Vanishing theorems for representation homology and the derived cotangent complex

TL;DR

This paper establishes vanishing theorems for representation homology of certain spaces and groups, linking algebraic topology, derived algebraic geometry, and representation theory, with implications for virtual fundamental classes.

Contribution

It introduces new vanishing theorems for representation homology of spaces and groups, and connects these results to the existence of a K-theoretic virtual fundamental class.

Findings

Vanishing of higher representation homology for virtually free groups.

Vanishing bounds for Riemann surfaces depending on genus.

Existence of a K-theoretic virtual fundamental class for derived representation schemes.

Abstract

Let $G$ be a reductive affine algebraic group defined over a field $k$ of characteristic zero. In this paper, we study the cotangent complex of the derived $G$-representation scheme $ {\rm DRep}_G(X)$ of a pointed connected topological space $X$. We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of $ {\rm DRep}_G(X) $ to the representation homology $ {\rm HR}_*(X,G) := \pi_*{\mathcal O}[{\rm DRep}_G(X)] $ to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in $ {\mathbb R}^3 $ and generalized lens spaces. In particular, for any f.g. virtually free group $ \Gamma $, we show that $\, {\rm HR}_i({\rm B}\Gamma, G) = 0 \,$ for all $ i > 0 $. For a closed Riemann surface $\Sigma_g $ of genus $ g \ge 1 $, we have $\, {\rm HR}_i(\Sigma_g, G) = 0 \,$ for all $ i > \dim G $. The sharp vanishing bounds for $ \Sigma_g $ depend actually on the genus: we conjecture that if $ g = 1 $, then $\, {\rm HR}_i(\Sigma_g, G) = 0 \,$ for $ i > {\rm rank}\,G $, and if $ g \ge 2 $, then $\, {\rm HR}_i(\Sigma_g, G) = 0 \,$ for $ i > \dim\,{\mathcal Z}(G) \,$, where $ {\mathcal Z}(G) $ is the center of $G$. We prove these bounds locally on the smooth locus of the representation scheme $ {\rm Rep}_G[\pi_1(\Sigma_g)]\,$ in the case of complex connected reductive groups. One important consequence of our results is the existence of a well-defined $K$-theoretic virtual fundamental class for $ {\rm DRep}_G(X)$ in the sense of Ciocan-Fontanine and Kapranov. We give a new `Tor formula' for this class in terms of functor homology.