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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 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) := πO[DRepG(X)] to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in 3 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.

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