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 pyra- midally 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 Gr¨unbaum 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 G¨unter 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. G¨unter 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 care- fully 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 dur- ing 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 stu- dent all the more enjoyable. In a very particular order (beauty? height? in- telligence? age?), they are Axel Werner, Thilo R¨orig, 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 Sch¨oneberg, and playing tennis with Till Plumbaum, Axel Werner, Andreas Profous, and Sebastian Stiller), learning Spanish (at the Sprach- und Kulturb¨orse der TU Berlin in the highly enjoyable courses by Florencia, Carmen, Liliana, and Miguel), the brilliant pieces by Les Luthiers, un grupo de m´usica 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 ix 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 .