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

V. Manuilov

Abstract

Given a graph $Γ$, 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 $Γ$ and their $K$-theory counterparts -- the $K$-theory of the (uniform) Roe algebra of the metric space $X$ of vertices of $Γ$. We construct here a natural map from homology of $Γ$ to the $K$-theory of the Roe algebra of $X$, and its uniform version. We show that, when $Γ$ is the Cayley graph of $\mathbb Z$, the constructed maps are isomorphisms.

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

Abstract

Given a graph , one may conside the set of its vertices as a metric space by assuming that all edges have length one. We consider two versions of homology theory of and their -theory counterparts -- the -theory of the (uniform) Roe algebra of the metric space of vertices of . We construct here a natural map from homology of to the -theory of the Roe algebra of , and its uniform version. We show that, when is the Cayley graph of , the constructed maps are isomorphisms.
Paper Structure (5 sections, 9 theorems, 6 equations)

This paper contains 5 sections, 9 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. map from $H_0$ to $K_0$
  4. Map from $H_1$ to $K_1$
  5. Examples

Key Result

Lemma 1

If $c$ is uniformly finite then $f_c,g_c\in M_n(C^*_u(X))$ for some $n$.

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Corollary 6
  • ...and 8 more