Abstract
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
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The author acknowledges support by the RNF grant 23-21-00068.
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Manuilov, V. Mapping Graph Homology to \(K\)-Theory of Roe Algebras. Russ. J. Math. Phys. 31, 132–136 (2024). https://doi.org/10.1134/S106192084010102
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DOI: https://doi.org/10.1134/S106192084010102