Localization for involutions in Floer cohomology

TL;DR

This paper proves a localization theorem for Lagrangian Floer cohomology in symplectic manifolds with involutions, leading to inequalities relating Floer cohomology of the manifold and its fixed points, with applications to symplectic Khovanov cohomology.

Contribution

It introduces a new localization theorem for Floer cohomology under symplectic involutions, extending Smith-type inequalities and applying to symplectic Khovanov cohomology.

Findings

Established a localization theorem for Floer cohomology in involutive symplectic manifolds.

Derived Smith-type inequalities relating Floer cohomology of the manifold and fixed point set.

Applied results to symplectic Khovanov cohomology, demonstrating practical implications.

Abstract

We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M. Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality for the Floer cohomology groups in M and its fixed point set. Two applications to symplectic Khovanov cohomology are included.