The Johnson cokernel and the Enomoto–Satoh invariant

Abstract

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.