Mapping Graph Homology to \(K\)-Theory of Roe Algebras

Given a graph \(\Gamma\) , one may consider 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 mapping 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 mappings are isomorphisms.

DOI 10.1134/S106192084010102