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Projective naturality in Heegaard Floer homology

Michael Gartner

Algebraic & Geometric Topology 23 (2023) 963–1054

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 functors

HF: Man 𝔽2[U] – 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 𝔽2[U] – Mod associated to a graph of embedded Heegaard diagrams representing the 3–manifold. We show that the Heegaard Floer invariants yield functors

HF: Man Trans (P([U] – Mod ))

to the category of transitive systems in a projectivized category of [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 [U] – Mod . We discuss an application to involutive Heegaard Floer homology, and potential generalizations of our results.

Heegaard Floer homology, $3$–manifolds, geometric topology
Mathematical Subject Classification 2010
Primary: 57M27, 57R58
Received: 8 January 2020
Revised: 26 July 2021
Accepted: 21 September 2021
Published: 6 June 2023
Michael Gartner
Philadelphia, PA
United States

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