A Symplectic Instanton Homology via Traceless Character Varieties

TL;DR

This paper constructs a new symplectic invariant for 3-manifolds using traceless character varieties, linking gauge theory and symplectic geometry, and establishes a surgery exact triangle relating these invariants.

Contribution

It introduces a Lagrangian Floer homology for 3-manifolds based on traceless character varieties, providing a symplectic counterpart to gauge theoretic instanton homology.

Findings

Defined the symplectic instanton homology SI(Y) for closed 3-manifolds.

Established a surgery exact triangle relating SI(Y) for different surgeries.

Connected the geometry of Lefschetz fibrations with Floer homology exact triangles.

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.