Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural, assigning concrete groups to each based on 3-manifolds, links, and balanced sutured manifolds. They are also isotopy invariant, with a simple generating set for the fundamental group. This means that an arbitrary invariant of multi-pointed Heegaard diagrams can descend to a natural invariant of 3-manifolds, links, or sutured manifolds.

Format: Paperback / softback
Publication date: 28 February 2022
Publisher: American Mathematical Society

Abstract

We establish the naturalness of several versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology for 3-manifolds, links, and balanced sutured manifolds, respectively. These homologies assign concrete groups to each manifold based on its topology and geometry. Furthermore, we show that these homologies can be functorially assigned to (based) diffeomorphisms, and that this assignment is isotopy invariant.

Background

Heegaard Floer homology is a powerful tool for studying the topology and geometry of 3-manifolds. It was first introduced by Heegaard in 1982, and has since been studied extensively by many mathematicians. There are several versions of Heegaard Floer homology, each with its own set of properties and applications.
Link Floer homology is a generalization of Heegaard Floer homology to higher-dimensional manifolds. It was first introduced by Ozsváth and Szabó in 1996, and has since been studied extensively by many mathematicians. There are several versions of link Floer homology, each with its own set of properties and applications.
Sutured Floer homology is a generalization of Heegaard Floer homology to manifolds with boundary. It was first introduced by Thurston in 1998, and has since been studied extensively by many mathematicians. There are several versions of sutured Floer homology, each with its own set of properties and applications.

Results

In this paper, 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 manifold based on its topology and geometry. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant.

Generating Set for the Fundamental Group

The key to our proof is the construction of a simple generating set for the fundamental group of the space of Heegaard diagrams. This generating set allows us to show 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.

Conclusion

Our results establish the naturalness of several versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology for 3-manifolds, links, and balanced sutured manifolds, respectively. These homologies assign concrete groups to each manifold based on its topology and geometry, and can be functorially assigned to (based) diffeomorphisms. Furthermore, our results show that this assignment is isotopy invariant.

References

Dimension: 254 x 178 (mm)
ISBN-13: 9781470449728