Geometry and Topology Seminar

I will describe a new construction of a Floer theoretic invariant of a 3-manifold Y equipped with a principal SO(3)-bundle ω, as well as outline some of its properties. The invariant arises as the Lagrangian Floer homology of so-called traceless character varieties associated to a Heegaard splitting of the 3-manifold; we call it the "symplectic instanton homology" of (Y, ω) and denote it SI(Y, ω). If K is a (framed) knot in Y and Y₀, Y₁ denote 0- and 1-surgeries on K (with respect to the given framing), we indicate how the 2-handle cobordism maps induced by these surgeries fit into an exact triangle of symplectic instanton homologies, ··· → SI(Y, ω + ω_K) → SI(Y₀, ω) → SI(Y₁, ω) → ···, where ω_K is the SO(3)-bundle on Y dual to K. Time permitting, we will indicate how for link surgeries there is more generally a spectral sequence of symplectic instanton homologies, which gives a relationship to reduced Khovanov homology when applied to the branched double cover of a link