Symplectic embeddings of polydisks into polydisks

If I am not mistaken, the methods in the previous post (plus a conjecture about the ECH chain complex of perturbed boundaries of convex toric domains) can be used to show that if a,b,c are real numbers with a,b\ge 1 and c>0, and if P(a,1) symplectically embeds into P(bc,c), and if a\le 2b, then a\le bc. In other words, if you include one four-dimensional polydisk into another, such that the long sides are the same, and the short side of the domain is at least half the short side of the target, then this symplectic embedding is optimal.

The conjecture needed is that in the ECH chain complex of the (perturbed) boundary of a convex toric domain, a generator with only elliptic orbits represents a nontrivial homology class. (This would follow from a conjectural description of the differential in terms of rounding corners.) Without this conjecture, one can still prove a version of the above theorem in which the hypothesis a\le 2b is strengthened somewhat. (When b=1 you can still just assume a\le 2.)

I’m working on writing this up cleanly.

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Hind-Lisi and more

References for this post:

[HL] R. Hind and S. Lisi, Symplectic embeddings of polydisks

[CONCAVE] K. Choi, D. Cristofaro-Gardiner, D. Frenkel, M. Hutchings, and V. Ramos, Symplectic embeddings into four-dimensional concave toric domains

[T3] M. Hutchings and M. Sullivan, Rounding corners of polygons and the ECH of T^3

[CC2] M. Hutchings and C. H. Taubes, Proof of the Arnold chord conjecture in three dimensions II

[BUDAPEST] M. Hutchings, Lecture notes on ECH

0. Introduction

Hind and Lisi [HL] recently proved that if the polydisk P(2,1) symplectically embeds into the ball B(b) then b\ge 3. (I’ll review what this notation means below.) This bound is optimal, because the polydisk P(a,1) symplectically embeds into the ball B(1+a) by inclusion. Also, the 2 is as large as possible for this sort of result, because whenever a>2, according to Schlenk’s book, one can use “symplectic folding” to symplectically embed P(a,1) into B(b) for certain b < 1+a.

So this result of Hind-Lisi is very nice. For a while I was wondering if one can reprove it using ECH methods; previously I couldn’t, but I was missing a couple of details, and I think I now know how to do this. Moreover the new method gives further obstructions to symplectically embedding polydisks into ellipsoids, and more generally symplectic embeddings of “convex toric domains”, which go beyond the obstructions coming from ECH capacities. Here are a couple of sample results.

First, we can extend the Hind-Lisi result to give some seemingly new obstructions to symplectic embeddings of skinnier polydisks into balls (I have no idea whether these are sharp).

Theorem 1. Let a\in[1,8] and suppose that P(a,1) symplectically embeds into B(b). Then:

  • If 1\le a\le 2 then b\ge 1+a.
  • If 2\le a\le 4 then b\ge (10+a)/4.
  • If 4\le a \le 9/2 then b\ge 7/2.
  • If 9/2 \le a \le 7 then b\ge (13+a)/5.
  • If 7\le a\le 8 then b\ge 4.

Of course the third and fifth lines follow trivially from the lines above them. I have written the fifth line because the inequality b\ge 4 is significant until a=8, at which point it ties the volume constraint b\ge\sqrt{2a}.

We can also extend the Hind-Lisi result in a different direction by replacing balls with integral ellipsoids:

Theorem 2. Let a\in[1,2] and let b be a positive integer. Then P(a,1) symplectically embeds into E(bc,c) if and only if bc \le a + b.

Note that P(a,1) includes into E(bc,c) whenever bc \ge a + b. Thus Theorem 2 asserts that the inclusion is sharp when b is an integer and 1\le a\le 2. It is unclear why integrality of b should be relevant here, but the method I am using works better for rational ellipsoids and best for integral ellipsoids.

I would now like to give an introduction to all of this, with a more formal writeup to come later.

1. Convex toric domains

Recall that if \Omega is a domain in the (closed) first quadrant of the plane, we define the “toric domain”

X_\Omega = \{z\in{\mathbb C}^2 \mid (\pi|z_1|^2,\pi|z_2|^2)\in\Omega\}.

Let us now define a “convex toric domain” to be a domain X_\Omega where \Omega=\{(x,y)\mid 0\le x \le A, 0 \le y \le f(x)\} where f is a convex function such that f(0)=A and f'(0)\le 0. (Sorry for the confusing terminology, but the region below the graph of a concave function is convex!) For example, the polydisk

P(a,b) = \{z\in{\mathbb C}^2\mid \pi|z_1|^2\le a, \pi|z_2|^2\le b\}

is a convex toric domain where A=a and f\equiv b so that \Omega is a rectangle. Also, the ellipsoid

E(a,b) = \{z\in{\mathbb C}^2\mid \pi|z_1|^2/a + \pi|z_2|^2/b\}

is a convex toric domain where A=a and f is linear so that \Omega is a triangle.

Convex toric domains should be contrasted with the “concave toric domains” considered in [CONCAVE]. For a concave toric domain, f is convex and f(A)=0. Note that an ellipsoid is both a convex toric domain and a concave toric domain; nothing else is.

Recall that ECH capacities give obstructions to symplectically embedding any symplectic four-manifold (typically with boundary) into another. McDuff showed that ECH capacities give a sharp obstruction to symplectically embedding one (four-dimensional) ellipsoid into another. More generally, at the Simons Center in June, Dan Cristofaro-Gardiner presented a proof of the remarkable result that ECH capacities give a sharp obstruction to symplectically embedding any concave toric domain into any convex toric domain.

On the other hand, ECH capacities do not give very good obstructions to symplectically embedding a convex toric domain into a concave toric domain, e.g. a polydisk into an ellipsoid. For example they only imply that if P(2,1) symplectically embeds into B(b)=E(b,b) then b\ge 2, although we actually want b\ge 3.

2. Introduction to the strategy.

To improve on the obstructions given by ECH capacities, the idea is a follows. First of all, the obstruction given by ECH capacities works as follows: if one symplectic four-manifold (with contact boundary) embeds into another, then we get a strong symplectic cobordism between the two contact manifolds, which induces a cobordism map on ECH. Whenever this cobordism map is nontrivial, a (possibly broken) holomorphic curve in the (completed) cobordism must exist, and the fact that this holomorphic curve has positive symplectic area gives us an inequality.

Now in general if we do not know very much about the holomorphic curve that will exist, then we will not get a very sharp inequality. The idea is to get restrictions on which holomorphic curves can exist in the cobordism, and thus improve the inequality. To do so we will use two facts:

(a) For the symplectic cobordism given by a symplectic embedding of one convex toric domain into another (the target of the embedding can also be more general), the holomorphic currents that one wants to count to define the ECH cobordism map are better behaved than usual (in particular there are no negative ECH index multiple covers).

(b) The quantity J_0 (a variant on the ECH index) bounds the topological complexity of holomorphic curves (when they are not multiply covered) and thus can be used to show that certain kinds of holomorphic curves cannot contribute to the ECH cobordism map (e.g. because the genus of a holomophic curve cannot be negative).

That was a bit vague, so let us now explain some details.

3. Convex ECH generators

Let X_\Omega be a convex toric domain and let Y denote its boundary. If Y is smooth, then it has a natural contact form, given by the restriction of

\lambda=\frac{1}{2}\sum_{i=1}^2(x_idy_i - y_idx_i).

The claim is now that one can perturb X_\Omega so that Y is smooth, \lambda is nondegenerate, and the generators of the ECH chain complex (up to large symplectic action) have the following combinatorial description.

Definition 3. A convex generator is a polygonal (i.e. comprised of finitely many line segments) path \Lambda in the plane such that:

  • \Lambda starts at (0,y) and ends at (x,0) where x and y are nonnegative integers.
  • The vertices of \Lambda (the points where it changes direction) are at lattice points.
  • The edges of \Lambda (i.e. line segments between vertices) have nonpositive slope (possibly -\infty), and as one moves to the right, the slope of each edge is less than that of the preceding edge.
  • Each edge of \Lambda is labeled `e’ or `h’.
  • Horizontal and vertical edges can only be labeled `e’.

If \Lambda is a convex generator, define its combinatorial ECH index by

I(\Lambda) = 2(L(\Lambda)-1) - h(\Lambda)

where L(\Lambda) is the number of lattice points in the region bounded by \Lambda and the axes (including lattice points on the boundary), and h(\Lambda) is the number of edges labeled `h’.

So far, the definition of a “convex generator” \Lambda and its combinatorial ECH index do not depend on the convex domain X_\Omega. However there is also a combinatorial notion of symplectic action of \Lambda, which does depend on \Omega; it is given by

A_\Omega(\Lambda) = \sum_{e\in Edges(\Lambda)}v_e\times p_e.

Here v_e is the vector determined by e, that is the difference between the final and initial endpoints; \times denotes the cross product of vectors; and p_e denotes a point on \partial\Omega such that a tangent line to \partial\Omega at p_e is parallel to e. (If p_e is a corner of \partial\Omega, this means that \Omega is on the lower left of the line through p_e parallel to e.)

The more precise statement is that for any L>0 large and \epsilon>0 small, one can perturb Y by a perturbation of size less than \epsilon in an appropriate sense, such that there is a bijection between ECH generators of (actual) action less than L with convex generators of (combinatorial) action less than L, such that the actual ECH index agrees with the combinatorial ECH index, and the action agrees with the combinatorial action up to \epsilon error. This is analogous to Lemma 3.3 in [CONCAVE] and proved similarly. (If you actually read this proof and try to compare, note that here we are first perturbing \Omega so that its boundary is nearly horizontal at the beginning and nearly vertical at the end.)

4. The chain complex

Now let us consider the ECH of the boundary of a convex toric domain with {\mathbb Z}/2 coefficients. We know that the homology of the chain complex has one generator in each nonnegative even degree. What do the homology generators actually look like? And before discussing this question we should maybe first ask if there is a combinatorial formula for the differential.

I conjecture that if \Lambda is any path in which all edges are labeled `e’, then \Lambda is a cycle which represents the homology generator of grading I(\Lambda). This would follow from a more general conjecture about the differential. Namely, extend each convex path to a closed polygon by appending “virtual” edges along the axes, with all of the virtual edges labeled `e’. The conjecture is then that the differential acts on these extended generators by rounding corners and locally losing one `h’ (possibly plus some relatively unimportant “double rounding” terms), just as in [T3]. Keon Choi may know how to prove this, related to his thesis work.

Anyway, to study symplectic embeddings into ellipsoids, we don’t need to know all of that. We just need to know the following fact, which can be proved more quickly:

Lemma 4. Suppose \Omega is the triangle with vertices (0,0), (a,0), and (0,b), so that X_\Omega = E(a,b). Let P be a line in the plane of slope -b/a which passes through at least one lattice point in the first quadrant. Let \Lambda be the maximal convex generator to the left of P, with all edges labeled `e’. Then \Lambda is a cycle which represents a nontrivial homology class in the ECH of (perturbed) \partial E(a,b).

Proof. Let k=I(\Lambda)/2. One can check that \Lambda uniquely minimizes the combinatorial action among generators of index 2k, and its combinatorial action agrees with the kth ECH capacity of E(a,b). This is similar to Example 1.23 in [CONCAVE]. The claim follows immediately (because the degree 2k homology class is represented by a cycle which is a sum of generators with action less than or equal to the kth ECH capacity, and \Lambda is the only such generator.)

5. The cobordism map

Suppose now that a convex toric domain X_\Omega with boundary Y symplectically embeds into (the interior of) another convex toric domain X_{\Omega'} with boundary Y'. Then X_\Omega minus the interior of the image of X_{\Omega'} is a strong symplectic cobordism from Y' to Y. We further get a strong symplectic cobordism between the perturbed contact forms as above. This induces a map ECH(Y')\to ECH(Y), which is an isomorphism, because the cobordism is diffeomorphic to the product [0,1]\times S^3. We know from my work with Taubes [CC2] using Seiberg-Witten theory that given an appropriate almost complex structure J on the cobordism, this ECH cobordism map is induced by a chain map, such that whenever a coefficient of the chain map is nonzero, there is a (possibly broken) J-holomorphic current with ECH index zero between the corresponding ECH generators.

In particular, the chain map decreases the symplectic action. All of our symplectic embedding obstructions come from this fact.

Now the key observation which gives us some control over this chain map is:

Lemma 5. The chain map (up to any given symplectic action L) can be chosen so that it only counts I=0 unbroken J-holomophic currents, such that the components are disjoint and have I=0, and the somewhere injective curve underlying each component is embedded and also has I=0.

To prove this, one needs to do a calculation, using the special nature of the contact forms in question, to show that if J is generic, then multiply covered curves with negative ECH index cannot arise here. I will explain this some other time. I will just remark for now that this lemma also holds for symplectic embeddings of convex or concave toric domains into concave toric domains. (It doesn’t work for concave toric domains into convex toric domains, but this is the case where Dan showed that ECH capacities already give a sharp symplectic embedding obstruction.)

6. Controlling topological complexity

For the above cobordism arising from a symplectic embedding of one convex toric domain into another, we now want to control the topological complexity of J-holomorphic currents with ECH index index I=0. To do so, we use the quantity J_0 (this is an integer similar to the ECH index, not an almost complex structure), see e.g. Section 5.2 of the lecture notes [BUDAPEST].

In our situation, J_0 can be computed combinatorially as follows. If \Lambda is a convex generator, going from (0,y) to (x,0), then

J_0(\Lambda) = I(\Lambda) - 2x - 2y - e(\Lambda).

Here e(\Lambda) denotes the number of edges of \Lambda that are labeled `e’, plus the number of edges that are labeled `h’ and contain at least one interior lattice point. (In other words, e(\Lambda) is the number of elliptic orbits in the orbit set corresponding to \Lambda.)

Now the significance of J_0 is that if C is an I=0 holomophic curve from \Lambda' to \Lambda which does not have any multiply covered components, then

J_0(\Lambda') - J_0(\Lambda) = -\chi(C) + O(C).

Here O(C) is the sum, over all Reeb orbits in \Lambda or \Lambda', of the number of ends of C at covers of that orbit minus one. This is proved as in Exercise 5.9 of [BUDAPEST].

We can be a bit more specific as follows. By the partition conditions, we know that all positive ends of C are at simple orbits, and C can have at most one negative end at covers of any given orbit. To make use of this fact, let n(\Lambda) denote the total number of Reeb orbits in \Lambda, namely the number of edges, plus the number of `h’ edges with at least one interior lattice point. Let m(\Lambda) denote the total multiplicity of all Reeb orbits in \Lambda, namely the sum over all edges of the number of interior lattice points minus one. Then

O(C) = m(\Lambda') - n(\Lambda').

Also, if C is connected with genus g, then

\chi(C) = 2 - 2g - m(\Lambda') - n(\Lambda).

Putting the above together, we get

2x(\Lambda) + 2y(\Lambda) = 2g - 2 + 2x(\Lambda') + 2y(\Lambda') + 2m(\Lambda') + h(\Lambda) - h(\Lambda').

Since g\ge 0 and h(\Lambda)\ge 0, we conclude the following:

Lemma 6. Suppose there exists an I=0, connected, embedded curve from \Lambda' to \Lambda. Suppose that all edges of \Lambda' are labeled `e’. Then

x(\Lambda) + y(\Lambda) \ge x(\Lambda') + y(\Lambda') + m(\Lambda') - 1.

(OK, I hate the writing style where the proof comes before the statement, sorry.)

Anyway, to prove Theorems 1 and 2, we will apply Lemma 6 repeatedly. Let’s see now how this works.

7. Proof of the first part of Theorem 1.

To start, let us prove the first part of Theorem 1, namely that if P(a,1) symplectically embeds into B(b), and if 1\le a \le 2, then b\ge 1 + a.

Assume that P(a,1) symplectically embeds into B(b), assume that a\ge 1, and assume that b < 1+a. We will show that b\ge 3.

Step 1. Consider the convex generator \Lambda' consisting of the straight line from (0,1) to (1,0), labeled `e’. Then I(\Lambda') = 4 and A_{B(b)}(\Lambda') = b. By Lemmas 4 and 5, there is an I=0 current from \Lambda' to some convex generator \Lambda with I(\Lambda)=4 and A_{P(a,1)}(\Lambda)\le b.

There are in fact only three convex generators \Lambda with I(\Lambda)=4: the horizontal line from (0,0) to (2,0), labeled `e’, which has action 2; the line from (0,1) to (1,0), labeled `e’, which has action 1+a; and the vertical line from (0,2) to (0,0), labeled `e’, which has action 2a. If there is a holomorphic curve from \Lambda' to either of the latter two generators, then we immediately get b\ge 1+a or b\ge 2a, which contradicts our assumption that b< 1+a. Thus I=0 curves from \Lambda' can only go to the horizontal line.

Step 2. Consider the convex generator \Lambda' consisting of the straight line from (0,2) to (2,0), labeled `e’. Then I(\Lambda')=10 and A_{B(b)}(\Lambda')=2b. By Lemmas 4 and 5, there is an I=0 current C from \Lambda' to some convex generator \Lambda with I(\Lambda)=10 and A_{P(a,1)}(\Lambda)\le 2b.

Now the current C must actually be a connected, embedded holomorphic curve. Why? Well, if C were disconnected, then each component would have I=0, and so by the end of Step 1, each component can only go to the horizontal line of length 2. Hence \Lambda is a horizontal line of length 4. But this only has I=8 which is not big enough. Likewise, if C is multiply covered, then again by the end of Step 1, the embedded curve underlying C can only map to the horizontal line of length 2, so again \Lambda is the horizontal line of length 4 which is impossible.

Since C is connected and embedded, we can apply Lemma 6 to conclude that

x(\Lambda) + y(\Lambda) \ge 5.

Now the action of \Lambda is given by

A_{P(a,1)}(\Lambda) = x(\Lambda) + a y(\Lambda).

If y(\Lambda)>0, then A_{P(a,1)}(\Lambda)\ge 4+a, so 4+a\le 2b. Combining this with our assumption b < 1+a then gives b > 3.

Assume now that 2b < 4+a. Then \Lambda' can only be the horizontal line of length 5, labeled `e’. In particular this gives b\ge 5/2. We need one more step to obtain b\ge 3.

Step 3. We are now assuming b<1+a and 2b<4+a. Consider the convex generator \Lambda' given by the straight line from (0,3) to (3,0), labeled `e’. This has I=18 and A_{B(b)}=3b. By Lemmas 4 and 5, there is a holomorphic current C from \Lambda' to a convex generator \Lambda with I(\Lambda) = 18 and A_{P(a,1)}(\Lambda) \le 3b.

By Steps 1 and 2, C is connected and embedded. (Otherwise \Lambda is a horizontal line of length at most 7, which only has I=14, which is too small.) Thus Lemma 6 applies to give

x(\Lambda) + y(\Lambda) \ge 8.

If y(\Lambda)>0, then A_{P(a,1)}(\Lambda)\ge 7+a, so 7+a\le 3b. Combining this with our assumption that b<1+a then gives b>3.

If y(\Lambda)=0, then \Lambda must be a horizontal line of length 9, since its ECH index is 18. Thus A_{P(a,1)}(\Lambda)=9, so b\ge 3.

QED

Which holomorphic curve is giving the obstruction? If you analyze the above proof, what it is saying is that if a\ge 1 and if P(a,1) symplectically embeds into B(b), then at least one of the following holomorphic curves must exist:

  • A pair of pants from the “diagonal” Reeb orbit of action b to the “horizontal” and “vertical” simple orbits for the polydisk, implying that b\ge 1+a.
  • A cylinder from \Lambda' as in Step 1 to the vertical line of length 2, implying that b\ge 2a.
  • A holomorphic curve with two positive ends at the “diagonal” orbit of action b, and negative ends with total action at least 4+a, implying that 2b\ge 4+a.
  • A holomorphic curve with three positive ends at the “diagonal” orbit of action b, and either one negative end with action 9, or with negative ends of total action at least 7+a.

Either way, we obtain b\ge 1+a if a\le 2. In any case, this looks quite different from the Hind-Lisi argument, which studies curves of “degree” d and takes the limit as d\to\infty.

8. Proof of the rest of Theorem 1.

To prove the rest of Theorem 1, we continue to play the above game. Namely:

Step 4. Assume a\ge 2, b<1+a, 2b<4+a, and 3b<7+a. (We can make this last assumption without loss of generality because (7+a)/3 > (10+a)/4 when a\ge 2.) Let \Lambda' be the straight line from (0,4) to (4,0), labeled `e’. This has I=28 and A_{B(b)}=4b. Then there is a holomorphic current C from \Lambda' to a convex generator \Lambda with I(\Lambda)=28 and A_{P(a,1)}(\Lambda) \le 4b. The above assumptions imply that C is connected and embedded, so we can apply Lemma 6 to deduce that x(\Lambda) + y(\Lambda) \ge 11. Thus

4b\ge \min(10+a,14).

This gives the lower bound on b in Theorem 1 for 2\le a \le 4.

Step 5. Now assume a\ge 4, b<1+a, 2b<4+a, 3b<7+a, and 4b<10+a. Let \Lambda' be the straight line from (0,5) to (5,0). Then we similarly obtain

5b\ge \min(13+a,20).

This gives the lower bound on b in Theorem 1 for 9/2\le a \le 7.

QED.

9. Proof of Theorem 2.

We now prove that if 1\le a\le 2 and b is a positive integer, and if P(a,1) symplectically embeds into E(bc,c), then bc \le a + b.

Theorem 1 covered the case where b=1.

To prove the rest, suppose that b>1 is an integer and that P(a,1) symplectically embeds into E(bc,c). Assume that a\ge 1 and a+b>bc. We need to show that a>2. This is actually a little simpler than the proof of Theorem 1 and only requires two steps.

Step 1. Let \Lambda' be the convex generator given by the straight line from (0,1) to (b,0), labeled `e’. Then I(\Lambda') = 2b and A_{E(bc,c)}(\Lambda') = bc. By Lemmas 4 and 5, there is a holomorphic current C from \Lambda' to a convex generator \Lambda with I(\Lambda) = 2b and A_{P(a,1)}(\Lambda) \le bc. Now C must be connected and embedded, since \Lambda' consists only of a single, simple Reeb orbit. Thus Lemma 6 implies that x(\Lambda) + y(\Lambda) \ge b+1. If y(\Lambda)>0 then A_{P(a,1)}(\Lambda) = x(\Lambda) + ay(\Lambda) \ge a+b, contradicting our assumption that a+b>bc. The conclusion is that \Lambda can only be the horizontal line of length b+1, labeled `e’.

Step 2. Now let \Lambda' be the convex generator given by the straight line from (0,2) to (2b,0), labeled `e’. Then I(\Lambda')= 6b+4 and A_{E(bc,c)}(\Lambda')=2bc. By Lemmas 4 and 5, there is a holomorphic current C from \Lambda' to a convex generator \Lambda with I(\Lambda)= 6b+4 and A_{P(a,1)}(\Lambda)\le 2bc. Now C must be connected and embedded, since otherwise, by Step 1, \Lambda is a horizontal line of length 2b+2, which has index 4b+4<6b+4. Thus Lemma 6 applies to give

x(\Lambda) + y(\Lambda) \ge 2b+3.

If y(\Lambda)>0 then A_{P(a,1)}(\Lambda) = x(\Lambda) + a y(\Lambda) \ge 2b+2+a. Thus 2b+2+a\le 2bc. By assumption 2bc < 2a+2b, so 2 < a and we are done.

If on the other hand y(\Lambda)=0, then x(\Lambda)=3b+2, so 3b+2\le 2bc. By assumption 2bc<2a+2b, so we get b+2 < 2a. Since we are assuming b\ge 2, it follows that a>2 so we are done again.

QED

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Gluing a flow line to itself

References for this post:

[TORSION] “Reidemeister torsion in generalized Morse theory

[LEE] Yi-Jen Lee, “Reidemeister torsion in Floer-Novikov theory and counting pseudoholomorphic tori I, II”

[GLUING] With Cliff Taubes, “Gluing pseudoholomorphic curves along branched covered cylinders II

In this post, inspired by a question of Jiayong Li and Katrin Wehrheim (and other people have asked me about this earlier), I want to discuss (modulo most analytic details) what happens when you glue multiple copies of a flow line in circle-valued Morse theory to itself. This is a nice example of the obstruction bundle gluing technique which I learned from Cliff Taubes during our work on gluing along branched covers of trivial cylinders [GLUING]. It’s maybe a bit too simple an example since the obstruction bundle in this case is trivial, but it’s fun. (Did I just say that gluing is fun? I must be going nuts.) A similar story holds for holomorphic cylinders, but we will stick with Morse theory to keep things as simple as possible. Also, we will work over {\mathbb Z}/2 in order to avoid worrying about orientations.

1. Setup.

The setup is this. Given a circle-valued Morse function f on a closed manifold X and a generic metric g, one can define the Novikov complex of (f,g), with coefficients in the ring of Laurent series ({\mathbb Z}/2)((T)). The simplest way to describe this is as follows: The chain complex is the free ({\mathbb Z})/2)(T) module generated by the critical points. To define the differential, let \Sigma be a level set of f which does not contain any critical points. If p is an index i critical point, then

\partial p = \sum_q \sum_{k=0}^\infty n_k(p,q)T^kq

where the sum is over index i-1 critical points q, and n_k(p,q) denotes the mod 2 count of flow lines from p to q that cross \Sigma exactly k times. (There are better ways to say this, but I want to focus on gluing and not get confused about algebra at the same time.)

Suppose you want to prove that the Novikov homology depends only on the homotopy class of the map f:X\to S^1 and does not depend on the metric g. One way to do so is to consider a generic one-parameter family \{(f_t,g_t)\}_{t\in[-1,1]} of pairs of circle-valued functions and metrics. For some t, called “bifurcation times”, the pair (f_t,g_t) will fail to be Morse-Smale. You then want to understand how the chain complex changes as you pass through this value of t and see that the homology is invariant. I worked out this bifurcation theory long ago in [TORSION] (for other purposes involving Reidemeister torsion), sometimes “cheating” by using some finite dimensional techniques which make things a lot easier. Yi-Jen Lee subsequently extended this to Hamiltonian Floer theory in her papers [LEE].

However there is one bifurcation which I only analyzed indirectly using a “finite cylic cover trick”, and I believe that Yi-Jen used more or less the same trick. The goal of this post is to use obstruction bundle gluing to analyze more explicitly what is going on in this bifurcation.

The bifurcation in question is where at some time, say t=0, the function f_0 is Morse, but for the metric g_0, we have a flow line u_0 from an index i critical point q to itself. If there is a flow line u_+ from an index i+1 critical point p to q at time t=0, then we “know” that u_+ and u_0 can be glued to a flow line from p to q either before or after the bifurcation. The difficult part is that at time t=0 we also have, for each positive integer k, broken configurations consisting of the flow line u_+ from p to q, together with the concatenation of k copies of the flow line u_0 from q to itself. The question is, how many ways can we glue these to flow lines before and after the bifurcation? Same question for the concatenation of k copies of u_0 together with a flow line u_- from q to an index i-1 critical point r.

For definiteness suppose that u_0 crosses \Sigma once. Let \partial_- and \partial_+ denote the differentials before and after the bifurcation. (One has to be a bit careful to make sense of this since the set of bifurcation times is not necessarily discrete. See Section 2.1 of [TORSION].) Lemma 3.7 in [TORSION] shows that there is a power series A=1 + T + \cdots such that

\partial_+ = A_q \circ \partial_- \circ A_q^{-1},

where A_q denotes the operator which multiplies q by A and fixes all other critical points. This is all we need to know for most purposes, such as showing that Novikov homology is invariant, or proving invariance of the product of Reidemeister torsion and the zeta function as in [TORSION].

However maybe we want to know what this power series actually is. An extremely obscure reference in Remark 3.12 of [TORSION] implies that

A = (1+T)^{\pm1}.

I will now sketch a direct proof that A=(1+T)^{\pm1}, which should generalize to Floer theory, and in which we actually see how the gluing works.

2. Warmup: Gluing one copy of u_0.

To warm up, let us first consider the simpler problem of gluing u_+ to u_-. I will do this following the obstruction bundle gluing technique in [GLUING], which will then generalize to glue multiple copies of u_0. [GLUING] is not exactly light reading, and one could probably write a more readable account of the present simpler situation. However I am just going to explain the basic idea without going into analytic details such as which Banach space completions to choose, how to bound the error terms, etc.

As usual, we will first “preglue” u_+ and u_- and then try to perturb the preglued curve to an actual flow line. To preglue, there are two “gluing parameters”, which we can denote by R and t. Here R is a large positive real number, and t is a small positive or negative real number. To preglue, we translate u_+ up and u_0 down so that the total translation distance is R; and we change the time from 0 to t. We then patch the translated curves together using cutoff functions \beta_+ on u_+ and \beta_0 on u_0, which have derivatives of order 1/R.

Now let \psi_+ be a section of u_+^*TX, and let \psi_0 be a section of u_0^*TX. We can then perturb the preglued curve by pushing it off via the exponential of \beta_+\psi_+ + \beta_0\psi_0. Let F_t denote the equation to be a gradient flow line of (f_t,g_t). We can then write F_t of the perturbed preglued curve in the form

F_t(\beta_+\psi_+ + \beta_0\psi_0) = \beta_+\Theta_+(\psi_+,\psi_0,t) + \beta_0\Theta_0(\psi_+,\psi_0,t)

Here \Theta_+ = D_+\psi_+ + \cdots where D_+ denotes the deformation operator associated to u_+, and the additional terms arise from the patching and do not concern us right now. Likewise \Theta_0 = D_0\psi_0 + \cdots where D_0 is the deformation operator associated to u_0.

As in Section 5 of [GLUING], one can show using the contraction mapping theorem that for R large and t small, there is a unique pair (\psi_+,\psi_0) in appropriate Banach spaces such that \psi_+ is orthogonal to Ker(D_+), \psi_0 is orthogonal to Ker(D_0), \Theta_+ is orthogonal to Im(D_+), and \Theta_0 is orthogonal to Im(D_0).

Now since we are in a generic one-parameter family, D_+ is surjective and D_0 has a one-dimensional cokernel. Thus we have achieved \Theta_+=0, while \Theta_0 lives in a one-dimensional space. If in fact \Theta_0=0, then we have successfully glued. If not, then we can think of \Theta_0 as the obstruction to gluing. Let us denote this gluing obstruction by

o(R,t) \in Coker(D_0).

We can think of o as a section of the “obstruction bundle” over the space of gluing parameters (R,t) whose fiber is Coker(D_0) (it is a trivial bundle in this case).

As in Section 7 of [GLUING], one can show that there is a bijection between gluings of u_+ and u_-, and pairs (R,t) such that the obstruction o(R,t)=0. And for the purpose of understanding the differentials \partial_+ and \partial_-, we want to know: For fixed small t, how many solutions are there to the equation o(R,t)=0?

The next step is to approximate o(R,t). As in Section 8 of [GLUING], one has an approximation

o(R,t) \approx e^{-\lambda_-R}\eta + tF'(u_0)

Here \lambda_- is the smallest positive eigenvalue of the Hessian of q, \eta is determined by the “leading asymptotic coefficient” of the negative end of u_+, and F'(u_0) denotes the projection to Coker(D_0) of \frac{d}{ds}|_{s=0}F_s(u_0). What I mean by “approximation” is that for fixed small t, the left and right hand side have the same count of zeroes (on the set of sufficiently large R). Note that since we are in a generic one-parameter family, we have F'(u_0)\neq 0. Thus, to make things a little simpler, we can choose an identification Coker(D_0)\simeq {\mathbb R} such that F'(u_0) corresponds to 1. Then we can rewrite the above approximation as

o(R,t) \approx e^{-\lambda_- R}\eta + t

where now \eta is a real number determined by the asymptotics of u_+. This real number is nonzero assuming suitable genericity.

Thus, for fixed small t we want to count solutions R to the equation

e^{-\lambda_-R}\eta + t = 0.

We see immediately that if t has the same sign as \eta, then there is no solution, while if t and \eta have opposite signs then there is one solution.

In conclusion, u_+ and u_0 glue to one flow line. Whether this glued flow line exists before or after the bifurcation depends on the asymptotics of the negative end of u_+.

3. Gluing two copies of u_0.

Now let’s try to glue u_+ to two copies of u_0. In this situation we have three gluing parameters R_1, R_2, and t. As before, t is a small positive or negative real number by which we shift time. Furthermore, R_1 is the translation distance between u_+ and the upper copy of u_0, while R_2 is the translation distance between the two copies of u_0. We now have to count solutions to the equations

e^{-\lambda_- R_1}\eta + e^{-\lambda_+R_2}\eta_+ + t=0,

e^{-\lambda_- R_2}\eta_- + t=0.

Here \eta and \lambda_- are as before. In addition, \lambda_+ is minus the smallest negative eigenvalue of the Hessian of q, while \eta_+ and \eta_- are determined by the “leading asymptotic coefficients” associated to the positive and negative ends of u_0, respectively. Under suitable genericity assumptions, \lambda_+\neq \lambda_-.

For a fixed small t, there is exactly one pair (R_1,R_2) satisfying these equations, provided that all of the following four conditions hold (and otherwise there is no solution):

(i) \eta_- and t have opposite signs.

(ii) If \eta_+ and t have the same sign, then \eta and t have opposite sign.

(iii) If \eta_+ and t have opposite signs, and if \eta and t have opposite signs, then \lambda_+ > \lambda_-.

(iv) If \eta_+ and t have opposite signs, and if \eta and t have the same sign, then \lambda_- < \lambda_+.

Here (i) is necessary and sufficient to solve the second equation; and then assuming this, (ii)—(iv) are necessary and sufficient to solve the first equation.

In summary, sometime you can glue for positive or negative time, depending on a handful of signs.

4. Gluing three or more copies of u_0.

It’s slightly simpler to glue u_+ to k\ge 3 copies of u_0. Now we have gluing parameters (R_1,\ldots,R_k,t), where R_1 denotes the translation distance between u_+ and the top copy of u_0, while for j=2,\ldots,k, the gluing parameter R_j is the translation distance between the (j-1)^{st} copy of u_0 and the j^{th} copy, counting downward. The (approximate) gluing equations are

e^{-\lambda_-R_1}\eta + e^{-\lambda_+R_2}\eta_+ + t = 0,

e^{-\lambda_-R_j}\eta_- + e^{-\lambda_+R_{j+1}}\eta_+ + t=0 for j=2,\ldots,k-1,

e^{-\lambda_-R_k}\eta_- + t = 0.

For fixed small t, these equations have one solution (R_1,\ldots,R_k) when all of the following three conditions hold, and otherwise they have no solution:

(i) \eta_- and t have opposite signs.

(ii) If \eta_+ and t have opposite signs then \lambda_+>\lambda_-.

(iii) \eta and t have opposite signs.

Here (i) is needed to solve the last equation, then (ii) is needed to solve the middle equations, and then (iii) is needed to solve the first equation.

5. Putting it all together.

Let us now count and organize what is glued.

Given u_+, define a power series f_+(u_+) = 1 + \sum_{k=1}^\infty a_kT^k where a_k is the number of gluings of u_+ to k copies of u_0 for t>0. Likewise define f_-(u_+) to be the power series whose k^{th} coefficient is the number of gluings of u_+ to k copies of u_0 for t<0.

In addition, if u_- is a flow line from q to an index i-1 critical point r, define g_+(u_-) and g_-(u_-) to be the power series whose k^{th} coefficient is the number of gluings of k copies of u_0 to u_- for t>0 and t<0 respectively.

If we did this right, then we should have

f_+(u_+) / f_-(u_+) = g_-(u_-) / g_+(u_-) = A

where A does not depend on u_+ or u_- and is the power series we are looking for, which we expect to equal (1+T)^{\pm 1}. Let’s see.

First let’s calculate f_+(u_+) and f_-(u_+). There are various cases depending on the signs of \eta, \eta_+, \eta_-, and \lambda_+-\lambda_-.

Without loss of generality \eta_-<0.

Suppose \eta_+>0. If \eta>0 then f_+=1 (because no gluing is possible for positive time) and f_-=1+T (because gluing for negative time is possible only for k=1). If \eta<0 then f_+=(1+T)^{-1}=1+T+T^2+\cdots (because for positive time gluing is possible for any k) and f_-=1. Either way, we have

f_+(u_+)/f_-(u_+) = (1+T)^{-1}.

If \eta_+<0 and \lambda_+>\lambda_- then f_+ and f_- are the same as in the previous case.

The last and most interesting case is where \eta_+<0 and \lambda_+ < \lambda_-. If \eta>0, then gluing for positive time is only possible for k=2, so we get f_+=1+T^2; while gluing for negative time is possible only for k=1, so f_-=1+T. If \eta<0, then we have f_+=1+T and f_-=1. Either way,

f_+(u_+) / f_-(u_+) = 1+T.

One can compute g_+(u_-) and g_-(u_-) by symmetry arguments and check that g_-/g_+ = f_+/f_-; I will leave this as an exercise.

6. Bonus: closed orbits.

I also leave it as an exercise to similarly check that one (mod 2) simple closed orbit of the gradient flow is created or destroyed by this bifurcation, and this orbit intersects \Sigma once.

 

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Cylindrical contact homology for dynamically convex contact forms in three dimensions

My paper with Jo Nelson on cylindrical contact homology for dynamically convex contact forms in three dimensions is now posted here. This paper shows that you can define the differential \partial by counting J-holomorphic cylinders for a generic almost complex structure J, without any abstract perturbation of the Cauchy-Riemann equation. (We make one additional technical assumption, which can probably be dropped with some more work on the asymptotics of holomorphic curves, and which automatically holds when \pi_1 of the three-manifold has no torsion.) We also prove that \partial^2=0, filling a gap in an earlier published proof of this, using the methods in this blog post.

The proof of invariance (i.e. the proof that cylindrical contact homology is an invariant of three-manifolds with contact structures that admit a dynamically convex contact form) is postponed to a subsequent paper. When there are no contractible Reeb orbits, the proof of invariance involves adapting the methods of Bourgeois-Oancea as explained in this blog post. In the presence of contractible Reeb orbits, there is an additional wrinkle; one has to count nontrivial contributions from holomorphic buildings consisting of an index 0 pair of pants branched covered over a trivial cylinder together with an index 2 plane. For an introduction to this, see the second half of this video.

 

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Erratum to “The ECH index revisited”

You may have noticed that there has been a long hiatus in postings on this blog. Last semester was extraordinarily busy with teaching. Anyway I am now thinking about some exciting (to me at least) research topics, and this led me to notice that there is a small mistake in my paper “The ECH index revisited”.

The mistake is that Proposition 6.14 is false as stated; to get a true statement, one should delete the N term when C and C' have a component in common. (The reason why this is a mistake is that the proof of Proposition 6.14 follows the proof of Theorem 5.1, and in the first line of Case 2 of the proof of Theorem 5.1, which is where the N term appears, one is assuming that a\neq b.) Moreover, Proposition 6.14 is used in the proof of Theorem 6.6.

One can easily correct Proposition 6.14 by deleting the N term.

Theorem 6.6 and its proof are then still fine if you add the hypothesis that C goes between orbit sets in which negative hyperbolic orbits never have multiplicity greater than one, for example ECH generators. I’m not sure if Theorem 6.6 still works if you allow orbit sets in which negative hyperbolic orbits have multiplicity greater than one. I could figure this out, but I’m not sure if anyone will ever need it.

The only time I have ever used Theorem 6.6 is in the appendix to “Algebraic torsion in contact manifolds” by Latschev and Wendl. And, there we only considered orbit sets that are ECH generators. Hence, this mistake has no effect on what is written in that appendix.

Phew!

The reason I noticed this error is that I have found a way to use J_0 and J_+ to get new symplectic embedding obstructions in some cases which go beyond the obstructions given by ECH capacities. Thus I needed to go back and recall the details of how J_0 and J_+ work. (I like to think that although the math literature is full of mistakes, things that are actually useful get reworked and checked and corrected as needed.) Stay tuned for more about these symplectic embedding obstructions a little later.

Posted in ECH, Errata | 1 Comment

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.

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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.

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