Real Heegaard Floer homology with integral coefficients

Abstract

We treat the question of upgrading Guth-Manolescu's real Heegaard Floer theory to a $\mathbb Z/2$-graded invariant defined over $\mathbb Z$, specifically for the hat version with one basepoint on each component of the branching link. We provide if-and-only-if obstructions to the existence of relative $\mathbb Z/2$-gradings and $\mathbb Z$-lifts, in terms of purely homological information associated to the 3-manifold with involution. When the corresponding obstruction vanishes, we study the set of such $\mathbb Z$-lifts, which a priori is only an invariant up to a torsor action. The obstruction theory could be of independent interest in various Floer theories more broadly, particularly Lagrangian Floer theory.