Symplectic Heegaard splittings and linked abelian groups

Let $f$ be the gluing map of a Heegaard splitting of a 3-manifold $W$. The goal of this paper is to determine the information about $W$ contained in the image of $f$ under the symplectic representation of the mapping class group. We prove three main results. First, we show that the first homology group of the three manifold together with Seiferts linking form provides a complete set of stable invariants. Second, we give a complete, computable set of invariants for these linking forms. Third, we show that a slight augmentation of Birmans determinantal invariant for a Heegaard splitting gives a complete set of unstable invariants.

Calculating the image of the second Johnson-Morita representation

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.