Mapping graph homology to $K$-theory of Roe algebras

Abstract

Given a graph $\Gamma$, one may conside the set $X$ of its vertices as a metric space by assuming that all edges have length one. We consider two versions of homology theory of $\Gamma$ and their $K$-theory counterparts -- the $K$-theory of the (uniform) Roe algebra of the metric space $X$ of vertices of $\Gamma$. We construct here a natural map from homology of $\Gamma$ to the $K$-theory of the Roe algebra of $X$, and its uniform version. We show that, when $\Gamma$ is the Cayley graph of $\mathbb{Z}$, the constructed maps are isomorphisms.