  The aim of this article is to introduce and study certain topological
invariants for closed, oriented three-manifolds Y. These groups are relatively
Z-graded Abelian groups associated to SpinC structures over Y. Given a genus g
Heegaard splitting of Y, these theories are variants of the Lagrangian Floer
homology for the g-fold symmetric product of the surface relative to certain
totally real subspaces associated to the handlebodies.
