Symplectic instanton homology and Floer theory for correspondences

Christopher Woodward (Rutgers)


I will describe work with K Wehrheim, S. Mau and C. Manolescu. Instanton homology is an invariant of homology spheres Y whose differential counts rank two instantons on Y x R. A conjecture of Atiyah and Floer relates this with Lagrangian Floer homology in the moduli space of flat bundles on a Heegard surface X for Y. I will describe how to prove that the latter is a topological invariant without reference to gauge theory using "pseudoholomorphic quilts", and extend the definition of instanton homology to include three-manifolds with knots and bundles of arbitrary rank. The resulting theory is a symplectic version of Donaldson-Floer theory, and satisfies many of the axioms of tft with corners. Unfortunately the invariants are quite difficult to compute.