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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 ∘ : Man
∗ → 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 ∘ : Man
∗ → 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
Publication
Received: 8 January 2020
Revised: 26 July 2021
Accepted: 21 September 2021
Published: 6 June 2023
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