## Algebraic & Geometric Topology

### Vanishing theorems for representation homology and the derived cotangent complex

#### 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 $ℝ 3$ 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

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

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

https://projecteuclid.org/euclid.agt/1549940434

Digital Object Identifier
doi:10.2140/agt.2019.19.281

Mathematical Reviews number (MathSciNet)
MR3910582

Zentralblatt MATH identifier
07053575

#### Citation

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. https://projecteuclid.org/euclid.agt/1549940434

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