Algebraic & Geometric Topology

Vanishing theorems for representation homology and the derived cotangent complex

Yuri Berest, Ajay C Ramadoss, and Wai-kit Yeung

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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 ( X ) of a pointed connected topological space X . We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of DRep G ( X ) to the representation homology HR ( X , G ) : = π O [ DRep G ( 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 HR i ( B Γ , G ) = 0 for all i > 0 . For a closed Riemann surface Σ g of genus g 1 , we have HR i ( Σ g , G ) = 0 for all i > dim G . The sharp vanishing bounds for Σ g actually depend on the genus: we conjecture that if g = 1 , then HR i ( Σ g , G ) = 0 for i > r a n k G , and if g 2 , then HR i ( Σ g , G ) = 0 for i > dim Z ( G ) , where Z ( G ) is the center of G . We prove these bounds locally on the smooth locus of the representation scheme Rep G [ π 1 ( Σ g ) ] 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 ( 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.

Article information

Algebr. Geom. Topol., Volume 19, Number 1 (2019), 281-339.

Received: 15 January 2018
Revised: 20 August 2018
Accepted: 2 September 2018
First available in Project Euclid: 12 February 2019

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Zentralblatt MATH identifier

Primary: 14A20: Generalizations (algebraic spaces, stacks) 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 14L24: Geometric invariant theory [See also 13A50] 18G55: Homotopical algebra 57M07: Topological methods in group theory
Secondary: 14F17: Vanishing theorems [See also 32L20] 14F35: Homotopy theory; fundamental groups [See also 14H30]

representation variety representation homology cotangent complex derived moduli spaces


Berest, Yuri; Ramadoss, Ajay C; Yeung, Wai-kit. Vanishing theorems for representation homology and the derived cotangent complex. Algebr. Geom. Topol. 19 (2019), no. 1, 281--339. doi:10.2140/agt.2019.19.281.

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