[Paper Review] Naturality and mapping class groups in Heegaard Floer homology

András Juhász, Dylan P. Thurston+1 more|arXiv (Cornell University)|Oct 18, 2012

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TL;DR

This paper establishes the naturality of Heegaard Floer homology, link Floer homology, and sutured Floer homology by proving they define functors from the category of based 3-manifolds (or links or sutured manifolds) to the category of graded abelian groups, assigning isomorphisms to (based) diffeomorphisms in a way that is invariant under isotopy. The key contribution is showing that Heegaard Floer homology has no monodromy around a generating set of the fundamental group of the space of Heegaard diagrams, thereby ensuring functoriality and naturality across all versions of the theory.

ABSTRACT

We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant. The proof relies on finding a simple generating set for the fundamental group of the "space of Heegaard diagrams," and then showing that Heegaard Floer homology has no monodromy around these generators. In fact, this allows us to give sufficient conditions for an arbitrary invariant of multi-pointed Heegaard diagrams to descend to a natural invariant of 3-manifolds, links, or sutured manifolds.

Motivation & Objective

• To resolve the long-standing issue of naturality in Heegaard Floer homology, where invariants were previously only defined up to isomorphism.
• To establish that all versions of Heegaard Floer homology (including hat, plus, minus, infinity, and link/sutured variants) are functorial with respect to (based) diffeomorphisms.
• To provide a general criterion for when an invariant of multi-pointed Heegaard diagrams descends to a natural invariant of 3-manifolds, links, or sutured manifolds.
• To show that the assignment of isomorphisms to diffeomorphisms is isotopy-invariant, thereby enabling the construction of cobordism maps and the study of specific elements like contact invariants.

Proposed method

• Identify a simple generating set for the fundamental group of the space of Heegaard diagrams, consisting of handleslides, stabilizations, destabilizations, and diffeomorphisms.
• Prove that Heegaard Floer homology has no monodromy around these generators by showing that the induced isomorphisms are independent of path choices in the diagram complex.
• Use the connectivity and simple connectivity of the 2-complex of handleslides (Y₂(B)) to ensure that isomorphisms defined along paths are well-defined and commute along rectangles.
• Construct a strong Heegaard invariant by extending weak invariants that satisfy axioms for α/β-equivalences, handleslide loops, and stabilization slides.
• Apply the theory of transitive systems and colimits to define a universal group from the system of groups assigned to diagrams, ensuring functoriality.
• Translate bifurcations of gradient vector fields on 3-manifolds into moves on Heegaard diagrams, using generic 1- and 2-parameter families of gradients to model isotopies and handle slides.

Experimental results

Research questions

• RQ1 Can Heegaard Floer homology be promoted from an invariant defined up to isomorphism to a truly functorial invariant assigning concrete groups to based 3-manifolds and isomorphisms to diffeomorphisms?
• RQ2 What conditions ensure that a diagram-based invariant descends to a natural invariant of 3-manifolds, links, or sutured manifolds?
• RQ3 Does Heegaard Floer homology exhibit monodromy around generators of the fundamental group of the space of Heegaard diagrams, and if not, why?
• RQ4 How can the isotopy invariance of diffeomorphism-induced isomorphisms be established in the context of Heegaard Floer theory?
• RQ5 What is the role of handleslide loops and stabilization slides in ensuring the consistency of the invariant across different diagram representatives?

Key findings

• All versions of Heegaard Floer homology—hat, plus, minus, infinity, link, and sutured—are natural invariants, assigning concrete groups to based 3-manifolds, links, and balanced sutured manifolds.
• The assignment of isomorphisms to (based) diffeomorphisms is functorial and isotopy-invariant, meaning it descends to a well-defined action on mapping class groups.
• Heegaard Floer homology has no monodromy around a generating set of the fundamental group of the space of Heegaard diagrams, which ensures that the invariant is well-defined and independent of path choices.
• The 2-complex of handleslides (Y₂(B)) is simply connected, which guarantees that isomorphisms defined via paths of handleslides are independent of the chosen path.
• A strong Heegaard invariant exists if and only if the invariant commutes along all handleslide loops and stabilization slides, providing a complete criterion for naturality.
• The extension of a weak Heegaard invariant to a strong one is unique if it satisfies the axioms for α/β-equivalences, handleslide loops, and stabilization slides.