Symplectic Heegaard splittings and linked abelian groups

Abstract

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 Seifert's 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 Birman's determinantal invariant for a Heegaard splitting gives a complete set of unstable invariants.

78 pages, 1 figure, final version; to appear in "Groups of Diffeomorphisms"

References in corpus (2)

• Stabilization of Heegaard splittings
• The Casson invariant and the word metric on the Torelli group