Floer Homology on the Extended Moduli Space

Abstract

AbstractStarting 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 \(\mathbb{Z}/8\mathbb{Z}\)-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.KeywordsFloer homologyThree-manifoldModuli spaceHeegaard surface