Symplectic Heegaard splittings and linked abelian groups

Details Symplectic Heegaard splittings and linked abelian groups Symplectic Heegaard splittings and linked abelian groups

2007-12-13

arxiv.org/abs/0712.2104v2

in "Groups of Diffeomorphisms", Adv. Stud. Pure Math., 52, Math. Soc. Japan, Tokyo, 2008, 135--220.

Mathematics

Geometric Topology

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

Scientific paper

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.

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