An alternate definition of local contact homology

This blog has been dormant for a while, but I have a backlog of topics to write about. The following is something I meant to write after the AIM workshop last December, where I suggested this in a discussion.

1. Introduction

The question is how to define local contact homology. The setting is that in a contact manifold (Y,\lambda_0), you have a (possibly degenerate) Reeb orbit \gamma_0, and a positive integer d, such that the k-fold cover of \gamma_0 is isolated in the space of Reeb orbits whenever k\le d. We then want to define the local contact homology LCH(Y,\lambda_0,\gamma_0,d) to be the contact homology of a nondegenerate perturbation \lambda of \lambda_0 in a tubular neighborhood N of \gamma_0, in the homotopy class of loops that wind d times around the tubular neighborhood N. How can we define this contact homology?

The basic approach would be to choose a generic almost complex structure J on {\mathbb R}\times N satisfying the usual compatibility conditions with \lambda, and count J-holomorphic cylinders (with rational coefficients) between good Reeb orbits of the nondegenerate perturbation \lambda in the homotopy class d. Unfortunately, when d>1, one usually cannot obtain the necessary transversality to define the differential for generic J. In the three-dimensional case one can obtain the requred transversality to define the differential (see for example my recent paper with Jo Nelson), but proving invariance requires another technique. Anyway, when sufficient transversality can be obtained to define this theory, let us denote it by LCH^{\mathbb Q}(Y,\lambda_0,\gamma_0,d).

One way to fix the above transversality difficulties is to use S^1-equivariant symplectic homology as done by Bourgeois-Oancea. There is also an alternate S^1-equivariant construction using contact geometry more directly, sketched in this blog post from January 2014 (details to appear in a forthcoming paper with Jo Nelson). These two S^1-equivariant theories are defined over the integers and are presumably isomorphic. When LCH^{\mathbb Q} can be defined, both S^1-equivariant theories are isomorphic to it after tensoring with {\mathbb Q}. However the S^1-equivariant theories (at least the contact version) have a lot of strange torsion in them, see the examples in the aforementioned blog post from January 2014.

Mohammed Abouzaid keeps asking me if there is an alternate definition of contact homology over the integers which is analogous to the homology of the quotient (of the loop space) by S^1, rather than to S^1-equivariant homology. There is in fact such a construction in this local setting, and this is what I am now going to explain. Moreover, the construction is very easy and avoids the usual technical difficulties. (However it is not clear to me right now how to fit this local construction into a global construction, without which the local construction might not be very useful.)

2. Preliminaries

Let (Y,\lambda) be a nondegenerate contact manifold. Assume either that Y is closed, or that Y is noncompact but we are in a setting where the usual Gromov compactness is applicable (for example in the tubular neighborhood N above). Assume that \lambda has no contractible Reeb orbits. Let a be a free homotopy class in Y such that all Reeb orbits in the class a are simple. In this very special situation, if J is a generic almost complex structure on {\mathbb R}\times Y satisfying the usual conditions, then we can define the cylindrical contact homology HC^a(Y,\lambda,J) to be the homology of the chain complex CC^a(Y,\lambda,J) freely generated over {\mathbb Z} by the Reeb orbits in the free homotopy class a, whose differential counts J-holomorphic curves between such Reeb orbits with signs. To be a little more precise, the chain complex is generated by pairs (\gamma,o) where \gamma is a Reeb orbit in the class a, and o is a choice of orientation data (which can be determined by choosing a base point on \gamma along with an orientation of the determinant line of a certain Fredholm operator associated to \gamma and the base point), modulo the relation (\gamma,-o)=-(\gamma,o). There are no transversality difficulties in defining the differential \partial and proving that \partial^2=0, because all Reeb orbits under consideration are simple (so that multiply covered holomorphic curves do not arise), and there are no contractible orbits (so that no planes can break off). Furthermore, if \lambda' is another contact form satisfying the same conditions (in the nonconpact case, assume that \lambda' agrees with \lambda on the complement of a compact set), and if J' is a generic almost complex structure for \lambda', then there is a canonical isomorphism

HC^a(Y,\lambda,J) = HC^a(Y,\lambda',J').

This isomorphism is defined by the usual cobordism construction, which again does not encounter any transversality difficulties. In particular, since this contact homology does not depend on J, we can denote it by HC^a(Y,\lambda).

Now let \phi:Y\to Y be a diffeomorphism (in the noncompact case, assume that it preserves \lambda outside a compact set). Then we have an induced isomorphism

HC(\phi): HC^a(Y,\lambda,J) \to HC^{\phi^*a}(Y,\phi^*\lambda,\phi^*J).

This is defined by an isomorphism of chain complexes which geometrically pulls everything back by \phi. Again, since the homologies do not depend on the almost complex structure, we get an isomorphism

HC(\phi): HC^a(Y,\lambda) \to HC^{\phi^*a}(Y,\phi^*\lambda).

Moreover, this is functorial under composition of diffeomorphisms:

HC(\phi_1\circ\phi_2) = HC(\phi_2)\circ HC(\phi_1).

3. Defining local contact homology

We now have all the ingredients we need to give a definition of local contact homology. Return to the setting (N,\lambda_0,d) from the introduction. Let \lambda be a nondegenerate perturbation of \lambda_0. Let \widetilde{N} denote the connected degree d cover of N, and let \pi:\widetilde{N}\to N denote the covering projection. By the construction in Section 2, the degree 1 contact homology of (\widetilde{N},\pi^*\lambda) is defined; let us denote this by HC^1(\widetilde{N},\pi^*\lambda).

Next, let \phi:\widetilde{N}\to\widetilde{N} be a generator of the group of deck transformations. Then by the construction in Section 2, \phi induces an isomorphism

HC(\phi): HC^1(\widetilde{N},\pi^*\lambda) \to HC^1(\widetilde{N},\pi^*\lambda).

We remark that since \phi^d=1, by functoriality we have HC(\phi)^d=1.

Now define the local contact homology LCH(Y,\lambda_0,\gamma_0,d) to be the fixed point set of HC(\phi) acting on HC^1(\widetilde{N},\pi^*\lambda). This is defined over {\mathbb Z}, and by the invariance properties in Section 2, it is depends only on the contact form \lambda_0 in a neighborhood of \gamma_0, and on the positive integer d. That’s it! We’re done. If only math were always this easy…

4. Why this is a reasonable definition

To see that the above definition is reasonable, suppose one can choose a perturbation \lambda of \lambda_0 and an almost complex structure J on {\mathbb R}\times N compatibly with \lambda satisfying sufficient transversality so that the degree d contact homology of (\lambda,J) in N is defined, call this HC^{\mathbb Q}(N,\lambda,d,J). As mentioned in the introduction, in this situation ths usual approach is to define the local contact homology LCH^{\mathbb Q}(Y,\lambda_0,\gamma_0,d) to agree with the above contact homology. Now let LCH(Y,\lambda_o,\gamma_0,d) denote the alternate, integral definition in Section 3.

Claim. In the above situation,

LCH^{\mathbb Q}(Y,\lambda_0,\gamma_0,d) = LCH(Y,\lambda_0,\gamma_0,d) \otimes {\mathbb Q}.

The proof is to define an isomorphism of chain complexes from the chain complex on the right (for the pullback \pi^*J) to the chain complex on the left, by projecting a Reeb orbit (with its orientation data) from \widetilde{N} to N. To see that this is an isomorphism on chains, we just need to note that if \gamma is a Reeb orbit in N, if \widetilde{\gamma} is a lift of \gamma to \widetilde{N}, and if o is any choice of orientation data for \widetilde{\gamma}, then the sum from k=1 to d of (\phi^k)^*(\widetilde{\gamma},o) is zero if \gamma is bad, and projects to d times \gamma with appropriate orientation data if \gamma is good. To see that this map is a chain map, we observe that the holomorphic curves counted are the same, so we just need to check that the orientations and combinatorial factors work out. I think that all of the relevant arguments are contained in Section 6 of the paper by Hryniewicz-Macarini on local contact homology (where they were proving something very similar but slightly different).

5. A quick example

In the three-dimensional case, suppose that \gamma_0 is a nondegenerate elliptic orbit (whose covers are also nondegenerate). What is its local contact homology in degree 2? If we use the S^1-equivariant definition, we get lots of 2-torsion, as explained in the blog post from January 2014. However if we use the definition in Section 3 above, we just get {\mathbb Z}. So that’s maybe nicer.

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