The cohomology groups of the outer Whitehead automorphism group of a free product

Abstract

ji 2 Aut() induced by gj i (gk )= g k if k 6= i g g j k if k = i (where gk 2 Gk and g g j k is shorthand for conjugating gk by gj )i s anelementary Whitehead automorphism. The Whitehead automorphism group, Wh(), is the subgroup of Aut() generated by the g j i . If none of the Gi are infinite cyclic, then Wh() is the kernel of the map Aut() Out(G1 ◊···◊ Gn). In particular, if the Gi are finite, then Wh() is a finite-index subgroup of Aut(). In this paper we compute the cohomology groups of OWh(), the quotient of Wh() in the outer-automorphism group, with field coecients. Our approach is to analyze the equivariant spectral sequence associated to the action of OWh() on a contractible, simplicial complex introduced by McCullough and Miller [10]. Throughout the paper, cohomology groups will be assumed to have field coecients unless otherwise indicated, and () n denotes the n-fold product ◊ ···◊ | {z } n copies