Cohomology of automorphism groups of free groups with twisted coefficients

Abstract

We compute the groups $H^{\ast }(\mathrm{Aut}(F_n);M)$ and $H^{\ast }(\mathrm{Out}(F_n);M)$ in a stable range, where $M$ is obtained by applying a Schur functor to $H_{\mathbb{Q}}$ or $H_{\mathbb{Q}}^{\ast }$, respectively the first rational homology and cohomology of $F_n$. For reasons which are not conceptually clear, taking coefficients in $H_{\mathbb{Q}}$ and its related modules behaves in a far less trivial way than taking coefficients in $H_{\mathbb{Q}}^{\ast }$ and its related modules. The answer may be described in terms of stable multiplicities of irreducibles in the plethysm ${\mathrm{Sym}}^k\circ {\mathrm{Sym}}^l$ of symmetric powers. We also compute the stable integral cohomology groups of $\mathrm{Aut}(F_n)$ with coefficients in $H$ or $H^{\ast }$, respectively the first integral homology and cohomology of $F_n$, and compute the stable cohomology with coefficients in Schur functors of $H$ or $H^{\ast }$ modulo small primes.

arXiv comment20 pages. Supersedes arXiv:1012.1433. v2 21 pages, to appear in Selecta Mathematica