Instanton Floer homology with Lagrangian boundary conditions

Abstract

In this paper we define instanton Floer homology groups for a pair consisting of a compact oriented $3$–manifold with boundary and a Lagrangian submanifold of the moduli space of flat $SU\left(2\right)$–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 $3$–manifold and the handle bodies is an integral homology $3$–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 $3$–manifold in our main example and thus can be used as a starting point for an adiabatic limit argument.