Mapping graph homology to $K$-theory of Roe algebras
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916108406095872 |
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| author | Manuilov, V. |
| author_facet | Manuilov, V. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15353 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mapping graph homology to $K$-theory of Roe algebras Manuilov, V. K-Theory and Homology Operator Algebras 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. |
| title | Mapping graph homology to $K$-theory of Roe algebras |
| topic | K-Theory and Homology Operator Algebras |
| url | https://arxiv.org/abs/2401.15353 |