Jakob Jonsson: On the topology of independence complexes of triangle-free graphs

For a graph G with vertex set V , the independence complex of G is the simplicial complex I_G on the vertex set V with the property that a set σ ⊆ V is a face of I_G if and only if there are no edges in G between the vertices in σ. It is well-known that any simplicial complex is homotopy equivalent, even homeomorphic, to I_G for some graph G. The goal of the talk is to show that a simplicial complex Δ is homotopy equivalent to I_G for some bipartite graph G if and only if ∆ is homotopy equivalent to the suspension of some simplicial complex. In particular, for any finitely generated abelian group A and any
degree d ≥ 2, we may find a bipartite graph G such that the homology of I_G in degree d is isomorphic to A. This answers a question by Engstrom regarding the existence of torsion in the homology of independence complexes of triangle-free graphs. We also examine independence complexes of graphs with a given girth and present some partial results about possible homotopy types of such complexes.