Abstract:
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 $\sigma \subseteq V$ is a face of $I_G$ if and only if there are no edges in $G$ between the vertices in $\sigma$. 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 $\Delta$ is homotopy equivalent to $I_G$ for some bipartite graph $G$ if and only if $\Delta$ is homotopy equivalent to the suspension of some simplicial complex. In particular, for any finitely generated abelian group $A$ and any degree $d \ge 2$, 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 Engström 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.