DOCSLIB.ORG

Some Algebraic Topology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Some Algebraic Topology Appendix A Some Algebraic Topology This appendix is an introduction to the basic notions of algebraic topology that are needed for Sections 9.6 and 9.7. To keep the new vocabulary limited, we will not use the language of category theory here, though those versed in it will easily be able to identify the functors, etc., behind the results. Most of the standard concepts from algebraic topology have been taken from [295], Chapter 4, Sections 1–4. Proofs that were omitted are either short or a reference to the literature is given. For a more current introduction, consider [124]. Notation A.1. We know how to assign to an ordered set P a graph GC.P/,the comparability graph, see Definition 6.6. We also know how to assign to a graph G a simplicial complex KG, see Example 9.13. Finally, we know who to assign to a simplicial complex K a topological space jKj, see Definition 9.23. In this appendix, we will learn how to assign to a simplicial complex K a chain complex C.K/,see Definition A.4. All these notions allow constructions that naturally arise in their respective settings. These constructions can then also be executed for structures induced by other structures: For example, we can consider the homology complex for a chain complex induced by a simplicial complex induced by a comparability graph of an ordered set. To reduce the amount of notation and without spelling out all the necessary definitions, which would be a large task indeed, we will sometimes use language formally “out of turn”: For example, we will talk about the homology complex of an ordered set, rather than about the lengthy description above. We shall also abbreviate symbols accordingly: For example, we will write H.P/ rather than H.C.K.GC.P////. This practice should not cause confusion, as we will always assume that the construction is performed for the appropriate induced structure. © Springer International Publishing 2016 357 B. Schröder, Ordered Sets, DOI 10.1007/978-3-319-29788-0 358 A Some Algebraic Topology A.1 Chain Complexes The key idea to using homology in fixed point theory is to connect algebra with simplicial complexes. In this section, we will translate/embed simplicial complexes and their morphisms into the theory of chain complexes, which is the first step. Throughout this section, let us consider a four chain P Df0; 1; 2; 3g with its natural order as an example. The chain P, its comparability graph and the clique complex of the comparability graph, which is also called the “P-chain complex,” are pictured in Figure A.1. Definition A.2. A chain complex C D .fCngn2Z; f@qgq2Z/ is an ordered pair of a family fCngn2Z of abelian groups and a family of functions @q W Cq ! Cq1, called boundary maps, such that @q@qC1 D 0 for all q 2 Z. C is called finitely generated iff Cq D 0 for all but finitely many q and all nonzero Cq have a finite set of generators. To obtain a chain complex from a simplicial complex, we consider the topolog- ical realization of the simplicial complex. For the four chain in Figure A.1 this is the tetrahedron given in Figure A.2. All simplicial complexes in this appendix are assumed to be finite. Definition A.3. Let S Dfv0;:::;vqg be a finite set of points. Two linear orders v <v < <v v <v < <v j0 j1 jq and k0 k1 kq are called equivalently oriented iff the permutation such that ı .k0;:::;kq/ D .j0;:::;jq/ is even. (Equivalent 4 3 K4 TCL(K4) 3 2 2 1 01 0 0123 Fig. A.1 The four chain 4, the corresponding comparability graph K4, and the corresponding clique complex visualized as TCL.K4/ Fig. A.2 The topological 1 realization of a four chain, which is a tetrahedron. The orientations of the “hidden” boundary pieces (triangles) of the tetrahedron are indicated. 3 The front triangle is positively oriented (counterclockwise) 0 2 A Some Algebraic Topology 359 orientation is an equivalence relation and it has two equivalence classes for q >0.) v <v < <v Œv ;v ;:::;v The equivalence class of j0 j1 jq will be denoted by j0 j1 jq and will be called an orientation of S. If v0;:::;vq are vertices of a simplex and vi D vj for some i 6D j, then we let Œv0;:::;vq WD 0. Although the orientations of the geometric objects we visualize are certainly subject to some choices, the connections between higher-dimensional and lower- dimensional objects via the boundary map are determined by the following. Definition A.4. Let K D .V; S/ be a simplicial complex. For q 2 Z let Cq.K/ be the free abelian group generated by the orientations of the q-simplices with different orientations of the same simplex being additive inverses of each other. @K . / . / Let the function q W Cq K ! Cq1 K be the homomorphism defined on the generators by Xq @K.Œv ;:::;v / . 1/iŒv ;:::;vb;:::;v ; q 0 q D 0 i q iD0 where, as is often customary, the hat indicates that the vertex under the hat is to @K be dropped. (We will show that q is well-defined via this definition.) With this . / . / ; @K / definition, C K WD fCq K gq2Z f q gq2Z is a chain complex, called the oriented chain complex of K. The orientations of @2.f0; 1; 2; 3g/ are indicated in Figure A.2. Note that the boundaries of the solid tetrahedron (the triangular faces) are oriented in such a way that their boundaries (sides) in turn are traversed once in each direction by anyone who travels the boundaries of the triangles in the direction indicated in the triangle. From my (most certainly limited) view of this subject, the motivation for this par- ticular definition of the boundary operator seems to lie deeply in considerations of differential geometry somewhere near Stokes’ theorem. Algebraically, the following proof shows that the right way to line up alternating signs is what makes the oriented chain complex a chain complex. Proof that the oriented chain complex really is a chain complex. To see that @K v ;v the q are well-defined, note that a transposition of adjacent elements j jC1 and a subsequent transposition of elements vk;vkC1 (in the new indexing after the first transposition) in Œv0;:::;vq do not affect the right-hand side of the definition of @K.Œv ;:::;v / q 0 q . Every even permutation is a composition of an even number of @K transpositions of adjacent elements. Hence this shows that the functions q are well- defined. / . / ; @K / To prove that C K D fCq K gq2Z f q gq2Z truly is a chain complex, we must @K @K 0 1 <1 @K @K show that q1 q D for all q .(Forq the maps q1 q map into the group Cq1.K/ Df0g.) To do this, we can limit ourselves to investigating the action of @K @K q1 q on the generators. 360 A Some Algebraic Topology ! Xq @K @K.Œv ;:::;v / @K . 1/iŒv ;:::;vb;:::;v q1 q 0 q D q1 0 i q iD0 Xq . 1/i@K .Œv ;:::;vb;:::;v / D q1 0 i q iD0 0 Xq Xi1 i @ j D .1/ .1/ Œv0;:::;vbj;:::;vbi;:::;vq iD0 jD0 1 Xq j1 A C .1/ Œv0;:::;vbi;:::;vbj;:::;vq jDiC1 Xq Xi1 iCj D .1/ Œv0;:::;vbj;:::;vbi;:::;vq iD0 jD0 Xq Xq iCj1 C .1/ Œv0;:::;vbi;:::;vbj;:::;vq iD0 jDiC1 Xq Xi1 iCj D .1/ Œv0;:::;vbj;:::;vbi;:::;vq iD1 jD0 Xq Xj1 iCj1 C .1/ Œv0;:::;vbi;:::;vbj;:::;vq jD1 iD0 D 0: As with any of the objects we have defined so far, we are interested in the natural maps between these objects. For chain complexes, the natural maps are the chain maps. Definition A.5. A chain map (see Figure A.3)f W C ! C0 from the chain complex . ; @ / 0 . 0 ; @0 / C D fCqgq2Z f qgq2Z to the chain complex C D fCngn2Z f qgq2Z is a family 0 Z f Dffqgq2Z of group homomorphisms fq W Cq ! Cq such that, for all q 2 ,we @ @0 have fq1 ı q D q ı fq. Chain maps can be induced by simplicial maps as shown below. Proposition A.6. Let K D .V; S/ and K0 D .V0; S0/ be simplicial complexes and let f W K ! K0 be a simplicial map. Then the function f ch W C.K/ ! C.K0/ ch.Œ ;:::; / Œ . /;:::; . / defined by fq s0 sq WD f s0 f sq on the generators and extended in the natural fashion is a chain map. A Some Algebraic Topology 361 Fig. A.3 A chain map . fq - Cq Cq ∂q ∂q ? fq−1 ? - Cq−1 Cq−1 . ch Proof. First note that fq is well-defined. Indeed, an even permutation of the s0;:::;sq translates into an even permutation of the f .s0/;:::;f .sq/. @K0 ch ch @K Now we must show that q ı fq D fq1 ı q . Yet this is quite trivial, because, for all generators, we have the following. ! Xq ch @K.Œv ;:::;v / ch . 1/iŒv ;:::;vb;:::;v fq1 ı q 0 q D fq1 0 i q iD0 Xq i b D .1/ Œf .v0/;:::;f .vi/;:::;f .vq/ iD0 @K0 .Œ .v /;:::; .v // D q f 0 f q @K0 ch.Œv ;:::;v /: D q fq 0 q We thus have embedded/translated the objects that we care about, simplicial complexes and their morphisms, into the theory of chain complexes and chain maps.
Load more

Recommended publications
  • Spectral Gap Bounds for the Simplicial Laplacian and an Application To
    Spectral gap bounds for the simplicial Laplacian and an application to random complexes Samir Shukla,∗ D. Yogeshwaran† Abstract In this article, we derive two spectral gap bounds for the reduced Laplacian of a general simplicial complex. Our two bounds are proven by comparing a simplicial complex in two different ways with a larger complex and with the corresponding clique complex respectively. Both of these bounds generalize the result of Aharoni et al. (2005) [1] which is valid only for clique complexes. As an application, we investigate the thresholds for vanishing of cohomology of the neighborhood complex of the Erdös- Rényi random graph. We improve the upper bound derived in Kahle (2007) [15] by a logarithmic factor using our spectral gap bounds and we also improve the lower bound via finer probabilistic estimates than those in Kahle (2007) [15]. Keywords: spectral bounds, Laplacian, cohomology, neighborhood complex, Erdös-Rényi random graphs. AMS MSC 2010: 05C80. 05C50. 05E45. 55U10. 1 Introduction Let G be a graph with vertex set V (G) (often to be abbreviated as V ) and let L(G) denote the (unnormalized) Laplacian of G. Let 0 = λ1(G) ≤ λ2(G) ≤ . ≤ λ|V |(G) denote the eigenvalues of L(G) in ascending order. Here, the second smallest eigenvalue λ2(G) is called the spectral gap. The clique complex of a graph G is the simplicial complex whose simplices are all subsets σ ⊂ V which spans a complete subgraph of G. We shall denote the kth k arXiv:1810.10934v2 [math.CO] 9 Oct 2019 reduced cohomology of a simplicial complex X by H (X).
    [Show full text]
  • Message Passing Simplicial Networks
    Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks Cristian Bodnar * 1 Fabrizio Frasca * 2 3 Yu Guang Wang * 4 5 6 Nina Otter 7 Guido Montufar´ * 4 7 Pietro Lio` 1 Michael M. Bronstein 2 3 v0 Abstract v6 v10 v9 The pairwise interaction paradigm of graph ma- v chine learning has predominantly governed the 8 v v modelling of relational systems. However, graphs 1 5 v3 alone cannot capture the multi-level interactions v v present in many complex systems and the expres- 4 7 v2 sive power of such schemes was proven to be lim- ited. To overcome these limitations, we propose Figure 1. Message Passing with upper and boundary adjacencies Message Passing Simplicial Networks (MPSNs), illustrated for vertex v2 and edge (v5; v7) in the complex. a class of models that perform message passing on simplicial complexes (SCs). To theoretically anal- 1. Introduction yse the expressivity of our model we introduce a Simplicial Weisfeiler-Lehman (SWL) colouring Graphs are among the most common abstractions for com- procedure for distinguishing non-isomorphic SCs. plex systems of relations and interactions, arising in a broad We relate the power of SWL to the problem of range of fields from social science to high energy physics. distinguishing non-isomorphic graphs and show Graph neural networks (GNNs), which trace their origins that SWL and MPSNs are strictly more power- to the 1990s (Sperduti, 1994; Goller & Kuchler, 1996; Gori ful than the WL test and not less powerful than et al., 2005; Scarselli et al., 2009; Bruna et al., 2014; Li et al., the 3-WL test.
    [Show full text]
  • Some Properties of Glued Graphs at Complete Clone
    Sci.Int.(Lahore),27(1),39-47,2014 ISSN 1013-5316; CODEN: SINTE 8 39 SOME PROPERTIES OF GLUED GRAPHS AT COMPLETE CLONE IN THE VIEW OF ALGEBRAIC COMBINATORICS Seyyede Masoome Seyyedi , Farhad Rahmati Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran. Corresponding Author: Email: [email protected] ABSTRACT: A glued graph at complete clone is obtained from combining two graphs by identifying edges of of each original graph. We investigate how to change some properties such as height, big height, Krull dimension, Betti numbers by gluing of two graphs at complete clone. We give a sufficient and necessary condition so that the glued graph of two Cohen-Macaulay chordal graphs at complete clone is a Cohen-Macaulay graph. Moreover, we present the conditions that the edge ideal of gluing of two graphs at complete clone has linear resolution whenever the edge ideals of original graphs have linear resolution. We show when gluing of two independence complexes, line graphs, complement graphs can be expressed as independence complex, line graph and complement of gluing of two graphs. Key Words: Glued graph, Height, Big height, Krull dimension, Projective dimension, Linear resolution, Betti number, Cohen-Macaulay. INTRODUCTION The concept edge ideal was first introduced by Villarreal in Our paper is organized as follows. In section 2, we give an [23], that is, let be a simple (no loops or multiple edges) explicit formula for computing the height of glued graph at graph on the vertex set and the edge set complete clone. Also, we present a lower bound for the big .
    [Show full text]
  • Topological Graph Persistence
    CAIM ISSN 2038-0909 Commun. Appl. Ind. Math. 11 (1), 2020, 72–87 DOI: 10.2478/caim-2020-0005 Research Article Topological graph persistence Mattia G. Bergomi1*, Massimo Ferri2*, Lorenzo Zuffi2 1Veos Digital, Milan, Italy 2ARCES and Dept. of Mathematics, Univ. of Bologna, Italy *Email address for correspondence: [email protected] Communicated by Alessandro Marani Received on 04 28, 2020. Accepted on 11 09, 2020. Abstract Graphs are a basic tool in modern data representation. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological constructions can be used to gain information otherwise concealed by the low-dimensional nature of graphs. We do this by extending previous work in homological persistence, and proposing novel graph-theoretical constructions. Beyond cliques, we use independent sets, neighborhoods, enclaveless sets and a Ramsey-inspired extended persistence. Keywords: Clique, independent set, neighborhood, enclaveless set, Ramsey AMS subject classification: 68R10, 05C10 1. Introduction Currently data are produced massively and rapidly. A large part of these data is either naturally organized, or can be represented, as graphs or networks. Recently, topological persistence has proved to be an invaluable tool for the exploration and understanding of these data types. Albeit graphs can be considered as topological objects per se, given their low dimensionality, only limited information can be obtained by studying their topology. It is possible to grasp more information by superposing higher-dimensional topological structures to a given graph. Many research lines, for ex- ample, focused on building n-dimensional complexes from graphs by considering their (n+1)-cliques (see, e.g., section 1.1).
    [Show full text]
  • Persistent Homology of Unweighted Complex Networks Via Discrete
    www.nature.com/scientificreports OPEN Persistent homology of unweighted complex networks via discrete Morse theory Received: 17 January 2019 Harish Kannan1, Emil Saucan2,3, Indrava Roy1 & Areejit Samal 1,4 Accepted: 6 September 2019 Topological data analysis can reveal higher-order structure beyond pairwise connections between Published: xx xx xxxx vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and undirected networks using persistent homology. Leveraging on the features of discrete Morse theory, our method not only captures the topology of the clique complex of such graphs via the concept of critical simplices, but also achieves close to the theoretical minimum number of critical simplices in several analyzed model and real networks. This leads to a reduced fltration scheme based on the subsequence of the corresponding critical weights, thereby leading to a signifcant increase in computational efciency. We have employed our fltration scheme to explore the persistent homology of several model and real-world networks. In particular, we show that our method can detect diferences in the higher-order structure of networks, and the corresponding persistence diagrams can be used to distinguish between diferent model networks. In summary, our method based on discrete Morse theory further increases the applicability of persistent homology to investigate the global topology of complex networks. In recent years, the feld of topological data analysis (TDA) has rapidly grown to provide a set of powerful tools to analyze various important features of data1. In this context, persistent homology has played a key role in bring- ing TDA to the fore of modern data analysis.
    [Show full text]
  • Arxiv:2006.02870V1 [Cs.SI] 4 Jun 2020
    The why, how, and when of representations for complex systems Leo Torres Ann S. Blevins [email protected] [email protected] Network Science Institute, Department of Bioengineering, Northeastern University University of Pennsylvania Danielle S. Bassett Tina Eliassi-Rad [email protected] [email protected] Department of Bioengineering, Network Science Institute and University of Pennsylvania Khoury College of Computer Sciences, Northeastern University June 5, 2020 arXiv:2006.02870v1 [cs.SI] 4 Jun 2020 1 Contents 1 Introduction 4 1.1 Definitions . .5 2 Dependencies by the system, for the system 6 2.1 Subset dependencies . .7 2.2 Temporal dependencies . .8 2.3 Spatial dependencies . 10 2.4 External sources of dependencies . 11 3 Formal representations of complex systems 12 3.1 Graphs . 13 3.2 Simplicial Complexes . 13 3.3 Hypergraphs . 15 3.4 Variations . 15 3.5 Encoding system dependencies . 18 4 Mathematical relationships between formalisms 21 5 Methods suitable for each representation 24 5.1 Methods for graphs . 24 5.2 Methods for simplicial complexes . 25 5.3 Methods for hypergraphs . 27 5.4 Methods and dependencies . 28 6 Examples 29 6.1 Coauthorship . 29 6.2 Email communications . 32 7 Applications 35 8 Discussion and Conclusion 36 9 Acknowledgments 38 10 Citation diversity statement 38 2 Abstract Complex systems thinking is applied to a wide variety of domains, from neuroscience to computer science and economics. The wide variety of implementations has resulted in two key challenges: the progenation of many domain-specific strategies that are seldom revisited or questioned, and the siloing of ideas within a domain due to inconsistency of complex systems language.
    [Show full text]
  • FACE VECTORS of FLAG COMPLEXES 1. Introduction We
    FACE VECTORS OF FLAG COMPLEXES ANDY FROHMADER Abstract. A conjecture of Kalai and Eckhoff that the face vector of an arbi- trary flag complex is also the face vector of some particular balanced complex is verified. 1. Introduction We begin by introducing the main result. Precise definitions and statements of some related theorems are deferred to later sections. The main object of our study is the class of flag complexes. A simplicial complex is a flag complex if all of its minimal non-faces are two element sets. Equivalently, if all of the edges of a potential face of a flag complex are in the complex, then that face must also be in the complex. Flag complexes are closely related to graphs. Given a graph G, define its clique complex C = C(G) as the simplicial complex whose vertex set is the vertex set of G, and whose faces are the cliques of G. The clique complex of any graph is itself a flag complex, as for a subset of vertices of a graph to not form a clique, two of them must not form an edge. Conversely, any flag complex is the clique complex of its 1-skeleton. The Kruskal-Katona theorem [6, 5] characterizes the face vectors of simplicial complexes as being precisely the integer vectors whose coordinates satisfy some par- ticular bounds. The graphs of the \rev-lex" complexes which attain these bounds invariably have a clique on all but one of the vertices of the complex, and sometimes even on all of the vertices. Since the bounds of the Kruskal-Katona theorem hold for all simplicial com- plexes, they must in particular hold for flag complexes.
    [Show full text]
  • Percolation and Stochastic Topology NSF/CBMS Conference
    Lecture 9: Percolation and stochastic topology NSF/CBMS Conference Sayan Mukherjee Departments of Statistical Science, Computer Science, Mathematics Duke University www.stat.duke.edu/ sayan ⇠ May 31, 2016 Stochastic topology “I predict a new subject of statistical topology. Rather than count the number of holes, Betti numbers, etc., one will be more interested in the distribution of such objects on noncompact manifolds as one goes out to infinity,” Isadore Singer. Random graphs Erd˝os-R´enyi G G(n, p) is a draw of a graph G from the graph distribution. ⇠ Consider p(n)andn and ask questions about thresholds of !1 graph properties. Erd˝os-R´enyi random graph model G(n, p) is the probability space of graphs on vertex set [n]= 1,...n with the probability of each edge p independently. { } Consider p(n)andn and ask questions about thresholds of !1 graph properties. Erd˝os-R´enyi random graph model G(n, p) is the probability space of graphs on vertex set [n]= 1,...n with the probability of each edge p independently. { } G G(n, p) is a draw of a graph G from the graph distribution. ⇠ Erd˝os-R´enyi random graph model G(n, p) is the probability space of graphs on vertex set [n]= 1,...n with the probability of each edge p independently. { } G G(n, p) is a draw of a graph G from the graph distribution. ⇠ Consider p(n)andn and ask questions about thresholds of !1 graph properties. Matthew Kahle (Ohio State University) Random topology and geometry AMS Short Course Matthew Kahle (Ohio State University) Random topology and geometry AMS Short Course Matthew
    [Show full text]
  • Matroid Representation of Clique Complexes∗
    Matroid representation of clique complexes¤ x Kenji Kashiwabaray Yoshio Okamotoz Takeaki Uno{ Abstract In this paper, we approach the quality of a greedy algorithm for the maximum weighted clique problem from the viewpoint of matroid theory. More precisely, we consider the clique complex of a graph (the collection of all cliques of the graph) which is also called a flag complex, and investigate the minimum number k such that the clique complex of a given graph can be represented as the intersection of k matroids. This number k can be regarded as a measure of “how complex a graph is with respect to the maximum weighted clique problem” since a greedy algorithm is a k-approximation algorithm for this problem. For any k > 0, we characterize graphs whose clique complexes can be represented as the intersection of k matroids. As a consequence, we can see that the class of clique complexes is the same as the class of the intersections of partition matroids. Moreover, we determine how many matroids are necessary and sufficient for the representation of all graphs with n vertices. This number turns out to be n - 1. Other related investigations are also given. Keywords: Abstract simplicial complex, Clique complex, Flag complex, Independence system, Matroid intersec- tion, Partition matroid 1 Introduction An independence system is a family of subsets of a nonempty finite set such that all subsets of a member of the family are also members of the family. A lot of combinatorial optimization problems can be seen as optimization problems on the corresponding independence systems. For example, in the minimum cost spanning tree problem, we want to find a maximal set with minimum total weight in the collection of all forests of a given graph, which is an independence system.
    [Show full text]
  • The Intersection of a Matroid and a Simplicial Complex 1
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 358, Number 11, November 2006, Pages 4895–4917 S 0002-9947(06)03833-5 Article electronically published on June 19, 2006 THE INTERSECTION OF A MATROID AND A SIMPLICIAL COMPLEX RON AHARONI AND ELI BERGER Abstract. A classical theorem of Edmonds provides a min-max formula re- lating the maximal size of a set in the intersection of two matroids to a “cov- ering” parameter. We generalize this theorem, replacing one of the matroids by a general simplicial complex. One application is a solution of the case r = 3 of a matroidal version of Ryser’s conjecture. Another is an upper bound on the minimal number of sets belonging to the intersection of two matroids, needed to cover their common ground set. This, in turn, is used to derive a weakened version of a conjecture of Rota. Bounds are also found on the dual parameter—the maximal number of disjoint sets, all spanning in each of two given matroids. We study in detail the case in which the complex is the complex of independent sets of a graph, and prove generalizations of known results on “independent systems of representatives” (which are the special case in which the matroid is a partition matroid). In particular, we define a no- tion of k-matroidal colorability of a graph, and prove a fractional version of a conjecture, that every graph G is 2∆(G)-matroidally colorable. The methods used are mostly topological. 1. Introduction The point of departure of this paper is a notion which has been recently developed and studied, that of an independent system of representatives (ISR), which is a generalization of the notion of a system of distinct representatives (SDR).
    [Show full text]
  • Structural Parameters in Combinatorial Objects
    DISS. ETH No. 15901, 2005 Structural Parameters in Combinatorial Objects A dissertation submitted to the Swiss Federal Institute of Technology Zurich¨ for the degree of Doctor of Technical Sciences presented by Yoshio Okamoto Master of Systems Science, the University of Tokyo, Japan born July 22, 1976 in Hekinan, Japan citizen of Japan submitted to Prof. Emo Welzl, ETH Zurich,¨ examiner Prof. Komei Fukuda, ETH Zurich,¨ co-examiner ii iii Acknowledgements First of all, I would like to express my sincere gratitude to my supervi- sor Emo Welzl. It was a great opportunity for me to work at his teach- ing/research group in Zurich. His persistent encouragement and sup- port were really helpful, and his way of looking at problems, way of presenting materials, and everything were very exciting to me. He was also a speaker of the Berlin/Zurich Joint Graduate Program “Combina- torics, Geometry, and Combinatorics” (CGC) financed by ETH Zurich and German Science Foundation. Without this program, I would not have imagined how I completed the study. My sincere gratitude also goes to Komei Fukuda who kindly ad- mitted to be a co-referee. Indeed, he was the first person who I made a contact when I tried to move to Zurich. At that time we did not know each other; Nevertheless he was so kind that he gave a lot of informa- tion and encouraged me to come to Zurich. Some of the chapters in the thesis are based on joint papers with the following people: Vladimir De˘ıneko, Michael Hoffmann, Kenji Kashi- wabara, Masataka Nakamura, Takeaki Uno and Gerhard Woeginger.
    [Show full text]
  • TOPOLOGY of RANDOM D-CLIQUE COMPLEXES 3
    TOPOLOGY OF RANDOM d-CLIQUE COMPLEXES DEMET TAYLAN Abstract. For a simplicial complex X, the d-clique complex ∆d(X) is the simplicial complex having all subsets of vertices whose (d + 1)-subsets are contained by X as its α −1 d+1 −1 faces. We prove that if p = n , with α< max{ k−d+1 , − k } or α> 2k+2 , then the k-th (d) ( d ) reduced homology group of the random d-clique complex ∆d(Gd(n,p)) is asymptotically −1 −1 (d+1)(k+1) −1 almost surely vanishing, and if t <α< t+1 where t = ( (d+1)(k+1) − k ) , then the ( d+1 ) ( +1) (kd + d − 1)-st reduced homology group of ∆d(Gd(n,p)) is asymptotically almost surely nonvanishing. This provides a partial answer to a question posed by Eric Babson. 1. Introduction logn One of the famous results in random graph theory establishes that p = n is a threshold for the connectivity of Erd˝os and R´enyi random graphs [5]. A 2-dimensional analogue of Erd˝os and R´enyi’s result was obtained by Linial-Meshulam in [7]. Meshulam-Wallach, in [10], presented a d-dimensional analogue for d ≥ 3. In particular, Linial-Meshulam- dlogn Wallach theorem yields that p = n is the threshold for the vanishing of the (d − 1)-st homology of the random simplicial complex Gd(n, p) with coefficients in a finite abelian group. Here, the random simplicial complex Gd(n, p) is a d-dimensional simplicial complex on [n] with a full (d − 1)-dimensional skeleton and with d-dimensional faces are chosen independently each with probability p.
    [Show full text]