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^*.
Highlights
Galatius [12] has proved the remarkable theorem that the natural homomorphisms n −→ Aut(Fn) −→ Out(Fn) both induce homology isomorphisms in degrees 2∗ ≤ n − 3 with integral coefficients
His approach is to model BOut(Fn) as the space Gn of graphs of the homotopy type of ∨n S1, and BAut(Fn) as the space Gn1 of pointed graphs of the same homotopy type. He produces a natural map from such spaces of graphs to the infinite loop space
Our goal is to show that the stable cohomology of Aut(Fn) and Out(Fn) with twisted coefficients may be approached with the geometric techniques used by Galatius, along with a little representation theory
Summary
Galatius [12] has proved the remarkable theorem that the natural homomorphisms n −→ Aut(Fn) −→ Out(Fn) both induce homology isomorphisms in degrees 2∗ ≤ n − 3 with integral coefficients. His approach is to model BOut(Fn) as the space Gn of graphs of the homotopy type of ∨n S1, and BAut(Fn) as the space Gn1 of pointed graphs of the same homotopy type. As long as n ≥ 4|λ| + 3, where ρ(λ) denotes the set of Young diagrams which may be obtained from λ by removing at most one box from each row
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