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(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.].
Keywords
Cite
@article{arxiv.1509.03236,
title = {Hopf Algebras and Invariants of the Johnson Cokernel},
author = {Jim Conant and Martin Kassabov},
journal= {arXiv preprint arXiv:1509.03236},
year = {2016}
}