Floer homology on the extended moduli space

TL;DR

This paper constructs a three-manifold invariant using Lagrangian Floer homology on the extended moduli space of flat SU(2)-connections, aiming to support the symplectic approach to the Atiyah-Floer Conjecture.

Contribution

It introduces a Floer homology framework on the extended moduli space, including a compactification method, to define a three-manifold invariant relevant to the Atiyah-Floer Conjecture.

Findings

Defined a Z/8-graded abelian group as an invariant

Developed Floer homology on a semipositive symplectic manifold

Provided a method to handle non-compactness via symplectic cutting

Abstract

Starting from a Heegaard splitting of a three-manifold, we use Lagrangian Floer homology to construct a three-manifold invariant, in the form of a relatively Z/8-graded abelian group. Our motivation is to have a well-defined symplectic side of the Atiyah-Floer Conjecture, for arbitrary three-manifolds. The symplectic manifold used in the construction is the extended moduli space of flat SU(2)-connections on the Heegaard surface. An open subset of this moduli space carries a symplectic form, and each of the two handlebodies in the decomposition gives rise to a Lagrangian inside the open set. In order to define their Floer homology, we compactify the open subset by symplectic cutting; the resulting manifold is only semipositive, but we show that one can still develop a version of Floer homology in this setting.