Spaces of homomorphisms, formality and Hochschild homology

TL;DR

This paper explores the structure of representation spaces of finitely generated groups, establishing formality and homology stability results, and explicitly computing equivariant homology and Poincaré polynomials for certain cases.

Contribution

It introduces new formality results for homomorphism spaces and connects these to Hochschild homology, advancing understanding of their algebraic and topological structures.

Findings

Homomorphism spaces exhibit formality under certain conditions.

Homology stability is established for specific homomorphism varieties.

Explicit computations of equivariant homology and Poincaré polynomials are provided.

Abstract

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{\'e} polynomial.