  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.
