Atiyah-Floer type conjecture and Virtual fundamental chain

Abstract: This is a report on my work in progress with Aliakbar Daemi

We are studying an SO(3) version of Atiyah-Floer conjecture relating Instanton Floer homology to Lagrangian Floer homology, via cobordism method. In the case when the moduli space of flat connections on 3 manifold is an {\it embedded} Lagrangian submanifold of the space of flat connections on 2 manifold, we can perturb Lipyanskiy type mixed moduli space using geometric perturbation. In the case it is an immersed Lagrangian submanifold, we need abstract perturbation and virtual technique. The singularity of the instanton moduli space is wilder than the case of pseudo-holomorphic curve and we need certain `stratified' version of Kuranishi structure. I will explain how we can define such a notion, show the existence of such structure and use it to obtain virtual fundamental chain.

algebraic geometrydifferential geometrygeometric topologysymplectic geometry

Audience: researchers in the topic

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