Spaces of homomorphisms, formality and Hochschild homology

2025/07/23 by Simon Gritschacher, Gritschacher, Simon
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.2507.17683

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 𝔼_∞-space K^def(G). We show that if G and A are finitely generated groups and A is abelian, then K^def(G× A)≃ K^def(G)⊗ \widehatA as 𝔼_∞-spaces, where \widehatA is the Pontryagin dual of A. We deduce a homology stability result for the homomorphism varieties Hom(G× 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 K^def(F) are formal as an 𝔼_∞-k-algebra. Using this we describe the equivariant homology of Hom(F × 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 Hom(F× ℤ^r,U(2)) is U(2)-equivariantly formal for every r and we compute the Poincaré polynomial.

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