Atiyah–Floer conjecture relating instanton and Lagrangian Floer homologies

Establish the Atiyah–Floer conjecture: given a Heegaard splitting of a closed 3‑manifold Y into handlebodies glued along a surface Σ, prove that the instanton Floer homology of Y is related—precisely, equivalent up to canonical isomorphism—to the Lagrangian Floer homology of a pair of Lagrangian submanifolds in the moduli space of flat connections over Σ associated to the two handlebodies.

Background

The paper surveys connections between gauge-theoretic Floer invariants of 3‑manifolds and Lagrangian Floer theory. The Atiyah–Floer conjecture posits a deep relationship between the instanton Floer homology of a closed 3‑manifold and a Lagrangian Floer theory built from moduli spaces of flat connections on a Heegaard surface. The author cites progress but indicates the conjecture remains unresolved.