Vanishing theorems for representation homology and the derived cotangent complex

Berest, Yuri; Ramadoss, Ajay; Yeung, Wai-kit

doi:10.2140/agt.2019.19.281

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Vanishing theorems for representation homology and the derived cotangent complex

Yuri Berest, Ajay C Ramadoss and Wai-kit Yeung

Algebraic & Geometric Topology 19 (2019) 281–339

DOI: 10.2140/agt.2019.19.281

Abstract

Let $G$ be a reductive affine algebraic group defined over a field $k$ of characteristic zero. We study the cotangent complex of the derived $G$ –representation scheme ${DRep}_{G} (X)$ of a pointed connected topological space $X$ . We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of ${DRep}_{G} (X)$ to the representation homology ${HR}_{*} (X, G) : = π_{*} O [{DRep}_{G} (X)]$ to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in $ℝ^{3}$ and generalized lens spaces. In particular, for any finitely generated virtually free group $Γ$ , we show that ${HR}_{i} (B Γ, G) = 0$ for all $i > 0$ . For a closed Riemann surface $Σ_{g}$ of genus $g \geq 1$ , we have ${HR}_{i} (Σ_{g}, G) = 0$ for all $i > dim G$ . The sharp vanishing bounds for $Σ_{g}$ actually depend on the genus: we conjecture that if $g = 1$ , then ${HR}_{i} (Σ_{g}, G) = 0$ for $i > r a n k G$ , and if $g \geq 2$ , then ${HR}_{i} (Σ_{g}, G) = 0$ for $i > dim Z (G)$ , where $Z (G)$ is the center of $G$ . We prove these bounds locally on the smooth locus of the representation scheme ${Rep}_{G} [π_{1} (Σ_{g})]$ 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 ${DRep}_{G} (X)$ in the sense of Ciocan-Fontanine and Kapranov (Geom. Topol. 13 (2009) 1779–1804). We give a new “Tor formula” for this class in terms of functor homology.

Keywords

representation variety, representation homology, cotangent complex, derived moduli spaces

Mathematical Subject Classification 2010

Primary: 14A20, 14D20, 14L24, 18G55, 57M07

Secondary: 14F17, 14F35

References

Publication

Received: 15 January 2018

Revised: 20 August 2018

Accepted: 2 September 2018

Published: 6 February 2019

Authors

Yuri Berest
Department of Mathematics Cornell University Ithaca, NY United States

Ajay C Ramadoss
Department of Mathematics Indiana University Bloomington, IN United States

Wai-kit Yeung
Department of Mathematics Indiana University Bloomington, IN United States

Volume 19, issue 1 (2019)