Calculating the image of the second Johnson-Morita representation

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 $\frac{1}{2} \wedge^3 H \rtimes \mathrm{S_p}(H)$. This Johnson-Morita homomorphism is not surjective, but its image is finite index in $\frac{1}{2} \wedge^3 H \rtimes \mathrm{S_p}(H)$ [11]. 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.