In this paper we define instanton Floer homology groups for a pair consisting of a compact oriented –manifold with boundary and a Lagrangian submanifold of the moduli space of flat –connections over the boundary. We carry out the construction for a general class of irreducible, monotone boundary conditions. The main examples of such Lagrangian submanifolds are induced from a disjoint union of handle bodies such that the union of the –manifold and the handle bodies is an integral homology –sphere. The motivation for introducing these invariants arises from our program for a proof of the Atiyah–Floer conjecture for Heegaard splittings. We expect that our Floer homology groups are isomorphic to the usual Floer homology groups of the closed –manifold in our main example and thus can be used as a starting point for an adiabatic limit argument.
"Instanton Floer homology with Lagrangian boundary conditions." Geom. Topol. 12 (2) 747 - 918, 2008. https://doi.org/10.2140/gt.2008.12.747