Mapping Graph Homology to -Theory of Roe Algebras

Given a graph , one may consider 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 nat...

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Bibliographic Details
Published in:Russian journal of mathematical physics 2024, Vol.31 (1), p.132-136
Main Author: Manuilov, V.
Format: Article
Language:English
Subjects:
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Algebra
Apexes
Graph theory
Homology
Isomorphism
Mapping
Mathematical and Computational Physics
Metric space
Physics
Physics and Astronomy
Theoretical
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Summary:Given a graph , one may consider 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 mapping 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 mappings are isomorphisms. DOI 10.1134/S106192084010102
ISSN:1061-9208
1555-6638
DOI:10.1134/S106192084010102