Hopf algebras and invariants of the Johnson cokernel

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

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Keywords

Johnson homomorphism, Hopf algebras, automorphism groups of free groups

Mathematical Subject Classification 2010

Primary: 20F65, 20J06, 16T05, 17B40

Secondary: 20C15, 20F28

References

Publication

Received: 17 September 2015

Revised: 20 January 2016

Accepted: 24 January 2016

Published: 12 September 2016

Authors

Jim Conant
Department of Mathematics University of Tennessee Knoxville, TN 37996 United States
http://www.math.utk.edu/~jconant/
Martin Kassabov
Department of Mathematics Cornell University Ithaca, NY 14853 United States
https://www.math.cornell.edu/m/People/bynetid/mdk35

Volume 16, issue 4 (2016)

Jim Conant and Martin Kassabov

Abstract