Spaces of homomorphisms, formality and Hochschild homology
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916859188609024 |
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| author | Gritschacher, Simon |
| author_facet | Gritschacher, Simon |
| contents | Let $G$ be a discrete group. The topological category of finite dimensional unitary representations of $G$ is symmetric monoidal under direct sum and has an associated $\mathbb{E}_\infty$-space $\mathcal{K}^{\mathrm{def}}(G)$. We show that if $G$ and $A$ are finitely generated groups and $A$ is abelian, then $\mathcal{K}^{\mathrm{def}}(G\times A)\simeq \mathcal{K}^{\mathrm{def}}(G)\otimes \widehat{A}$ as $\mathbb{E}_\infty$-spaces, where $\widehat{A}$ is the Pontryagin dual of $A$. We deduce a homology stability result for the homomorphism varieties $\mathrm{Hom}(G\times \mathbb{Z}^r,U(n))$ using the local-to-global principle for homology stability of Kupers--Miller. For a finitely generated free group $F$ and a field $k$ of characteristic zero, we show that the singular $k$-chains in $\mathcal{K}^{\mathrm{def}}(F)$ are formal as an $\mathbb{E}_\infty$-$k$-algebra. Using this we describe the equivariant homology of $\mathrm{Hom}(F \times A,U(n))$ for every $n$ in terms of higher Hochschild homology of an explicitly determined commutative $k$-algebra. As an example we show that $\mathrm{Hom}(F\times \mathbb{Z}^r,U(2))$ is $U(2)$-equivariantly formal for every $r$ and we compute the Poincar{é} polynomial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17683 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spaces of homomorphisms, formality and Hochschild homology Gritschacher, Simon Algebraic Topology Let $G$ be a discrete group. The topological category of finite dimensional unitary representations of $G$ is symmetric monoidal under direct sum and has an associated $\mathbb{E}_\infty$-space $\mathcal{K}^{\mathrm{def}}(G)$. We show that if $G$ and $A$ are finitely generated groups and $A$ is abelian, then $\mathcal{K}^{\mathrm{def}}(G\times A)\simeq \mathcal{K}^{\mathrm{def}}(G)\otimes \widehat{A}$ as $\mathbb{E}_\infty$-spaces, where $\widehat{A}$ is the Pontryagin dual of $A$. We deduce a homology stability result for the homomorphism varieties $\mathrm{Hom}(G\times \mathbb{Z}^r,U(n))$ using the local-to-global principle for homology stability of Kupers--Miller. For a finitely generated free group $F$ and a field $k$ of characteristic zero, we show that the singular $k$-chains in $\mathcal{K}^{\mathrm{def}}(F)$ are formal as an $\mathbb{E}_\infty$-$k$-algebra. Using this we describe the equivariant homology of $\mathrm{Hom}(F \times A,U(n))$ for every $n$ in terms of higher Hochschild homology of an explicitly determined commutative $k$-algebra. As an example we show that $\mathrm{Hom}(F\times \mathbb{Z}^r,U(2))$ is $U(2)$-equivariantly formal for every $r$ and we compute the Poincar{é} polynomial. |
| title | Spaces of homomorphisms, formality and Hochschild homology |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2507.17683 |