Floer homology
The Atiyah–Floer conjecture connects the instanton Floer homology with the Lagrangian intersection Floer homology. Consider a 3-manifold Y with a Heegaard splitting along a surface formula_6. Then the space of flat connections on formula_6 modulo gauge equivalence is a symplectic manifold formula_8 of dimension 6"g" − 6, where "g" is the genus of the surface formula_6. In the Heegaard splitting, formula_6 bounds two different 3-manifolds; the space of flat connections modulo gauge equivalence on each 3-manifold with boundary embeds into formula_8 as a Lagrangian submanifold. One can consider the Lagrangian intersection Floer homology. Alternately, we can consider the Instanton Floer homology of the 3-manifold Y. The Atiyah–Floer conjecture asserts that these two invariants are isomorphic. Salamon-Wehrheim and Daemi-Fukaya are working on their programs to prove this conjecture.