Projective naturality in Heegaard Floer homology

Gartner, Michael

doi:10.2140/agt.2023.23.963

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Projective naturality in Heegaard Floer homology

Michael Gartner

Algebraic & Geometric Topology 23 (2023) 963–1054

DOI: 10.2140/agt.2023.23.963

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 functors

H F^{\circ} : {Man}_{*} \to 𝔽_{2} [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 $𝔽_{2} [U] – Mod$ associated to a graph of embedded Heegaard diagrams representing the $3$ –manifold. We show that the Heegaard Floer invariants yield functors

H F^{\circ} : {Man}_{*} \to Trans (P (ℤ [U] – Mod))

to 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.

Keywords

Heegaard Floer homology, $3$–manifolds, geometric topology

Mathematical Subject Classification 2010

Primary: 57M27, 57R58

References

Publication

Received: 8 January 2020

Revised: 26 July 2021

Accepted: 21 September 2021

Published: 6 June 2023

Authors

Michael Gartner
Philadelphia, PA United States

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Volume 23, issue 3 (2023)