Ph.D Thesis

Ph.D StudentWallach Nathan Shimon
SubjectTopological and Combinatorial Properties of Random
Simplicial Complexes
DepartmentDepartment of Mathematics
Supervisor PROF. Roy Meshulam


This research considers d-dimensional simplicial complexes which have a complete (d-1)-skeleton. We define the model K(n,d,p) of random complexes of this type on n vertices. In this model, we study the (d-1)-dimensional homological connectivity of a random complex. We determine that the probability p = (d log n) / n  is the threshold probability at which the homology group Hd-1(K;Z) almost surely becomes a finite group whose order is not divisible by any prime number smaller that n d / (d+1) - c, for any c > 0. This is also the threshold for the last isolated (d-1) -simplex to be covered. In the complex process model, where the d-simplices are added in a random order, these two events occur almost surely at the same step. These results give a high-dimensional analogue of the Erdős-Rényi results on the connectivity of random graphs.

Our approach builds upon recent results of Linial and Meshulam, who showed such a result in two dimensions for the homology group H1(K,Z2). We extend this to all dimensions and to homology groups with other coefficient groups.

The model we consider is quite general. In the two dimensional case we show how to construct a subcomplete complex whose fundamental group is any desired finitely presented group. A consequence of this result is that in the two-dimensional case, for a sufficiently large value of n, the fundamental group (the homology group H1), can be any possible finitely presented group (finitely generated abelian group) at the time when the last edge (1-simplex) is covered in the complex process model.

We also determine the minimal degree of a (d-1) -simplex which suffices to guarantee that the complex has a trivial (d-1)-dimensional homology group. This generalizes a classic result of this nature for graphs.

In the last chapter of this work, we give an elementary approach to the Steinberg representation of the general linear group GLn(Fq) over a finite field. The standard approaches to this subject are quite complicated. Our approach uses elementary tools from linear algebra, poset topology, topological combinatorics, and algebraic topology. Our results show that simplicial homology and these `simple' tools suffice to prove a significant portion of the results on the character of the Steinberg representation.