Let be a reductive affine algebraic group defined over a field of characteristic zero. We study the cotangent complex of the derived –representation scheme of a pointed connected topological space . We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of to the representation homology to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in and generalized lens spaces. In particular, for any finitely generated virtually free group , we show that for all . For a closed Riemann surface of genus , we have for all . The sharp vanishing bounds for actually depend on the genus: we conjecture that if , then for , and if , then for , where is the center of . We prove these bounds locally on the smooth locus of the representation scheme in the case of complex connected reductive groups. One important consequence of our results is the existence of a well-defined –theoretic virtual fundamental class for 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.
"Vanishing theorems for representation homology and the derived cotangent complex." Algebr. Geom. Topol. 19 (1) 281 - 339, 2019. https://doi.org/10.2140/agt.2019.19.281