Real bordered Floer homology

TL;DR

This paper introduces a new module-based approach to compute real bordered Floer homology for 3-manifolds with boundary and involution, extending previous invariants and providing practical algorithms.

Contribution

It develops modules over bordered Floer algebra satisfying a gluing theorem, extending Guth-Manolescu's real Heegaard Floer homology and enabling practical computations.

Findings

Describes a method to associate modules to Heegaard diagrams of 3-manifolds with involution.

Provides a gluing theorem for these modules, facilitating computations.

Offers an algorithm to compute the extended real Heegaard Floer homology for real 3-manifolds.

Abstract

Fix a 3-manifold $Y$ with boundary $F\amalg F$ and an orientation-preserving involution $\tau: Y\to Y$ exchanging the boundary components, with nonempty fixed set. To an appropriate kind of Heegaard diagram for $Y$, we describe how to associate a module over the bordered Heegaard Floer algebra of $F$. These modules satisfy a gluing, or pairing, theorem, and extend the "hat" variant of Guth-Manolescu's real Heegaard Floer homology, $\widehat{HFR}(Y,\tau)$. Using these modules, we give a practical algorithm to compute $\widehat{HFR}(Y,\tau)$ for real 3-manifolds $(Y,\tau)$ with connected fixed set.