# Cohomology of automorphism groups of free groups with twisted coefficients

Research paper by Oscar Randal-Williams

Indexed on: 04 May '18Published on: 01 Apr '18Published in: Selecta Mathematica

#### Abstract

We compute the groups $$H^*(\mathrm {Aut}(F_n); M)$$ and $$H^*(\mathrm {Out}(F_n); M)$$ in a stable range, where M is obtained by applying a Schur functor to $$H_\mathbb {Q}$$ or $$H^*_\mathbb {Q}$$ , respectively the first rational homology and cohomology of $$F_n$$ . 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 $$H^*$$ .