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On the stable cohomology of the IA-automorphism groups of free groups

Kazuo Habiro, Mai Katada

Abstract

Borel's stability and vanishing theorem gives the stable cohomology of $\mathrm{GL}(n,\mathbb{Z})$ with coefficients in algebraic $\mathrm{GL}(n,\mathbb{Z})$-representations. By combining the Borel theorem with the Hochschild-Serre spectral sequence, we compute the twisted first cohomology of the automorphism group $\mathrm{Aut}(F_n)$ of the free group $F_n$ of rank $n$. We also study the stable rational cohomology of the IA-automorphism group $\mathrm{IA}_n$ of $F_n$. We propose a conjectural algebraic structure of the stable rational cohomology of $\mathrm{IA}_n$, and consider some relations to known results and conjectures. We also consider a conjectural structure of the stable rational cohomology of the Torelli groups of surfaces.

On the stable cohomology of the IA-automorphism groups of free groups

Abstract

Borel's stability and vanishing theorem gives the stable cohomology of with coefficients in algebraic -representations. By combining the Borel theorem with the Hochschild-Serre spectral sequence, we compute the twisted first cohomology of the automorphism group of the free group of rank . We also study the stable rational cohomology of the IA-automorphism group of . We propose a conjectural algebraic structure of the stable rational cohomology of , and consider some relations to known results and conjectures. We also consider a conjectural structure of the stable rational cohomology of the Torelli groups of surfaces.
Paper Structure (37 sections, 34 theorems, 164 equations)

This paper contains 37 sections, 34 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. First cohomology of $\operatorname{Aut}(F_n)$ with coefficients in algebraic $\operatorname{GL}(n,\mathbb{Z})$-representations
  3. Conjectural structure of the stable rational cohomology of $\operatorname{IA}_n$
  4. Relation to Church and Farb's conjecture
  5. Conjectural structure of the stable rational cohomology of $\operatorname{IO}_n$
  6. Stable rational cohomology of the Torelli groups of surfaces
  7. Organization of the paper
  8. Algebraic $\operatorname{GL}(n,\mathbb{Z})$-representations
  9. Borel's stability and vanishing theorem for cohomology of $\operatorname{GL}(n,\mathbb{Z})$
  10. The stable cohomology of $\operatorname{GL}(n,\mathbb{Z})$
  11. Stable families of algebraic $\operatorname{GL}(n,\mathbb{Z})$-representations
  12. The low-degree twisted cohomology of $\operatorname{Aut}(F_n)$ and $\operatorname{Out}(F_n)$
  13. Known results about the first (co)homology of $\operatorname{Aut}(F_n)$ and $\operatorname{Out}(F_n)$
  14. First cohomology of $\operatorname{Aut}(F_n)$ and $\operatorname{Out}(F_n)$ with algebraic coefficients
  15. Second cohomology of $\operatorname{Aut}(F_n)$ and $\operatorname{Out}(F_n)$ with algebraic coefficients
  16. ...and 22 more sections

Key Result

Theorem 1.1

We have

Theorems & Definitions (99)

  • Theorem 1.1: Satoh Satoh2013Satoh2006
  • Theorem 1.2: Theorem \ref{['H1Aut']}
  • Theorem 1.3: Theorem \ref{['H1HpHq']}
  • Conjecture 1.4: KatadaIA, see Conjecture \ref{['conjalb']}
  • Conjecture 1.5: Conjecture \ref{['conjIAn']}
  • Conjecture 1.6: Conjecture \ref{['conjectureHIAinv']}
  • Conjecture 1.7: Conjecture \ref{['conjIAnW']}
  • Theorem 1.8: Lindell Lindell22, see Theorem \ref{['conjKV']}
  • Conjecture 1.9: Church--Farb Church-Farb, see Conjecture \ref{['conjPi']}
  • Theorem 1.10: see Theorems \ref{['thmHIAinv']} and \ref{['conjIAprop']}
  • ...and 89 more