Vanishing theorems for representation homology and the derived cotangent complex

Berest, Yuri¹ ; Ramadoss, Ajay² ; Yeung, Wai-kit²

¹ Department of Mathematics, Cornell University, Ithaca, NY, United States
² Department of Mathematics, Indiana University, Bloomington, IN, United States

Algebraic and Geometric Topology, Tome 19 (2019) no. 1, pp. 281-339

Résumé

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 DRepG(X) of a pointed connected topological space X. We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of DRepG(X) to the representation homology HR∗(X,G) := π∗O[DRepG(X)] to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in ℝ3 and generalized lens spaces. In particular, for any finitely generated virtually free group Γ, we show that HRi(BΓ,G) = 0 for all i > 0. For a closed Riemann surface Σg of genus g ≥ 1, we have HRi(Σg,G) = 0 for all i > dimG. The sharp vanishing bounds for Σg actually depend on the genus: we conjecture that if g = 1, then HRi(Σg,G) = 0 for i > rankG, and if g ≥ 2, then HRi(Σg,G) = 0 for i > dimZ(G), where Z(G) is the center of G. We prove these bounds locally on the smooth locus of the representation scheme RepG[π1(Σg)] 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 DRepG(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.

Détail

DOI : 10.2140/agt.2019.19.281

Classification : 14A20, 14D20, 14L24, 18G55, 57M07, 14F17, 14F35
Keywords: representation variety, representation homology, cotangent complex, derived moduli spaces