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2019 Vanishing theorems for representation homology and the derived cotangent complex
Yuri Berest, Ajay C Ramadoss, Wai-kit Yeung
Algebr. Geom. Topol. 19(1): 281-339 (2019). DOI: 10.2140/agt.2019.19.281

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 ( 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.

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Yuri Berest. Ajay C Ramadoss. Wai-kit Yeung. "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

Information

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

zbMATH: 07053575
MathSciNet: MR3910582
Digital Object Identifier: 10.2140/agt.2019.19.281

Subjects:
Primary: 14A20, 14D20, 14L24, 18G55, 57M07
Secondary: 14F17, 14F35

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.19 • No. 1 • 2019
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