#### Volume 16, issue 4 (2016)

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Hopf algebras and invariants of the Johnson cokernel

### Jim Conant and Martin Kassabov

Algebraic & Geometric Topology 16 (2016) 2325–2363
##### Abstract

We show that if $H$ is a cocommutative Hopf algebra, then there is a natural action of $Aut\left({F}_{n}\right)$ on ${H}^{\otimes n}$ which induces an $Out\left({F}_{n}\right)$ action on a quotient $\overline{{H}^{\otimes n}}$. In the case when $H=T\left(V\right)$ is the tensor algebra, we show that the invariant ${Tr}^{C}$ of the cokernel of the Johnson homomorphism studied in Algebr. Geom. Topol. 15 (2015) 801–821 projects to take values in ${H}^{vcd}\left(Out\left({F}_{n}\right);\overline{{H}^{\otimes n}}\right)$. We analyze the $n=2$ case, getting large families of obstructions generalizing the abelianization obstructions of Geom. Dedicata 176 (2015) 345–374.

##### Keywords
Johnson homomorphism, Hopf algebras, automorphism groups of free groups
##### Mathematical Subject Classification 2010
Primary: 20F65, 20J06, 16T05, 17B40
Secondary: 20C15, 20F28