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The stable cohomology of automorphisms of free groups with coefficients in the homology representation (1012.1433v1)
Published 7 Dec 2010 in math.AT and math.GR
Abstract: We study the cohomology of Aut(F_n) and Out(F_n) with coefficients in the modules \wedgeq H, \wedge H*, Symq H or Symq H*, where H is the Out(F_n)-module obtained by abelianising the free group F_n. 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.