Atiyah–Floer conjecture: relate instanton and Lagrangian Floer homologies

Prove the Atiyah–Floer conjecture by establishing that, for any Heegaard splitting of a closed 3-manifold Y along a surface Σ, the instanton Floer homology of Y is isomorphic to the Lagrangian Floer homology of the pair of Lagrangian submanifolds in the moduli space of flat connections over Σ determined by the two handlebodies of the splitting.

Background

The remark compares Floer-type invariants in low-dimensional topology with Lagrangian Floer theory and articulates the Atiyah–Floer conjecture as a bridge between gauge-theoretic and symplectic approaches. Given a Heegaard splitting of a 3-manifold Y, one obtains two Lagrangian submanifolds in a moduli space of flat connections over the splitting surface Σ, and the conjecture predicts that the instanton Floer homology of Y matches the Lagrangian Floer homology of this pair.

The text notes there has been substantial progress toward this conjecture (e.g., by Salamon–Wehrheim and Daemi–Fukaya–Lipyanskiy), but it remains unresolved in general.