Local contact homology with integer coefficients I

In a recent preprint, Bourgeois and Oancea define $S^1$-equivariant symplectic homology, and show that if the transversality needed to define linearized contact homology holds, then these two homologies are isomorphic.

For a while I was bothered by an apparent paradox. Namely, $S^1$-equivariant symplectic homology is defined with integer coefficients (and invariant). Moreover, in the absence of contractible Reeb orbits, one can imitate the Bourgeois-Oancea construction directly for a contact form, to define a version of contact homology with integer coefficients. However the usual definition of contact homology (when sufficient transversality holds to define it) is invariant only using rational coefficients.

I think I have now figured out what is going on here. Namely:

1) For nondegenerate contact forms without contractible Reeb orbits, one can indeed imitate the Bourgeois-Oancea construction to define a version of contact homology with integer coefficients. (For example one can do this in a neighborhood of a possibly degenerate Reeb orbit to define local contact homology.)

2) This is an invariant of contact structures that admit nondegenerate contact forms without contractible Reeb orbits (and it usually has lots of torsion).

3) In the above situation, if sufficient transversality holds to define contact homology in the usual way, then this agrees with the tensor product of the above invariant with the rationals.

This seems really bizarre, and I could be mistaken, so let me try to explain it. At the end I will show some examples of this integer coefficient version of local contact homology and its invariance.

1. $S^1$-equivariant Floer theory in general

As a warmup, we first need to recall how to define $S^1$-equivariant Morse or Floer theory in general, following Bourgeois-Oancea.

Suppose we are trying to compute an example of some kind of Floer theory (given e.g. a specific contact manifold). The usual situation is that there is an infinite-dimensional contractible space $X$ of data (e.g. metrics, compatible almost complex structures, etc.) needed to define the theory. For a generic point in $x\in X$ there is a chain complex $C_*(x)$. For a generic path $\gamma$ between two generic points $x_0,x_1\in X$ there is a chain map $\Phi(\gamma): C_*(x_0)\to C_*(x_1)$ inducing an isomorphism on homology. For a generic homotopy of paths there is a chain homotopy between the corresponding chain maps, and so on. The homology of $C_*(x)$ for generic $x$ is then the Floer homology we are trying to compute.

Suppose now that the space $X$ has an $S^1$ action preserving all of the above. That is, if $x\in X$ is generic, then so is every point in its orbit, and all of the chain complexes in this orbit are canonically isomorphic. Furthermore, the chain maps and chain homotopies and higher chain homotopies induced by chains in $X$ are invariant under the $S^1$ action.

Example: $M$ is a closed smooth manifold with a smooth $S^1$ action, $X$ is the set of pairs $(f,g)$ where $f:M\to{\mathbb R}$ is a smooth function and $g$ is a metric on $M$, and $S^1$ acts on $X$ by pulling back the function and the metric via the $S^1$ action on $M$.

We now define $S^1$-equivariant Floer homology roughly as follows. Let us identify $ES^1=S^\infty$ and $BS^1={\mathbb C}P^\infty$. We choose a suitably generic $S^1$-equivariant map $ES^1\to X$ and a generic Morse-Smale pair on $BS^1$ (which we can take to be the usual one with one critical point of each nonnegative even index). In particular, we want each critical point on $BS^1$ to map to a generic orbit in $X$ for which the chain complex is defined. In fact, it is possible and convenient to arrange that each critical point on $BS^1$ maps to the same generic orbit in $X$. Let $x$ denote a point in this orbit.

One can now define a chain complex analogously to Section 6.1 of my paper “Floer homology of families I”. A generator is a pair $(p,q)$ where $p$ is a critical point on $BS^1$ and $q$ is a generator of the chain complex $C_*(x)$. The grading is the sum of the index of $p$ and the grading of $q$. That is, the chain complex is

$C_* = {\mathbb Z}[u]\otimes C_*(x)$

where $u$ is a formal variable of degree $2$.

The differential can be written as a sum

$\partial(u^k\otimes q) = \sum_{i=0}^k u^{k-i}\otimes \partial_iq$

where $\partial_i:C_*(x) \to C_{*+2i-1}(x)$ counts a kind of hybrid of gradient flow lines in $BS^1$ and Floer trajectories “above” them. (One needs to do a bit of geometric setup to arrange that the differential commutes with the operation that sends $u^k$ to $u^{k-1}$ for $k>1$ and $u^0$ to $0$.) In particular, $\partial_0$ is just the original differential on $C_*(x)$, and $\partial_1$ is called the “BV operator”. The homology of this complex is now the $S^1$-equivariant Floer homology.

Example. In the case where $X$ is the space of pairs $(f,g)$ on a compact manifold $M$ with an $S^1$ action, this construction computes the homology of the fiber bundle $M\to M\times_{S^1} ES^1 \to BS^1$, similarly to my paper Floer homology of families, so we recover the usual $S^1$-equivariant homology of $M$.

Remark 1. One can define $G$-equivariant Morse homology for other groups $G$ by a similar construction.

Remark 2. In the general setup, if $S^1$ acts freely on $X$, except for an infinite codimension subset $X_0$, then one can take $X\setminus X_0$ as a model for $ES^1$. This leads to a considerably more abstract version of the above theory, involving general chains in $X\setminus X_0$.

2. Morse-Bott theory the old-fashioned way

Now let $\lambda$ be a nondegenerate contact form with no contractible Reeb orbits, on a manifold $Y$ which is closed, or in a situation where Gromov compactness applies, such as in local contact homology. As I have explained in a previous post, given a generic $S^1$-family of almost complex structures on ${\mathbb R}\times Y$
one can define “non-equivariant contact homology”, which is analogous to symplectic homology. We would now like to apply the construction in Section 1 to define an $S^1$-equivariant version of this (which will then correspond to contact homology), using the $S^1$ action which rotates the $S^1$-family of almost complex structures.

However, the definition of non-equivariant contact homology previously discussed here has a flaw which makes it unsuitable for this construction. Namely, it depends on a choice of base point on each Reeb orbit. Now the space of $S^1$-families of almost complex stuctures is contractible, but if we also have to choose base points on the Reeb orbits, then contractibility goes out the window.

Fortunately, there is an alternate construction of non-equivariant contact homology which does not require any choice of base points on the Reeb orbits. This uses an older version of Morse-Bott theory, defined using generic chains on the manifolds of Reeb orbits (instead of Morse functions on the Reeb orbits and “cascades”, as in works of Bourgeois and Frauenfelder).

For simplicity let me first explain this in the finite dimensional case. Let $M$ be a closed smooth manifold and let $f:M\to {\mathbb R}$ be a Morse-Bott function, such that the critical points come in $S^1$-families. Let $g$ be a generic metric. We then define a version of Morse-Bott homology as follows.

Let $S$ be a critical submanifold of index $i$. There is then a finite set $Z(S)\subset S$ of points from which there is a gradient flow line to another critical submanifold of index $i$. [EDIT: The set $Z(S)$ needs to include some other points too. Not a big deal.] We now define $C_0(S)$ to be the set of generic singular $0$-chains in $S$, where “generic” here means not containing any points in the set $Z(S)$. We define $C_1(S)$ to be the set of generic singular $1$-chains in $S$, where “generic” now means that each $1$-simplex does not have any boundary point in $Z(S)$. Furthermore, we declare two elements of $C_1(S)$ to be equivalent if they represent the same current in $S$.

The above chains have coefficients in a local coefficient system given by the “orientation sheaf” of the bundle of unstable manifolds of critical manifolds of $S$. This local coefficient system is isomorphic to the constant local coefficient system ${\mathbb Z}$ if and only if the bundle of unstable manifolds of critical manifolds of $S$ is orientable.

The restriction of the differential on singular homology with local coefficients now defines a differential $\partial^+:C_1(S)\to C_0(S)$ whose homology is the usual homology of $S$, with coefficients in the above local coefficient system.

If $S'$ is another critical submanifold of the same index $i$, then there is also a map $\partial^+:C_1(S)\to C_0(S')$ counting gradient flow lines from $S$ to $S'$, modulo ${\mathbb R}$ translation as usual. More precisely, if $\sigma:[0,1]\to S$ is a generic chain (together with orientations of the unstable manifolds of the critical points it hits), then $\partial^+\sigma$ is a sum, over pairs $(t,\gamma)$ where $t\in[0,1]$ and $\gamma$ is a flow line from $\gamma(t)$ to some point $y\in S'$, of $y$, with an appropriate orientation.

If $S'$ is a critical submanifold of index $i-1$, then there are similar maps $\check{\partial}:C_0(S)\to C_0(S')$ and $\hat{\partial}:C_1(S)\to C_1(S')$. The map $\check{\partial}$ counts gradient flow lines from a given point on $S$ to $S'$, and the map $\hat{\partial}$ measures gradient flow lines from a given 1-simplex on $S$ to $S'$. Finally, if $S'$ is a critical submanifold of index $i-2$, then there is a map $\partial^-:C_1(S)\to C_0(S')$ counting gradient flow lines from a given $1$-simplex on $S$ to $S'$.

We now define a chain complex $C_*=\bigoplus_S(C_0(S)\oplus C_1(S))$ with differential $\partial = \partial^+ + \check{\partial} + \hat{\partial} + \partial^-$. A standard argument shows that $\partial^2=0$ and the homology of this chain complex agrees with the ordinary homology of $M$.

I think one can also show directly (without passing through the singular homology of $M$) that the homology of this complex agrees with the homology of the “cascade” version of Morse-Bott theory. To do so, one can filter the complex $C_*$ by the “action”, i.e. by the value of the Morse-Bott function $f$. The filtered complex then gives rise to a spectral sequence which computes its homology. I believe that if one thinks very carefully about what the spectral sequence of a filtered complex is doing, then one will recover the cascade picture. However I have not done this exercise. (Has someone done this?)

3. What’s next.

We now have all the ingredients in place to define and play with an analogue of the Bourgeois-Oancea construction for contact forms with no contractible Reeb orbits. However I am out of time to write at the moment, so I will explain this in the next post.