Hopf Algebras and Invariants of the Johnson Cokernel

TL;DR

This paper explores the action of automorphism groups on tensor powers of cocommutative Hopf algebras, linking invariants of the Johnson cokernel to cohomology of Out(F_n), and provides new obstructions in the case n=2.

Contribution

It introduces a natural Out(F_n) action on tensor powers of cocommutative Hopf algebras and connects invariants of the Johnson cokernel to top cohomology, extending previous obstruction results.

Findings

Established a natural Out(F_n) action on tensor powers of H.

Linked Johnson cokernel invariants to top cohomology of Out(F_n).

Provided new obstructions for the case n=2.

Abstract

We show that if H is a cocommutative Hopf algebra, then there is a natural action of Aut(F_n) on the nth tensor power of H which induces an Out(F_n) action on a quotient \overline{H^{\otimes n}}. In the case when H=T(V) is the tensor algebra, we show that the invariant Tr^C of the cokernel of the Johnson homomorphism studied in [J. Conant, The Johnson cokernel and the Enomoto-Satoh invariant, Algebraic and Geometric Topology, 15 (2015), no. 2, 801--821.] projects to take values in the top dimensional cohomology of Out(F_n) with coefficients in \overline{H^{\otimes n}}. We analyze the n=2 case, getting large families of obstructions generalizing the abelianization obstructions of [J. Conant, M. Kassabov, K. Vogtmann, Higher hairy graph homology, Journal of Topology, Geom. Dedicata 176 (2015), 345--374.].