The stable cohomology of automorphisms of free groups with coefficients in the homology representation
Abstract
We study the cohomology of Aut(Fn) and Out(Fn) with coefficients in the modules q H, H*, Symq H or Symq H*, where H is the Out(Fn)-module obtained by abelianising the free group Fn. 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.
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