Spaces of homomorphisms, formality and Hochschild homology

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.