Calculating the image of the second Johnson-Morita representation

TL;DR

This paper precisely characterizes the image of Morita's extension of Johnson's homomorphism from the mapping class group to a semi-direct product, including the handlebody subgroup, advancing understanding of the homomorphism's structure.

Contribution

It provides an exact description of the image of the Johnson-Morita homomorphism and computes its image on the handlebody subgroup.

Findings

Exact image of Morita's homomorphism described

Image of handlebody subgroup computed

Homomorphism's image has finite index in target group

Abstract

Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus $g$ with one boundary component to $\wedge^3 H$, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping class group to ${1/2} \wedge^3 H \semi \sp(H)$. This Johnson-Morita homomorphism is not surjective, but its image is finite index in ${1/2} \wedge^3 H \semi \sp(H)$. Here we give a description of the exact image of Morita's homomorphism. Further, we compute the image of the handlebody subgroup of the mapping class group under the same map.