  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.
