Symplectic Heegaard splittings and linked abelian groups

TL;DR

This paper investigates how the symplectic representation of the mapping class group encodes information about 3-manifolds from Heegaard splittings, providing complete invariants for stable and unstable cases.

Contribution

It introduces new complete invariants for the homology and linking forms of 3-manifolds derived from Heegaard splittings, enhancing understanding of their symplectic representations.

Findings

First homology and linking form form complete stable invariants.

A computable set of invariants for linking forms is provided.

A slight augmentation of Birman's invariant yields complete unstable invariants.

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.