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

2022/11/24 by Kazuo Habiro, Habiro, Kazuo, Mai Katada +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2211.13458

Abstract

Borel's stability and vanishing theorem gives the stable cohomology of GL(n,ℤ) with coefficients in algebraic GL(n,ℤ)-representations. By combining the Borel theorem with the Hochschild-Serre spectral sequence, we compute the twisted first cohomology of the automorphism group Aut(F_n) of the free group F_n of rank n. We also study the stable rational cohomology of the IA-automorphism group IA_n of F_n. We propose a conjectural algebraic structure of the stable rational cohomology of 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.

Related

• The walled Brauer category and stable cohomology of IA_n
• The stable cohomology of automorphisms of free groups with coefficients in the homology representation
• On Borel's stable range of the twisted cohomology of GL(n,ℤ)
• Stable rational homology of the IA-automorphism groups of free groups
• The stable Albanese homology of the IA-automorphism groups of free groups