Projective Naturality in Heegaard Floer Homology

Abstract

Let ${\text{Man}}_{\ast }$ denote the category of closed, connected, oriented and based $3$-manifolds, with basepoint preserving diffeomorphisms between them. Juhász, Thurston and Zemke showed that the Heegaard Floer invariants are natural with respect to diffeomorphisms, in the sense that there are functors $H{F}^{\circ }:{\text{Man}}_{\ast }\to {\mathbb{F}}_2[U]\text{-}\text{Mod}$ whose values agree with the invariants defined by Ozsváth and Szabó. The invariant associated to a based $3$-manifold comes from a transitive system in ${\mathbb{F}}_2[U]\text{-}\text{Mod}$ associated to a graph of embedded Heegaard diagrams representing the $3$-manifold. We show that the Heegaard Floer invariants yield functors $H{F}^{\circ }:{\text{Man}}_{\ast }\to \text{Trans}(P(\mathbb{Z}[U]\text{-}\text{Mod}))$ to the category of transitive systems in a projectivized category of $\mathbb{Z}[U]$-modules. In doing so, we will see that the transitive system of modules associated to a $3$-manifold actually comes from an underlying transitive system in the projectivized homotopy category of chain complexes over $\mathbb{Z}[U]\text{-}\text{Mod}$. We discuss an application to involutive Heegaard Floer homology, and potential generalizations of our results.

arXiv admin note: text overlap with arXiv:1210.4996 by other authors. Updated to accepted version. Modified the approach to the main results in section 8, which were not sound; extended applications section; improved and extended discussion of various aspects of orientation systems

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