Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mapping class group action on the cohomology of the $\mathrm{SL}_n$ character variety

Anne Larsen

Abstract

We describe the mapping class group action on the cohomology of the twisted $\mathrm{SL}_n$-character variety of a surface $Σ_g$ of genus $g$. Our main tool is a relative version of the endoscopic decomposition of Maulik-Shen; this allows us to reduce the problem to the mapping class group action on the cohomology of a canonical finite cover of $Σ_g$, which was studied by Looijenga.

Mapping class group action on the cohomology of the $\mathrm{SL}_n$ character variety

Abstract

We describe the mapping class group action on the cohomology of the twisted -character variety of a surface of genus . Our main tool is a relative version of the endoscopic decomposition of Maulik-Shen; this allows us to reduce the problem to the mapping class group action on the cohomology of a canonical finite cover of , which was studied by Looijenga.
Paper Structure (8 sections, 26 theorems, 95 equations)

This paper contains 8 sections, 26 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Relativizing the Maulik-Shen construction
  3. Set-up and notation
  4. Preparatory steps
  5. The Ngô-Yun correspondence
  6. The vanishing cycles trick
  7. Algebraic monodromy
  8. Back to the character variety

Key Result

Theorem 1.1

The kernel of the $\mathop{\mathrm{Mod}}\nolimits(\Sigma_g \setminus p)$ action on $H^*(M^{\operatorname{SL}}, \mathbb Z)$ is commensurable to the kernel of the $\mathop{\mathrm{Mod}}\nolimits(\Sigma_g \setminus p)$ action on $H^1(\widetilde{\Sigma}_g^n, \mathbb Z)$.

Theorems & Definitions (54)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 44 more