A note on real Heegaard Floer homology and localization

TL;DR

This paper establishes a localization spectral sequence for real Heegaard Floer homology, extending its applications to branched covers, invertible knots, and real Lagrangian Floer homology in symplectic geometry.

Contribution

It introduces a new localization spectral sequence for real Heegaard Floer homology and broadens its applicability to symplectic geometry and knot theory.

Findings

Existence of a localization spectral sequence for real Heegaard Floer homology.

Application to branched double covers and strongly invertible knots.

Extension to real Lagrangian Floer homology in symplectic manifolds.

Abstract

We prove the existence of a localization spectral sequence for the hat variant of Guth and Manolescu's recent construction of real Heegaard Floer homology, and apply it to branched double covers and strongly invertible knots. Our construction applies to real Lagrangian Floer homology in exact symplectic manifolds equipped with anti-symplectic involutions more generally, and may be of independent interest to symplectic geometers.