Naturality and mapping class groups in Heegaard Floer homology

TL;DR

This paper proves that all versions of Heegaard Floer homology and related theories are natural, assigning groups and isomorphisms functorially to diffeomorphisms, ensuring isotopy invariance and functoriality.

Contribution

It establishes the naturality of Heegaard Floer homology, link Floer homology, and sutured Floer homology, providing a framework for functorial assignments and invariance under isotopy.

Findings

Heegaard Floer homology has no monodromy around generators of the diagram space.

Sufficient conditions are given for invariants to descend to natural invariants.

All versions of Floer homology are shown to be functorial and isotopy invariant.

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.