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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 \(H^*\) .