Hom (G, H) is a polyhedral complex defined for any two undirected graphs G and H. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes.We prove that Hom (Km, Kn) is homotopy equivalent to a wedge of (n − m)-dimensional spheres, and provide an enumeration formula for the number of the spheres. As a corollary we prove that if for some graph G, and integers m ≥ 2 and k ≥ −1, we have ̟ k 1 (Hom (Km, G)) = 0, then χ(G) ≥ k + m; here Z 2 -action is induced by the swapping of two vertices in Km, and ̟ 1 is the first Stiefel-Whitney class corresponding to this action.

We study Linial-Meshulam random 2 2 -complexes Y ( n , p ) Y(n,p) , which are 2 2 -dimensional analogues of Erdős-Rényi random graphs. We find the threshold for simple connectivity to be p = n − 1 / 2 p = n^{-1/2} . This is in contrast to the threshold for vanishing of the first homology group, which was shown earlier by Linial and Meshulam to be p = 2 log ⁡ n / n p = 2 \log n / n . We use a variant of Gromov’s local-to-global theorem for linear isoperimetric inequalities to show that when p = O ( n − 1 / 2 − ϵ p = O( n^{-1/2 -\epsilon } ), the fundamental group is word hyperbolic. Along the way we classify the homotopy types of sparse 2 2 -dimensional simplicial complexes and establish isoperimetric inequalities for such complexes. These intermediate results do not involve randomness and may be of independent interest.

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