Hopf algebras and invariants of the Johnson cokernel

Abstract

We show that if H is a cocommutative Hopf algebra, then there is a natural action of Aut(Fn) on H⊗n which induces an Out(Fn) action on a quotient H⊗n. In the case when H = T(V) is the tensor algebra, we show that the invariant TrC of the cokernel of the Johnson homomorphism studied by the first author projects to take values in Hvcd(Out(Fn); H⊗n) We analyze the n=2 case, getting large families of obstructions generalizing the abelianization obstructions of the authors and Vogtmann.