## Local contact homology with integer coefficients II

Continuing the previous post, I will now (1) outline how, for a nondegenerate contact form with no contractible Reeb orbits, one can define a version of contact homology with integer coefficients, analogously to the Bourgeois-Oancea definition of $S^1$-equivariant symplectic homology (2) explain why the tensor product of this with ${\mathbb Q}$ recovers the cylindrical contact homology when the latter is defined, and (3) compute some examples to see how the integer coefficient version is actually an invariant of contact structures that admit nondegenerate contact forms without contractible Reeb orbits.

1. Non-equivariant contact homology revisited.

Let $\lambda$ be a nondegenerate contact form with no contractible Reeb orbits on a manifold $Y$ which is either compact, or for which Gromov compactness is applicable.

Let $X$ denote the space of one-parameter families $\{J_t\}_{t\in S^1}$ of almost complex structures $J_t$ on ${\mathbb R}\times Y$ satisfying the usual conditions, namely they are ${\mathbb R}$-invariant, send the derivative of the ${\mathbb R}$ coordinate to the Reeb vector field, and send the contact structure to itself, compatibly with $d\lambda$. The space $X$ is contractible.

For generic $\{J_t\}_{t\in S^1}$ in $X$, we define a version of “non-equivariant contact homology”, using the chain version of Morse-Bott theory recalled in the previous post, as follows. If $\gamma$ is a possibly multiply covered Reeb orbit, let $\overline{\gamma}$ denote the underlying embedded Reeb orbit. We now define a chain complex $(C_*,\partial)$ over ${\mathbb Z}$ as follows. The chain group $C_*$ is the direct sum over all (possibly bad) Reeb orbits $\gamma$ of $C_0(\overline{\gamma})\oplus C_1(\overline{\gamma})$, where $C_0(\overline{\gamma})$ denotes the generic $0$-chains in $\overline{\gamma}$ (namely those $0$-chains avoiding a certain finite set of “non-generic” points), $C_1(\overline{\gamma})$ denotes the generic $1$-chains in $\overline{\gamma}$ (namely those whose boundaries avoid the same set of nongeneric points), regarded as currents, and both $C_0(\overline{\gamma})$ and $C_0(\overline{\gamma})$ have coefficients in a certain local coefficient system (arising from coherent orientations), which is isomorphic to the constant local coefficient system ${\mathbb Z}$ when $\gamma$ is good, and the twisted local coefficient system (that is locally isomorphic to ${\mathbb Z})$ when $\gamma$ is bad.

To prepare to define the differential, if $\alpha$ and $\beta$ are two distinct Reeb orbits, let ${\mathcal M}(\alpha,\beta)$ denote the set of maps $u:{\mathbb R}\times S^1\to{\mathbb R}\times Y$ such that $\partial_su+J_t\partial_tu=0$, and $u$ is asymptotic to $\alpha$ and $\beta$ as $s$ goes to $+\infty$ and $-\infty$ respectively. We mod out by ${\mathbb R}$ translation in the domain.

If $J$ is generic, then each component of ${\mathcal M}(\alpha,\beta)$ is a smooth manifold of dimension $CZ(\alpha) - CZ(\beta) + 1$, where $CZ$ denotes the Conley-Zehnder index, computed with respect to trivializations of the contact structure over $\alpha$ and $\beta$ that are compatible with respect to the cylinders $u$. (To simplify the notation below, I will ignore this last subtlety and pretend that there is a single trivialization of the contact structure over each Reeb orbit which makes this dimension formula true. This is the case in the local contact homology situtation that we care about.)

There is an evaluation map $e_+:{\mathcal M}(\alpha,\beta)\to\overline{\alpha}$ sending $u$ to $\lim_{s\to\infty}\pi_Y(u(s,0))$. Likewise there is an evaluation map $e_-:{\mathcal M}(\alpha,\beta)\to\overline{\beta}$. Also ${\mathbb R}$ acts on ${\mathbb M}(\alpha,\beta)$ by translating the ${\mathbb R}$ coordinate in the target, and the evaluation maps are invariant under this action.

The differential now has four components, denoted by $\partial^+$, $\check{\partial}$, $\hat{\partial}$, and $\partial^-$. The component $\partial^+$ maps from $C_1(\overline{\alpha})$ to $C_0(\overline{\beta})$ when $CZ(\alpha)=CZ(\beta)$. When $\alpha=\beta$, this is just the singular homology differential $C_1(\overline{\alpha})\to C_0(\alpha)$ (with local coefficients).  If $\alpha\neq\beta$, and if $\sigma$ is a generic $1$-chain in $\overline{\alpha}$, then the $\beta$ component of $\partial^+\sigma$ is

$(\partial^+\sigma)_\beta = \sum_{u\in{\mathcal M}(\alpha,\beta)/{\mathbb R}}\epsilon(u) (e_+(u)\cdot\sigma) e_-(u).$

Here $\epsilon(u)$ is a sign (with respect to the local coefficient systems on $\alpha$ and $\beta$), and $e_+(u)\cdot\sigma$ denotes the intersection number of $e_+(u)$ with $\sigma$ in $\overline{\alpha}$. The component $\check{\partial}$ maps from $C_0(\overline{\alpha})$ to $C_0(\overline{\beta})$ when $CZ(\beta) = CZ(\alpha)-1$. If $p$ is a generic $0$-chain in $\overline{\alpha}$, then the $\beta$ component of $\check{\partial}p$ is

$(\check{\partial} p)_\beta = \sum_{u\in {\mathcal M}(\alpha,\beta)/{\mathbb R} | e_+(u)=p} \epsilon(u) e_-(u).$

The component $\hat{\partial}$ maps from $C_1(\overline{\alpha})$ to $C_1(\overline{\beta})$ when $CZ(\beta) = CZ(\alpha) - 1$. If $\sigma$ is a generic $1$-chain in $\overline{\alpha}$, then the $\beta$ component of $\hat{\partial}\sigma$ is

$(\hat{\partial}\sigma)_\beta = (e_-)_*[\overline{\{u\in {\mathcal M}(\alpha,\beta)/{\mathbb R} | e_+(u)\in\sigma\}}].$

Here the bar on the right hand side indicates the compactification of the indicated subset of ${\mathcal M}(\alpha,\beta)/{\mathbb R}$ obtained by adding one boundary point for each end, the square brackets indicate a “fundamental chain” on this moduli space (with respect to the local coefficient systems), and $(e_-)_*$ indicates the pushforward of this chain via $e_-$. Finally $\partial^-$ similarly maps from $C_0(\overline{\alpha})$ to $C_1(\overline{\beta})$ when $CZ(\beta) = CZ(\alpha) - 2$.

Let me emphasize that all of the above components of the differential are counting actual homolorphic maps, and not “cascades”.

Now I claim that $\partial^2=0$. I further claim that the Morse-Bott gluing analysis needed to prove this is exactly the same as the analysis needed to justify the “cascade” version of non-equivariant contact homology that I outlined in this earlier post (similarly to the paper by Bourgeois-Oancea in Duke). Finally, I claim that the homologies of these two chain complexes are canonically isomorphic, although I still haven’t done this exercise. Let us denote this homology by $SH(\lambda)$, since it is an analogue of symplectic homology.

2. Local contact homology with integer coefficients

Continuing with the above setup, observe that the space $X$ has an $S^1$-action by rotating the $S^1$-family of almost complex structures. That is, if $\{J_t\}_{t\in S^1}\in X$ and $t_0\in S^1$, then $t_0\cdot\{J_t\} = \{J_{t+t_0}\}$. The holomorphic maps for $\{J_t\}$ agree with the holomorphic maps for $\{J_{t+t_0}\}$, after rotating the $S^1$ coordinate on the domain. Thus there is a canonical bijection between the moduli spaces ${\mathcal M}(\alpha,\beta)$ for the two elements of $X$, which rotates the evaluation maps $e_+$ and $e_-$. So there is a canonical isomorpism between the two chain complexes, which for each Reeb orbit $\gamma$ rotates the chains in $C_0(\overline{\gamma})$ and $C_1(\overline{\gamma})$ at speed $d(\gamma)$ where $d(\gamma)$ denotes the covering multiplicity of $\gamma$ over $\overline{\gamma}$.

We are now in a position to define $S^1$-equivariant Floer theory a la Bourgeois-Oancea, as outlined in the previous post. We obtain a chain complex $C_*^{S^1}$ over ${\mathbb Z}$, and its homology is the promised “local contact homology with integer coefficients”. Let us denote this homology by $SH^{S^1}(\lambda)$, since it is analogous to the $S^1$-equivariant symplectic homology of Bourgeois-Oancea. Similar analysis shows that this is an invariant of contact structures that admit contact forms without contractible Reeb orbits.

3. Relation with cylindrical contact homology

In the situation above, suppose that one can choose $\{J_t\}$ to be independent of $t$ and equal to a fixed almost complex structure $J$ on ${\mathbb R}\times Y$, so that the transversality needed above (or equivalently, the transversality needed to define cylindrical contact homology) still holds. This is the case for example when $\dim(Y)=3$. Let us denote the cylindrical contact homology by $CH(\gamma)$. I will now compute $SH^{S^1}$ with rational coefficients and show that it agrees with $CH(\gamma)$. I think that Bourgeois-Oancea have very similar arguments in their paper but I haven’t read all of it. Anyway we are now going to see why you really need rational coefficients to recover cylindrical contact homology.

Recall that to define the $S^1$-equivariant theory, we need to define a suitably generic $S^1$-equivariant map $ES^1\to X$. In the present situation, the constant map $ES^1\to \{J\}$ is suitably generic in this sense.

I claim that the BV operator $\partial_1$ only has nontrivial components from $C_0(\overline{\gamma})$ to $C_1(\overline{\gamma})$ for the same Reeb orbit $\gamma$. What this map does is input a $0$-chain and sweep it all the way around the orbit $\gamma$ to obtain a $1$-chain. Consequently, if $\gamma$ is good, then $\partial_1$ of a point in $\overline{\gamma}$ is $d$ times the fundamental class of $\overline{\gamma}$, where $d$ denotes the covering multiplicity. If $\gamma$ is bad, then $\partial_1$ of a point in $\overline{\gamma}$ is zero.

At first glance one might think that the BV operator is zero for index reasons. However it is as above because when you act on $X$ by $S^1$, even though the point $\{J_t\equiv J\}$ is fixed, you have to rotate the chains to get the canonical isomorphisms of chain complexes that we are using. Anyway, index considerations do show that the higher differentials $\partial_i$ for $i>1$ are zero.

By the algebraic exercise which I didn’t do (which however is easier in this special case), we can now replace everything by a “cascade” picture. So our chain complex has generators $u^k\check{\alpha}$ and $u^k\hat{\alpha}$ for each nonnegative integer $k$ and each Reeb orbit $\alpha$. The differential sends $u^k\hat{\alpha}$ to $u^k\partial_0\hat{\alpha}$, it sends $\check{\alpha}\to\partial_0\check{\alpha}$, and if $k>0$ then it sends $u^k\check{\alpha}$ to $u^k\partial_0\check{\alpha} + u^{k-1}\hat{\alpha}$. Here $\partial_0$ is the differential on nonequivariant contact homology in the cascade version, which I described for example in this earlier post.

The differential now splits as a sum of two commuting differentials: the sum of the terms involving $\partial_0$, and the sum of the terms involving $\partial_1$. To compute the homology of the chain complex, we can first compute the homology of the latter differential, and then compute the homology of the former differential acting on that.

When we compute the homology of the part of the differential involving $\partial_1$, this kills $u^k\hat{\alpha}$ whenever $k\ge 0$ and $\hat{\alpha}$ is good, and it also kills $u^k\check{\alpha}$ whenever $k>0$ and $\hat{\alpha}$ is good. Here it is critical that we are using rational coefficients, because $\partial_1$ multiplies by the covering multiplicities of Reeb orbits. After we have passed to this homology, all that is left is generators of the form $\check{\alpha}$, as well as, for each bad orbit $\beta$, the generators $u^k\hat{\beta}$ for $k\ge 0$ and $u^k\check{\beta}$ for $k>0$.

We now have to compute the homology of $\partial_0$ acting on the above. It follows from the discussion in this previous post that $\partial_0$ sends $u^k\hat{\beta}$ to $2u^k\check{\beta}$ (possibly plus some terms of lower action) when $\beta$ is bad, and it sends $\check{\alpha}$ to $\delta\kappa\check{\alpha}$ when $\alpha$ is good. Again, since we are using rational coefficients, all the bad orbits are killed, and the homology is just the homology of $\delta\kappa$ acting between good Reeb orbits, which is the cylindrical contact homology.

4. Examples with integer coefficients.

So here is where we stand now. We have a homology $SH^{S^1}(\lambda)$ which is defined over ${\mathbb Z}$ whenever $\lambda$ is a nondegenerate contact form with no contractible Reeb orbits, and this depends only on the contact structure (assuming that it admits such a contact form). When furthermore $CH(\lambda)$ is defined, it agrees with $SH^{S^1}\otimes{\mathbb Q}$. However $CH(\lambda)$ is not an invariant with ${\mathbb Z}$ coeffficients. (The differential is actually defined over ${\mathbb Z}$, with two possible conventions $\delta\kappa$ and $\kappa\delta$. However in general, if one uses ${\mathbb Z}$ coefficients, then the homologies with these two conventions are neither invariant nor equal to each other.)

Can this be right? Let’s investigate a couple of examples to see what is going on.

Let $e$ be a nondegenerate embedded elliptic Reeb orbit in a contact three-manifold. I now want to consider the contact homology just in a tubular neighborhood of $e$, for Reeb orbits which have winding number two around this neighborhood. We will then consider some bifurcations and check that the integer coefficient theory $SH^{S^1}$ doesn’t change.

Let $E$ denote the double cover of $e$; this is a good orbit. Before any bifurcations happen, we just have the generators $u^k\check{E}$ and $u^k\hat{E}$ for all $k\ge 0$. The chain complex has a relative ${\mathbb Z}$ grading, and we can normalize it to an absolute ${\mathbb Z}$ grading so that $u^k\check{E}$ has grading $2k$, and $u^k\hat{E}$ has grading $2k+1$. By the discussion above, the differential for $SH^{S^1}$ sends $u^k\check{E}\to 2u^{k-1}\hat{E}$ for $k>0$, and all other generators to zero. Thus $SH_*^{S^1}$ is ${\mathbb Z}$ in grading $0$ (generated by $\check{E}$), ${\mathbb Z}/2$ in all positive odd gradings (generated by $u^k\hat{E}$), and zero in all other gradings.

Now suppose that $e$ undergoes a period-doubling bifurcation, so that it splits into a new elliptic orbit $e_2$ of period two, together with a negative hyperbolic orbit $h$ of period one. Let $H$ denote the double cover of $h$; this is a bad orbit. The chain complex for $SH^{S^1}$ now has generators $u^k\check{e}_2$, $u^k\hat{e}_2$, $u^k\check{H}$, and $u^k\hat{H}$, for each $k\ge 0$. There are different versions of the period-doubling bifurcation, but there is one version in which $\check{e}_2$ has grading zero and $\check{H}$ has grading one (and then changing a check to a hat increases the grading by one, and multiplying by $u$ increases the grading by two). I claim that after this bifurcation, $\partial_0$ sends $\hat{H}$ to $\hat{e}_2 + 2\check{H}$ and all other generators to zero. I can try to justify that later. Anyway if you believe this, then the homology of the chain complex is still ${\mathbb Z}$ in grading zero (generated by $\check{e}_2$), ${\mathbb Z}/2$ in every odd positive grading (generated by $u^k\check{H}$; note that $2u^k\check{H}=\partial(u^k\hat{H}-u^{k+1}\check{e}_2$), and zero in all other gradings.

To sum up: before the bifurcation, there was just an elliptic orbit, but the homology had a bunch of $2$-torsion. After the bifurcation, a bad orbit appeared, and now the $2$-torsion comes from the bad orbit. We can think of the $2$-torsion before the bifurcation as measuring the potential for a bad orbit to appear in a bifurcation.

Anyway, I did some other examples and it all seems to work, but I think I have written enough for now. This all seems kind of bizarre, so if you think that I have gotten it wrong, or just want clarification, please ask in the comments.