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
Johnsons 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 Moritas homomorphism. Further, we compute the image of the handlebody subgroup of the mapping class group under the same map.
