Calculating the image of the second Johnson-Morita representation

Birman, Joan S.; Brendle, Tara E.; Broaddus, Nathan

Calculating the image of the second Johnson-Morita representation

Authors

Birman, Joan S.

Brendle, Tara E.

Broaddus, Nathan

Description

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.
11 pages, 1 figure