Projective Naturality in Heegaard Floer Homology

Abstract

Let Man * 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 functorswhose values agree with the invariants defined by Ozsváth and Szabó. The invariant associated to a based 3-manifold comes from a transitive system in F 2 [U ]-Mod associated to a graph of embedded Heegaard diagrams representing the 3-manifold. We show that the Heegaard Floer invariants yield functorsto the category of transitive systems in a projectivized category of 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 Z[U ]-Mod. We discuss an application to involutive Heegaard Floer homology, and potential generalizations of our results. Contents 1. Introduction 1.1. Statement of Main Results 1.2. Further Directions and Applications 1.3. Organization of the Paper 1.4. Acknowledgements 2. Background 2.1.

Property G and the 4-genus

“…Since (v + 0 ) * and (h + 0 ) * have the same rank, we also have (h + 0 ) * = 0. Thus (h + d−1 ) * = 0 by (6). So both mapping cones are quasi-isomorphic to A + d−1 ⊕ B + , and Property G still follows.…”

“…The chain homotopy equivalence C{j ≥ 0} → C{i ≥ 0} is obtained by changing the basepoint from z to w. In [28], Ozsváth and Szabó said that the chain homotopy equivalence is canonical up to sign. This assertion is justified by [26, Theorem 2.1] and [6]. (If we use Z/2Z coefficients, which are enough for the applications in our paper, we can also refer to [9].…”

On naturality of the Ozsvath-Szabo contact invariant

“…Finally, we note that we work with Heegaard Floer homology with Z/2Z coefficients throughout. The naturality results of [19] have been extended to projective Z coefficients (i.e., Z/ ± 1) [7], but at the moment these extensions have not been established for the graph cobordism maps. Assuming they will be, the results at hand should immediately extend to the refined setting.…”