A Symplectic Instanton Homology via Traceless Character Varieties

Abstract

Since its inception, Floer homology has been an important tool in low-dimensional topology. Floer theoretic invariants of $3$-manifolds tend to be either gauge theoretic or symplecto-geometric in nature, and there is a general philosophy that each gauge theoretic Floer homology should have a corresponding symplectic Floer homology and vice-versa. In this article, we construct a Lagrangian Floer invariant for any closed, oriented $3$-manifold $Y$ (called the symplectic instanton homology of $Y$ and denoted $\mathrm{SI}(Y)$) which is conjecturally equivalent to a Floer homology defined using a certain variant of Yang-Mills gauge theory. The crucial ingredient for defining $\mathrm{SI}(Y)$ is the use of traceless character varieties in the symplectic setting, which allow us to avoid the debilitating technical hurdles present when one attempts to define a symplectic version of instanton Floer homologies. Furthermore, by studying the effect of Dehn surgeries on traceless character varieties, we establish a surgery exact triangle using work of Seidel that relates the geometry of Lefschetz fibrations with exact triangles in Lagrangian Floer theory.