The stable cohomology of automorphisms of free groups with coefficients in the homology representation

2010/12/07 by Oscar Randal‐Williams, Randal-Williams, Oscar
Mathematics · #20F28 #20J06 #57R20 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1012.1433

openalex publication_date 2010/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the cohomology of Aut(F_n) and Out(F_n) with coefficients in the modules \wedge^q H, \wedge H^*, Sym^q H or Sym^q H^*, where H is the Out(F_n)-module obtained by abelianising the free group F_n. For reasons which are not conceptually clear, taking coefficients in H and its related modules behaves in a far less trivial way than taking coefficients in H^* and its related modules. Based on a conjectural homology stability theorem for spaces of graphs labeled by a simply connected background space, we give a stable integral calculation of these groups in low degrees, and modulo a further conjecture a stable rational calculation in all degrees.

Citations

• Cohomological Aspects of Magnus Expansions

Related

• On the stable cohomology of the IA-automorphism groups of free groups
• Stable cohomology of Aut(F_n) with bivariant twisted coefficients
• Stable rational cohomology of automorphism groups of free groups and the integral cohomology of moduli spaces of graphs
• Cohomoloy of Aut(F_n)
• Stable homology of automorphism groups of free groups