Some Characteristics of the Second Betti Number of Random Two Dimensional Simplicial Complexes by Kang Tan B.Sc, Fudan University, 1989 A THESIS SUBMITTED TN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Mathematics) We accept this thesis as coiiforrriing to the required standard The University of British Columbia April 1996 © Kang Tan, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Da,e M IS,!** DE-6 (2/88) ABSTRACT In this thesis, through generating random two-dimensional simplicial complexes, we studied the event (b2=0) for some specific probabilities. We found when the probability of event (b2=0) takes on certain specific values, the pair (no,n2) lies on certain lines. However, this research is limited by our sample space ( i.e. for P(b2=0) « 10%, 10 < no < 80; for P(b2=0) « 50%, 12 < no < 100; for P(b2=0) « 90%, 12 < no < 145). The " linear behavior" may not hold asymptotically. In the same time, we endeavor to find the number of tetrahedra and 6-triangles in the simplicial complexes. When the event (b2=0) occurs in our specific probabilities, it seems the second Betti number should come from tetrahedra and 6-triangles with high probability. However, the expectation of the number of tetrahedra and 6-triangles goes to zero, when no goes to infinity and there exists linear relationships between the pair (no,n2). This evidence may also support that the " linear behavior" may not hold asymptotically. If n2 and n0 vary linearly with no going to infinity, then the probability that no(K) — no is extremely small in model MB for reasons are similar to the Coupon Collector's problem. Hence, the probability that we cannot find element in S(no,n2) is large, which indicates that the model may have problems. TABLE OF CONTENTS Abstract ii Table of Contents iii List of Tables iv List of Figures v Acknowledgment vi INTRODUCTION 1 Section One Algebraic Topology Concepts 2 Simplices and Simplicial Complexes 2 Boundary and Homology 3 The Betti Number and the Euler Number 4 Section Two The Probabilistic Model 5 Section Three The Random Simplicial Complex Generator and Experimental Outcomes 10 The Random Simplicial Complex Generator 10 The Data and Data Analysis 14 Section Four Tetrahedra and Six-triangles in a Simplicial Complex 18 The Expected Number of Tetrahedra and Six-triangles in S(no, n2) 19 The Process to Detect Tetrahedra and Six-triangles in a Simplicial Complex 21 Bibliography 36 iii LIST OF TABLES Table 1 The Upper Bounds of PMBO^I) for Some Specific Values of no and n2 8 Table 2 Computation Results of Floating Point and Exact Precision for Some Data 14 Table 3 Data Distribution for P(b2=0) « 10 14 Table 4 Data Distribution for P(b2=0) » 50% 15 Table 5 Data Distribution for P(b2=0) « 90% 15 Table 6 Regression Functions for P(b2=0) * 10% 16 Table 7 Regression Functions for P(b2=0) * 50% 17 Table 8 Regression Functions for P(b2=0) « 90% 17 Table 9 Portion of the Second Betti Number Coming from Tetrahedra and Six-triangles 23 iv LIST OF FIGURES Figure 1 Scatter Plot of n2 against no when P(b2=0) near 10% 24 Figure 2 Scatter Plot of n2 against no when P(b2=0) near 50% 25 Figure 3 Scatter Plot of n2 against no when P(b2=0) near 90% 26 Figure 4 Regression Line of n2 against no when P(b2=0) = 10% 27 Figure 5 Regression Line of n2 against no when P(b2=0) near 10% 28 Figure 6 Regression Line of n2 against no when P(b2=0) = 50% 29 Figure 7 Regression Line of n2 against no when P(b2=0) near 50% 30 Figure 8 Regression Line of n2 against no when P(b2=0) = 90% 31 Figure 9 Regression Line of n2 against no when P(b2=0) near 90% 32 Figure 10 Distribution of b2 when no=20 n2= 143 33 Figure 11 Distribution of b2 when no=30 n2=200 33 Figure 12 Distribution of b2 when no=50 n2=400 33 Figure 13 Distribution of b2 when no=45 n2=256 34 Figure 14 Distribution of b2 when no=50 n2=260 34 Figure 15 Distribution of b2 when no=80 n2=500 34 Figure 16 Distribution of b2 when no=30 n2=80 35 Figure 17 Distribution of b2 when no=75 n2=250 35 Figure 18 Distribution of D2 when no=l 00 n2=350 35 v ACKNOWLEDGMENT I wish to express gratefulness to my supervisor, Prof. Joel Friedman. During the 1 year study and research, he always gave me valuable suggestions, patiently explained anything and kindly guided me to finish this thesis. I am also thankful to Prof. Jack Snoeyink for his helpful comments on revising the thesis which led to a considerable improvement of my presentation. vi Introduction Distinguishing the topological isomorphism classes of topological spaces is one of the main interests in the field of algebraic topology. To distinguish these isomorphism classes, one of the methods used is to compute the homology groups of topological spaces. The homology groups of a topological space can be computed from those of an "approximating" geometric simplicial complex; the homology groups of a geometric simplicial complex are equal to those of its abstraction, an abstract simplicial complex, which is a finite combinatorial object (generalizing graphs). Throughout this thesis, when we refer to simplicial complexes we mean abstract simplicial complexes. A part of a homology group is its rank. The i-th Betti number, bi, is the rank of the i-th homology group. The integers, bo, bi, b„, are called Betti numbers, and are fundamental invariants of the n-dimensional topological spaces. The Betti numbers of a simplicial complex have intuitive meanings. For example, the 0-th Betti number, bo, is the number of connected components in the simplicial complex. The general goal of this line of research is to understand the Betti numbers of the simplicial complexes in any dimension. In a one-dimensional simplicial complex (i.e. graph), one can easily find bi for any graph, G, if bo is known, by the Euler characteristic formula bi - b0= x(G) = E(G) - V(G), where %(G), E(G), and V(G), are the Euler characteristic, number of edges, and number of vertices of graph G, respectively. A two-dimensional simplicial complex is a collection of subsets of a fixed set, V, which is closed under taking subsets, and where the largest faces are triangles1. In a two-dimensional simplicial complex, K, even if bo is given2, one cannot 'A triangle is a set of size three, namely {vi, v2, v3}. 2 If a simplicial complex has more than one components, we can study each component separately. 1 / 1 find b2 or bi from the Euler characteristic formula, b2 -bi +bo= %(K)=n2(K) - n\(K) + n0(K), where %(K), n2(K), ni(K), and no(K), are the Euler characteristic, number of triangles, number of edges, and number of vertices of K, respectively. For this reason, simplicial complexes in two dimensions are the lowest dimensional simplicial complexes of interest. When studying these two dimensional simplicial complexes, we will use a probabilistic model to study typical simplicial complexes. The remaining sections of this thesis are arranged as follows. In section 1, we review the necessary concepts of algebraic topology. We describe our probabilistic model in section 2. We explicate the process of generating a random simplicial complex, display computational data, and interpret the results in section 3. In section 4, we study some combinatorial quantities which seem to be related to the second Betti number, when certain conditions hold in our model. 1. Algebraic Topology Concepts Let us briefly review some necessary concepts from algebraic topology. All of our terminology follows that in [2]. 1.1 Simplices and Simplicial Complexes For 0 < k < n, a geometric k-simplex c in R" is the convex hull of a set T of k+1 geometrically independent points. The dimension of C is dim(c) = |T| - 1 = k. For every U c T , the simplex a defined by U is a face of C, and if TJ ^ T then a is called a proper face of c. In Rn, we use the terms vertex for 0-simplex, edge for 1-simplex, and triangle for 2-simplex. A collection of simplices, H, is a geometric simplicial complex if it satisfies 2 following two properties: (i) if a is a face of c and c sH, then asH. (ii) if a e if, o e if, then a n o either a face of both or an empty set. The largest dimension of any simplex in H is the dimension ofH. An abstract simplicial complex is a collection of subsets of a fixed set, K, which is closed under taking subsets. A simplex a ofK is an element of K. The dimension of a is one less than the number of vertices in it.