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(Fn) on Hn which induces an Out(Fn) action on a quotient Hn¯. 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 in Algebr. Geom. Topol. 15 (2015) 801–821 projects to take values in Hvcd(Out(Fn);Hn¯). 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
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