Volume 19, issue 1 (2019)

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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
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 ${DRep}_{G}\left(X\right)$ of a pointed connected topological space $X\phantom{\rule{0.3em}{0ex}}$. We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of ${DRep}_{G}\left(X\right)$ to the representation homology ${HR}_{\ast }\left(X,G\right):={\pi }_{\ast }\mathsc{O}\left[{DRep}_{G}\left(X\right)\right]$ 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 $\Gamma$, we show that ${HR}_{i}\left(B\Gamma ,G\right)=0$ for all $i>0$. For a closed Riemann surface ${\Sigma }_{g}$ of genus $g\ge 1$, we have ${HR}_{i}\left({\Sigma }_{g},G\right)=0$ for all $i>dimG\phantom{\rule{0.3em}{0ex}}$. The sharp vanishing bounds for ${\Sigma }_{g}$ actually depend on the genus: we conjecture that if $g=1$, then ${HR}_{i}\left({\Sigma }_{g},G\right)=0$ for $i>rank\phantom{\rule{0.3em}{0ex}}G\phantom{\rule{0.3em}{0ex}}$, and if $g\ge 2$, then ${HR}_{i}\left({\Sigma }_{g},G\right)=0$ for $i>dim\mathsc{Z}\left(G\right)$, where $\mathsc{Z}\left(G\right)$ is the center of $G\phantom{\rule{0.3em}{0ex}}$. We prove these bounds locally on the smooth locus of the representation scheme ${Rep}_{G}\left[{\pi }_{1}\left({\Sigma }_{g}\right)\right]$ 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 ${DRep}_{G}\left(X\right)$ 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