The Johnson Cokernel and the Enomoto-Satoh invariant

TL;DR

This paper investigates the cokernel of the Johnson homomorphism for surface mapping class groups, introducing a generalized trace map that detects a broad family of representations, extending prior invariants.

Contribution

It introduces a new graphical trace map generalizing previous trace maps, revealing a large family of representations in the Johnson cokernel.

Findings

The new trace map detects a vast family of representations.

The Enomoto-Satoh trace is the rank 1 component of the new trace.

The rank 2 component of the trace is also analyzed.

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-Satoh and Conant-Kassabov-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-Satoh. The Enomoto-Satoh trace is the rank 1 part of the new trace. The rank 2 part is also investigated.