Vanishing theorems for representation homology and the derived cotangent complex

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 $X$. We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of ${DRep}_G\left(X\right)$ to the representation homology ${HR}_{\ast }\left(X,G\right):={\pi }_{\ast }\mathcal{O}\left[{DRep}_G\left(X\right)\right]$ 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 $\mathrm{\Gamma }$, we show that ${HR}_i\left(B\mathrm{\Gamma },G\right)=0$ for all $i>0$. For a closed Riemann surface ${\mathrm{\Sigma }}_g$ of genus $g\ge 1$, we have ${HR}_i\left({\mathrm{\Sigma }}_g,G\right)=0$ for all $i>dimG$. The sharp vanishing bounds for ${\mathrm{\Sigma }}_g$ actually depend on the genus: we conjecture that if $g=1$, then ${HR}_i\left({\mathrm{\Sigma }}_g,G\right)=0$ for $i>\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}G$, and if $g\ge 2$, then ${HR}_i\left({\mathrm{\Sigma }}_g,G\right)=0$ for $i>dim\mathcal{Z}\left(G\right)$, where $\mathcal{Z}\left(G\right)$ is the center of $G$. We prove these bounds locally on the smooth locus of the representation scheme ${Rep}_G\left[{\pi }_1\left({\mathrm{\Sigma }}_g\right)\right]$ 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.