Volume 15, issue 2 (2015)

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The Johnson cokernel and the Enomoto–Satoh invariant

James Conant

Algebraic & Geometric Topology 15 (2015) 801–821

We study the cokernel of the Johnson homomorphism for the mapping class group of a surface with one boundary component. A graphical trace map simultaneously generalizing trace maps of Enomoto and Satoh and Conant, Kassabov and Vogtmann is given, and using technology from the author’s work with Kassabov and Vogtmann, this is is shown to detect a large family of representations which vastly generalizes series due to Morita and Enomoto and Satoh. The Enomoto–Satoh trace is the rank-1 part of the new trace, and it is here that the new series of representations is found. The rank-2 part is also investigated, though a fuller investigation of the higher-rank case is deferred to another paper.

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Johnson homomorphism, Enomoto–Satoh invariant, Johnson cokernel
Mathematical Subject Classification 2010
Primary: 17B40
Secondary: 20C15, 20F28
Received: 23 December 2013
Revised: 5 July 2014
Accepted: 7 July 2014
Published: 22 April 2015
James Conant
Department of Mathematics
University of Tennessee
227 Ayres Hall
1403 Circle Drive
Knoxville, TN 37996