realizability dimension positively of kernels...... 107 k-spanning...... 171 reconstruction of k-skeleton . . 104 positively spanning ...... 17 reconstruction of skeleta . 139, 140 prism over a polytope ...... 18 refinement ...... 10 problem by Bienia & reflexivity...... 10 Las Vergnas ...... vii, 181 regular cell compex product of polytopal lattices . 137 pure...... 54 product of two polytopes ...... 17 regular cell complex ...... 54 projectively unique ...... 95 antistar...... 130 proper subset of a sphere. . . . .126 cells...... 54 pseudo-graph manifold ...... 57 Cohen-Macaulay ...... 54 pseudo-manifold property . . . . . 56 cone ...... 124 pyramid...... 18 dimension...... 54 k-fold...... 18 face poset of...... 54 pyramidal...... 18, 136 faces...... 54 pyramidally inequivalent facets...... 54 complexes ...... 116 intersection property ...... 54 of polytopes on d + 2 vertices link...... 130 116 prism...... 124 of polytopes on d + 3 vertices pyramid...... 124 116 ridges...... 54 pyramidally k-equivalent . . . . . 105 star...... 130 pyramidally perfect ...... 136 strongly connected ...... 54 lattices...... 136 list of types on at most 8 el- strongly regular ...... 54 ements...... 137 regular cell decomposition . . . . . 54 versus relatively relation of face lattices to pyrami- complemented ...... 141 dally perfect lattices. .121 relative interior points...... 14 quadrilateral...... 14 relatively complemented . 12, 141, question by Larman & Mani on 182 linkages in polytopes . . 75 Björner’s criterion ...... 12 low dimensions ...... 75 – 76 versus pyramidally negative answer ...... 76 perfect...... 141 quotient versus diamond property . . 12 of a strong sphere . . . . 125 remarks by McMullen . . 166 – 167 PL of an empty face ...... 126 rigidity...... 1

215 Index

γ ...... 17 topological representation Sd semimodular...... 122 theorem...... 182, 186 sign vector...... 184 totally cyclic ...... 185 simple geometric statement. . . .70 transitivity...... 10 simple polytopes ...... 16 truncatable...... 49 simplicial polytopes ...... 16 underlying space ...... 20 size of minimal positive spanning of a geometric simplicial com- configurations ...... 172 plex...... 123 Gale’s Lemma...... 172 unique Mani polytopes ...... 163 lower bound...... 172 unneighborly polytope ...... 171 upper bounds...... 172 Marcus’ conjecture ...... 170 standard d-crosspolytope ...... 14 unneighborly polytopes...... 97 standard d-cube...... 14 upper bound on minimal positive standard d-simplex...... 14 k-spanning sets...... 187 star upper intervals...... 11 face structure ...... 60 (s, t)-linkage...... 77 values of the function k(d).....96 strong d-sphere ...... 123 vector configuration ...... 17 PL strong chain...... 56 vertex figure ...... 15 strong components...... 58 of a strong sphere . . . . 125 PL strong walk...... 61 vertex-induced ...... 103, 125 concatenation ...... 62 vertices...... 7 construction of...... 63 weak d-graph manifold ...... 56 induced components . . 61 – 62 weak graph manifold ...... 56 subcomplex...... 20 weakly normal ...... 57 subdirect sum...... 18 graph complex ...... 60 combinatorially...... 18 – 19 graph manifold ...... 59 geometrically ...... 18 subdivision...... 10 subgraph...... 8 subpath notation...... 78 sunflower...... 115 core...... 115 petals...... 115 theory of illumination of convex bodies...... 155 theory of minors ...... 75 topological minor ...... 10, 81

216