Naturality in Heegaard Floer homology

Andras Juhasz gave a talk yesterday afternoon on the naturality of HF. Naturality is a subtle issue that can be easily overlooked. The upcoming related paper is joint with Ozsvath and D. Thurston. The issue is that Heegaard Floer homology associates abelian groups to a 3-manifold which are only well defined up to isomorphism. If you want to be able to compare specific group elements of HF^{\circ}(Y) where the elements are defined by distinct Heegaard diagrams H_1,H_2, you need a canonical isomorphism from HF^{\circ}(H_1) to HF^{\circ}(H_2) that tells you whether the element defined using Heegaard diagram H_1 corresponds to the same element defined using Heegaard diagram H_2. Their project is to construct these isomorphims, and decide how much data must be fixed in order for these maps to be canonical. Fortunately for those using Heegaard Floer homology, there are such canonical isomorphisms when you fix some small amount of data. A particularly useful application of these canonical isomorphisms is that it would allow direct comparison of the contact invariant for different contact structures on a given 3-manifold. This would give stronger results when both contact structures had nonvanishing contact invariant, and no easily distinguishable algebraic properties.

The theorem is as follows. Let Man_* be the category of based 3-manifolds with basepoint preserving diffeomorphisms. Then there is a functor from Man_* to the category of \mathbb{F}_2 vector spaces, which is isomorphic to \widehat{HF} as originally defined by Ozsvath and Szabo. In other words, if H and H’ are two Heegaard diagrams for a 3-manifold Y, and d is a diffeomorphism of Y taking H to H’ fixing the basepoint, then there is a canonical isomorphism induced on the Heegaard Floer homology. Loops in the space of Heegaard diagrams with fixed basepoint induce the identity isomorphism. There is also a version for link Heegaard Floer homology.

Note that these Heegaard diagrams are considered as embedded into the 3-manifold Y, not just abstract Heegaard diagrams. For two Heegaard diagrams which are abstractly the same, but embedded in different ways, we must find a diffeomorphism of Y taking one embedding to the other, and there will be an induced isomorphism on the Heegaard Floer homology.

One important issue Juhasz mentioned, is that in the Heegaard diagram (\Sigma, \alpha, \beta, z), \Sigma should be considered an oriented surface, since there are examples of diffeomorphisms taking a Heegaard diagram to itself, but reversing orientation on \Sigma that induce a nontrivial isomorphism on the Heegaard Floer homology. The example Juhasz gave is for S^2\times S^1 with a torus Heegaard diagram. See the picture below.

Heegaard Diagram for S^2xS^1

Heegaard diagram for S^2xS^2

A 180^{\circ} rotation about the axis shown reverses the orientation of the torus, and switches the two generators of the Heegaard Floer homology, thus giving a nontrivial isomorphism induced by a loop in the space of Heegaard diagrams.

An additional requirement is that the diffeomorphism must fix the basepoint, at least for the hat version of Heegaard Floer. A simple example showing the necessity of this condition is a lens space L(p,1) with a standard Heegaard diagram on a torus (identified with \mathbb{R}^2/\mathbb{Z}^2) with a horizontal alpha curve and a beta curve of slope p. A horizontal translation by 1/p induces a permutation of the p intersection points which generate \widehat{HF}, however such translations do not fix the basepoint.

Given these requirements, they are able to show naturality of Heegaard Floer homology. Here is some idea of the proof. Given two Heegaard diagrams H and H’ and some diffeomorphism d: H\to H' we want to construct a unique map \phi_{H,H'}: HF(H) \to HF(H'). For any two Heegaard diagrams H and H’, we can find a sequence H\to H_1 \to \cdots \to H_n=H' where each arrow is one of the following:

  1. an isotopy of \alpha or \beta curves
  2. a handleslide
  3. a stabilization/destabiliation
  4. an isotopy of \Sigma in Y

They construct an isomorphism on the Heegaard Floer homology for each of these moves and then define \phi_{H,H'} as the composition of all these isomorphisms. To show naturality, they need to show that \phi_{H,H'} does not depend on intermediate choices. This means that any loop from a Heegaard diagram back to itself, should induce the identity map. To show this I think the idea is to find a set of generators of nontrivial loops through Heegaard diagrams, and prove that for each of these generators the map induced on Heegaard Floer homology is the identity. One example of a generator given at the end of the talk is shown in the picture in this link.

Heegaard Diagram Loop

A sequence of Heegaard diagram moves in a loop.

The first arrow going down and the second arrow pointing up to the right are handleslides, and the last arrow going up to the left is an isotopy back to the original Heegaard diagram. The idea is then to compute the isomorphisms corresponding to the handleslides and isotopies and prove that they compose to give back the identity.

Juhasz mentioned there are 14 cases to check like the above double handleslide-isotopy loop. Then it seems like you should use commutativity relations of the above four moves to determine that all loops can be reduced to products of these 14 loops.

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