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arXiv · 2401.15353

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

V. Manuilov
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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.

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V. Manuilov. 2024-01-27. Mapping graph homology to $K$-theory of Roe algebras. https://arxiv.org/abs/2401.15353

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