The space of Heegaard Splittings

TL;DR

This paper investigates the topological structure of the space of Heegaard splittings in 3-manifolds, revealing its relation to the Goeritz group and showing contractibility under certain conditions, with detailed results for low-genus cases.

Contribution

It provides a detailed description of the space of Heegaard splittings in terms of the diffeomorphism group and Goeritz group, including new results on its homotopy type and contractibility.

Findings

For hyperbolic manifolds, each path component is a classifying space for the Goeritz group.

When the Hempel distance exceeds 3, each path component of the splitting space is contractible.

Complete homotopy types are determined for genus 0 or 1 splittings, modulo the Smale Conjecture.

Abstract

For a Heegaard surface F in a closed orientable 3-manifold M, H(M,F) = Diff(M)/Diff(M,F) is the space of Heegaard surfaces equivalent to the Heegaard splitting (M,F). Its path components are the isotopy classes of Heegaard splittings equivalent to (M,F). We describe H(M,F) in terms of Diff(M) and the Goeritz group of (M,F). In particular, for hyperbolic M each path component is a classifying space for the Goeritz group, and when the (Hempel) distance of (M,F) is greater than 3, each path component of H(M,F) is contractible. For splittings of genus 0 or 1, we determine the complete homotopy type (modulo the Smale Conjecture for M in the cases when it is not known).