Author Archives: Peter Lambert-Cole

Follow-up: Physics and Mathematics of Link homology

For the past few weeks, I attended the SMS workshop “Physics and Mathematics of Link homologies” at the University of Montreal.  It was organized by Sergei Gukov, Mikhail Khovanov and Johannes Walcher.  The lectures were videotaped, although they don’t seem to have been posted to the website yet.  The main themes were: Chern-Simons theory, generalizations of the A-polynomial, categorification of homology theories, and the search for a unified picture.  In particular, how the Chern-Simons interpretation of link and 3-manifold invariants can lead to conjectures about the structure of these invariants.

Resources: If you’re interested in learning about the physics, here are some resources that were suggested to me over the course of the workshop

Lectures on knot homology and quantum curves, Sergei Gukov and Ingmar Saberi
Quantum Fields and Strings: A Course for Mathematicians, Deligne, Etinghof, Freed, Jeffrey, Kazhdan, Morgan, Morrison, Witten (collected lecture notes from a year-long series of talks in ’96-’97 at the IAS)
Conformal Field Theory and Topology, Toshitake Kohno
Chern-Simons Theory, Matrix Models, and Topological Strings, Marcos Marino
Mirror symmetry, Clay Mathematics monographs
Quantum Mechanics for Mathematicians, Leon Takhtajan
Supersymmetry and Supergravity, Julius Wess and Jonathan Bagger
Introduction to Superanalysis F.A. Berezin
Lecture notes on global analysis, JD Moore

Classic papers:
Quantum field theory and the Jones polynomial, Witten
Supersymmetry and Morse theory, Witten
The Yang-Mills equations over Riemann surfaces, Atiyah and Bott

(Note for those who attended the workshop: One of our discussion questions after-hours
was to write down a Math-Physics dictionary.  The IAS book has a great glossary at the beginning that does exactly this.

Similarly, another question was: What is the Wess-Zumino-Witten model?  The Kohno book has an involved, explicit description of this conformal field theory model.  It also elaborates on lots of the moments in Witten’s papers where he skips over what seem like crucial parts of the proof by appealing to “the general principles of quantum/conformal field theory”)

Again, if you have any other suggested resources, post a comment and I’ll update my list. I’ve added these to the resources page.

Moving forward,

The workshop had one great feature which I would like to continue here, as our target audience is grad students.

One of the participants pointed out that most of the talks could have been given anywhere from 10-25 years ago.   But one of the best parts of the conference was that the speakers mentioned lots of basic facts that, had you been around for the past two decades, you’d know, but that no one had ever mentioned to me.

For example, Gukov mentioned the hierarchy of U(1)-gauge theory equations:
dim 4: Seiberg-Witten
dim 3: Monopole
dim 2: Vortex
and that these are really just one set of equations (SW), modified by dimensional reduction.  I.e. The monopole equations describe t-invariant solutions to the SW-equations on Y^3 \times \mathbb{R}.  Further, if we assume Y^3 is just \Sigma \times \mathbb{R}, we get the vortex equations.

Presumably, I could have figured this out for myself after awhile, but mentioning it helped dissipate a lot of my confusion.

I suspect one reason for this phenomenon is that if you’ve been around for 20-30 years, then you’ve been saying these basic facts over and over again, and it just gets boring to repeat them.  But if you’re a grad student trying to make sense of the enormous, interconnected framework of mathematics motivated by these ideas, you weren’t around to get bored by hearing it ad nauseum and only showed up after everyone stopped mentioned it because they assumed everyone knows by now.  As a grad student, you may learn those bits integral to your area of specialization, but the big, intuitive picture is lost.

I’d like to return to the vortex equations here in the future, for several reasons (1) to get a feel for gauge theory in a lower-dimensional (n=2) setting, (2) to give intuition in terms of complex geometry for (a) the description the SW moduli on Kahler surfaces and (b) Taubes’ Gromov invariant, and (3) to give intuition for why Heegaard Floer homology, which ostensibly is given by the Lagrangian intersection Floer homology in \text{Sym}^g(\Sigma_g), is really an invariant arising from gauge theory and to motivate its equivalence to Monopole Floer and Embedded Contact Homology.

I’d also like to go back and blog some of the classic papers mentioned above, like Witten’s approach to the Jones polynomial.

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Consistent monodromies

Update: Andy Wand has posted a preprint of his proof on the arXiv.  Obviously, definitions and content there supercede what was originally written in this series of blog posts.

Update: (12 May 2014) Some minor technical details have been corrected (Thanks to Patrick Massot for pointing these out)

Here is the third and final post on the Legendrian surgery conjecture.

In the previous post, we went through H-K-M’s proof that a contact structure is overtwisted if and only if there is a supporting open book with a left-veering non-right-veering arc. This gives a criterion for establishing tight vs ovetwisted, but its not invariant under positive stabilization. So we can’t just check one supporting open book for \xi, we have to check every single one, and we have to check every possible arc in each open book to see if it is right veering or not.

But the problem might be somewhat manageable:

Let’s start with a simple example.  Suppose our monodromy is not efficient and some arc \phi(\gamma), starts off heading to the right but then immediately crosses back over \gamma, forming a trivial bigon that can be isotoped away. If we remove this bigon by an isotopy, then \phi(\gamma) appears to be left-veering non-right-veering now.

completed_bigon

The arc \phi(\gamma) is right-veering if and only if the curves \gamma, \phi(\gamma) form a second trivial bigon, allowing us to isotope \phi(\gamma) back across \gamma.

completed_bigon2

So, if we have a bigon B_+ formed by the pair \gamma, \phi(\gamma), we want to be able to find a second bigon B_- that completes it, meaning that the two cancel each other out and we preserve right-veering-ness.

Now a second example. Suppose we have a square region, as in the following local picture, and that the monodromy is efficient. Then \alpha_1,\alpha_2 are right-veering. However, suppose that we can destabilize along the right \alpha_2 arc. Remember, this means that \phi( \alpha_2), \alpha_2 do not intersect except at the boundary. The destabilization means that we cut the surface along \alpha_2 and then add a negative Dehn twist along a closed curve isotopic to \alpha_2 \cup \phi (\alpha_2). This is the green curve in the second picture.

positive_square

Notice what happens to \phi'(\alpha_1) after the destabilization: it forms a trivial bigon with \alpha-1. So we can isotope \phi'(\alpha_1) so that it becomes left-veering non-right-veering. If that was it, then this contact structure would be overtwisted.

destab_square

However, suppose that there is a second square region as in the following picture, forming a “singular annulus”

completed_square

Now, when we destabilize, the second square region becomes a trivial bigon that pulls all the way around the diagram to cancel out the bigon formed from the first square region.

destab_completed_square

So, if a contact structure is tight and if we have a square R_+ bounded by \alpha_1, \phi(\alpha_1), \alpha_2, \phi(\alpha_2) such that we can destabilize along one of the arcs, we need to be able to find a second square R_- bounded by the same four arcs and connected at opposite corners. In other words, we need to be able to find a second square R_- that (completes) R_+.

Lets go back to an example from the second post, where we saw how a positive stabilization can hide a left-veering non-right-veering arc. In that example, we can find such a square R_+ bounded by the original arcs and the new arcs added by the stabilization. Furthermore, notice that there is a completing square R_- if and only if \phi(\alpha) crosses back over \alpha.

stabilization_square

Before we continue, lets fix some terminology. A (region) is an immersed 2n-gon in a page whose boundary lies in the curves \{\alpha, \phi(\alpha)\}, alternating between an \alpha and a \phi(\alpha). Furthermore, let’s orient the curves \alpha_i, \phi(\alpha_i), subject to the condition that for each i, \alpha_i \cup \phi(\alpha_i) is a closed, oriented curve. With this orientation, we call a region (positive) if the induced orientation on the boundary agrees with the orientation on the arc basis and a region is (negative) otherwise. Note that an oriented arc basis does not necessarily induce a coherent orientation on the boundary of a region.

Third example: Now let’s take the local picture near a square region and stabilize again. After stabilization, we can find a hexagon region.Note that this region is “completed”, meaning there is a second hexagon region attached at the corners of the first region, if and only if the square region before the stabilization was “completed”.

hexagon_stabilization

We can repeat this process by positively stabilizing as many times as we want. The notion of a “completed” region, of any number of sides, persists after repeated positive stabilizations. And if we can destabilize along the various sides of our 2n-gon, we know we can reduce it back to the case of the canceling bigons in the first example above.

We can also look for uncompleted regions further into the surface: Let’s suppose we have a chain of regions as in the following picture:

support_chain

There are two square regions, R_1,R_2 connected to the boundary, and we can orient the arc basis so these regions are positive. Then implies that R_3,R_4 must be negative regions. Finally, R_5 is a positive region, that is not connected to the boundary at all.  Now let’s do something really tricky: stabilize along the green arc that passes through all five regions.

support_stabilization

Afterwards, we get the following picture:

octogon

In effect, the three positive regions R_1,R_2,R_5 have been amalgamated into a single octagonal region. The regions R_3, R_4 have been amalgamated as well and they have also been joined with the negative regions beyond R_5. The result is that the 8-gon region we find after stabilizing is “completed” if and only if R_5 was “completed” (meaning there is a square negative region forming a singular annulus, as in example 2 above).

Again, “completion” is a property that persists after positive stabilizations. So, if we have an open book for some contact structure, we can look for uncompleted regions in the interior that are connected to the boundary by a chain of regions; then by a stabilization, bring it to the boundary.  Andy refers to any a monodromy with no such uncompleted regions as consistent and it should be a stabilization-invariant characterization of tightness.

These are just some suggestive examples: it seems like a decent amount of work to clean this up into an algorithm that takes an open book with an uncompleted region and, by a sequence of positive (de)stabilizations, find an open book with a left-veering non-right-veering arc.

Finally, let’s get back to the Legendrian surgery conjecture. Andy never elaborated much on this aspect, leaving it as a corollary of the characterization of tightness in terms of open books.

However, the basic idea is that Legendrian surgery can be realized as adding to the monodromy a positive Dehn twist along some essential curve in some open book supporting \xi. The key step now is to prove that positive Dehn twist preserve the consistency of the monodromy or equivalently, that a positive Dehn twist does not create an uncompleted region.

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Resources page

I added a resources page to the blog to collect links to lecture notes, survey papers and books that cover in detail many of the topics that are mentioned here in the blog.  As of right now, it consists mostly of lecture notes from the courses at the 2011 and 2012 summer schools in Nantes and Budapest run by the European CAST network.  Feel free to post suggestions for other resources and we’ll add it to the list.

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Tightness and right-veering monodromies

Update: Andy Wand has posted a preprint of his proof on the arXiv.  Obviously, definitions and content there supercede what was originally written in this series of blog posts.

Update: (12 May 2014) Some minor technical details have been corrected (Thanks to Patrick Massot for pointing these out)

In this second post on the Legendrian surgery conjecture, I want to reinterpret the conjecture in terms of the Giroux correspondence and give an overview of Honda-Kazez-Matic’s criterion for tightness in terms of right-veering monodromies.

Giroux correspondence

The Legendrian surgery conjecture can be reinterpreted via the Giroux correspondence as a statement about the monodromy of an open book supporting a tight contact structure.

An open book decomposition of a 3-manifold M is a pair (B, \pi) of a link B \subset M and a fibration \pi: M \setminus B \rightarrow S^1, whose fibers are the interior of compact surface \Sigma with boundary B. Note that flowing once around the base S^1 gives a homeomorphism \phi: \Sigma \rightarrow \Sigma that is the identity near the boundary. So, we could instead define an abstract open book, which is a pair (\Sigma, \phi) of a compact surface with nonempty boundary and a monodromy map \phi. This defines a closed, oriented 3-manifold
M = \Sigma \times [0,1] / \Sigma \times \{0\} \cong \phi(\Sigma) \times \{1\} \cup \partial \Sigma \times D^2
with an obvious open book decomposition.

The same manifold admits many open book decompositions. For instance, take an abstract open book (\Sigma, \phi) and attach a 2-dimensional 1-handle to \Sigma any way you please. Now, choose any essential simple closed curve \gamma \subset \Sigma' = \Sigma \cup \text{1-handle} that intersects the cocore of the 1-handle exactly once. Then the abstract open books (\Sigma', D_{\gamma}^{\pm} \circ \phi) define the same 3-manifold as (\Sigma, \phi), where D_{\gamma} denotes performing a positive Dehn twist along \gamma. These are referred to as positive/negative stabilizations of the open book.

Undoing this procedure is called a destabilization. Let \alpha be an arc in \Sigma with boundary points in \partial \Sigma and suppose that \phi(\alpha), \alpha only intersect on the boundary. Let \gamma be the simple closed curve obtained by concatenating \alpha, \phi(\alpha). Then one of the mapping classes D_{\gamma}^{\pm} \circ \phi fixes \alpha up to isotopy and we can choose it to be the identity near \alpha. Thus,it restricts to a well-defined monodromy on the surface \Sigma' given by cutting \Sigma along \alpha and a new open book. Choosing \alpha to be the cocore of the 1-handle attached during a stabilization undoes that stabilization

Proposition. All open book decompositions for M are related by a sequence of (de)stabilizations.

There is a deep connection between open books and contact structures. An open book for M supports a contact structure \xi if there is a contact form \alpha for \xi such that (1) B is a positively transverse knot, and (2) d \alpha is a positive area form on all of the pages \Sigma_{\theta} = \pi^{-1}(\theta).

Theorem (Thurston-Winkelnkemper) Every open book supports a unique contact structure.

[Edit: Uniqueness of the contact structure is due to Giroux]

However, contact structures are not supported by a unique open book. Positively (de)stablizing an open book decomposition gives a new OB decomposition supporting the same contact structure. Intuitively, this is because positive stabilizations are essentially given by connecting summing with (S^3, \xi_{std}), endowed with some nontrivial open book.

That this is move is enough to classify supporting open books is due to Giroux and the following relationship is called the Giroux correspondence.

Theorem (Giroux) There is a 1-1 correspondence between (abstract) open book decompositions of a closed, oriented 3-manifold M, up to positive (de)stabilization and contact structures on M, up to isotopy isomorphism.

This relies on two facts: (1) every contact structure is supported by some OB decomposition, and (2) all supporting open book decompositions for the same contact structure are related by positive (de)stabilizations.

Thus, all that is needed to specify a contact structure is a compact surface \Sigma and a mapping class \phi for that surface. And this correspondence allows us to study contact geometry algebraically via mapping class groups.

So, to understand tightness in terms of open books, we would like to find some property of mapping classes that is (1) invariant under positive (de)stabilizations, and (2) is equivalent to tightness/overtwistedness. Honda-Kazez-Matic’s non-right-veering condition satisfies (2) but not (1), which is really good but not yet sufficient.

Legendrian surgery also has a nice characterization in terms of open book decompositions.

A curve \gamma embedded on a page \Sigma_{\theta} is really a knot K in M and the page determines a framing of K. Integral Dehn surgery on K can be described via the following modification of the monodromy map:

Lemma. The pair (\Sigma, D_{\gamma}^k \circ \phi) is an abstract open book for (M_K(-1/k)).

Proof. (Intuitive) Think of the surgery torus S^1 \times I \times I sitting in the cylinder \Sigma \times [0,1]. Now, take a piece of string (thought of as sitting in a page below the solid torus and transverse to \gamma) and pull it through the surgery torus.

Geometrically, for any almost all essential simple closed curves sitting in a page, the contact structure supported by that open book can be perturbed to make that curve Legendrian. Moreover, the page framing is exactly the Thurston-Bennequin framing. This is because d \alpha is a positive area form on the page, so by definition the Reeb vector field is always positively transverse to the pages.

Lemma. The contact structure (M_K(\pm 1), \xi_K(\pm 1)) is supported by the open book (\Sigma, D_{\gamma}^{\mp 1}) (when K lies in a page).

Finally, the following lemma follows easily from Giroux’s proof that every contact structure is supported by some open book.

Lemma. For any Legendrian link L in (M,\xi), there is a supporting open book such that L lies on some page.

This means that Legendrian surgery can be encoded by a tuple (\Sigma, \phi, \gamma) of a compact surface, mapping class and essential simple closed curve. To prove the Legendrian surgery conjecture, we will want our property to satisfy a third condition: (3) it is persistent after positive Dehn twists.

Conjecture. There is some property P of abstract open books (\Sigma, \phi) that is (1) invariant under positive (de)stabilizations, (2) invariant under composition \phi with positive Dehn twists along any essential simple closed curve in \Sigma, and is (3) equivalent to the tightness of the supported contact structure.

Right-veering monodromies

One way to study mapping classes [\phi] is by understanding the relation between some collection of arcs or curves \{\alpha_i\} in the surface and their images \{\phi(\alpha_i)\} under a nice representative \phi of the mapping class.
Take a compact surface \Sigma, an embedded arc \alpha \subset \Sigma with boundary points on the boundary of \Sigma and a mapping class [\phi]. We can also choose a representive \phi of the mapping class that is efficient with respect to \alpha, meaning the \alpha, \phi(\alpha) do not form any trivial bigons on the surface that can be isotoped away. Orient \alpha, \phi(\alpha) so that their concatenation is an oriented loop. We say that $latex\alpha$ is right-veering  if the orientation given by \alpha', \phi(\alpha)' at some both boundary point agrees with orientation on \Sigma and that \alpha is left-veering non-right-veering otherwise. See the picture.

[Edit: Non-right veering (i.e. left veering at some endpoint) does not imply left-veering (left-veering at both endpoints)]

left-right-veering

Theorem: (Honda-Kazez-Matic) A contact structure \xi is overtwisted if and only if there is an open book (\Sigma, \phi) supporting \xi with a left-veering non-right-veering arc.

So, to know that a contact structure \xi is overtwisted, all we need to know is that there is at least one open book decomposition supporting \xi with at least one single left-veering non-right-veering arc. And, for every overtwisted contact structure, we can find such and open book decomposition and left-veering non-right-veering arc. So this gives an OB decomposition/mapping class group characterization of tightness.

Proof. It’s fairly straightforward to show that every overtwisted contact structure admits some open book with a left-veering non-right-veering arc.

Recall that Eliashberg classified overtwisted contact structures by their homotopy types. Let (M,\xi) be an overtwisted contact structure [Edit: I’ve rewritten the following paragraph -PLC] and (S^3, \xi_{OT}) denote the standard overtwisted contact structure . Then since the contact structures (M, \xi), (M,\xi) \sharp (S^3, \xi_{OT}) are homotopic, they must be isotopic. [Edit: As Marco points out in the comments below, the overtwisted contact structure on S^3 supported by an open book with an annular page and monodromy a single Dehn twist is not homotopic to the standard tight contact structure \xi_{std}.  In order to get a contact structure on M homotopic to \xi, we need to connect sum with another contact structure \xi' on S^3 so that the algebraic topology works out correctly: (M,\xi) and (M,\xi) \sharp (S^3, \xi_{OT}) \sharp (S^3, \xi') are homotopic.]  Now, (S^3, \xi_{OT}) has an open book decomposition with an annular page and monodromy given by one negative Dehn twist along the core curve. The connect sum is equivalent to a Murasugi sum of the open books, which in this case is exactly a negative stabilization along some boundary parallel arc. The cocore of the new 1-handle is now a left-veering non-right-veering arc.

There is a unique overtwisted contact structure (S^3, \xi_{OT}) homotopic to the standard tight contact structure on S^3 .  Since for all contact structures, the connect sum (M,\xi) \# (S^3, \xi_{std}) is isomorphic to (M,\xi), this implies that if \xi is overtwisted then (M,\xi) \# (S^3,\xi_{OT}) are homotopic, hence isotopic by Eliashberg.

There is another familiar overtwisted contact structure (S^3, \xi_{Hopf}) that is supported by an open book with annular pages and monodromy given by one negative Dehn twist along the core curve of the annulus.  We can find a third contact structure (S^3, \xi') such that (S^3, \xi_{OT}) and (S^3, \xi') \# (S^3, \xi_{Hopf}) are isomorphic.  Thus (M,\xi) is isomorphic to the double connect sum (M,\xi) \# (S^3, \xi') \# (S^3, \xi_{Hopf}).  Connect sum is equivalent to a Murasugi sum of the open books, which for (S^3,\xi_{Hopf}) is exactly a negative stabilization and the cocore of the new 1-handle is now a non-right-veering arc.

The converse is not too hard and is a straighforward application of convex surface theory:

Recall that a convex surface is a surface S embedded in a contact 3-manifold such that there is a contact vector field \eta transverse to S. The contact structure near S can be completely understood in terms of the isotopy class of the dividing curves, which are given by the points in S where the contact planes contain the contact vector field: \eta_x \in \xi_x. Generic surfaces in contact 3-manifolds are convex with transversely cut-out a set of dividing curves.

For example, the horizontal planes in the standard overtwisted contact structure \xi = \text{ker}(\cos \pi r d z + r \sin \pi r d \theta) are convex, because the vertical vector field \partial_z is contact. Note that there are dividing curves when r = 1/2 + k for some nonnegative integer k, which is when \partial_z \in \xi.

overtwisted_disk

A bypass D for a convex surface S is a convex disk with Legendrian boundary and a single dividing curve, that intersects the surface S along an arc with boundary on the dividing set and intersecting the dividing set exactly 3 times.

bypass

Now, notice that a bypass is essentially half of an overtwisted disk; the idea is to find two bypasses along the same arc on opposite sides of a convex surface, then glue them together to find an overtwisted disk.

Each open book determines a Heegaard splitting of along the surface \Sigma = - \Sigma_0 \cup \Sigma_{1/2} given by gluing together two pages. The contact structure can be isotoped so that this Heegaard surface is convex, with dividing curves exactly given by the binding.  We can assume the compressing disks are convex and it follows that they each have exactly 1 dividing arc, so they are already essentially bypasses. To make it so, cut a little notch in the disk at the binding and push the boundary off a little along the binding.

compress_disk_bypass

There are lots of possibly bypasses, one for each compressing disk on each side, but they don’t line up exactly at the Heegaard surface. To achieve this, we need to do what H-K-M call a bypass rotation. Recall that isotoping across a bypass changes the dividing curves as in the picture.

bypass_dividing_curves

Suppose we have two potential arcs on which to attach bypasses, as in the figure below. Notice that if we first attach a bypass along the right arc, attaching a bypass on the left arc doesn’t change the dividing curves (up to isotopy). It’s trivial. And by what H-K-M refer to as the “Right to Life” principle, there always is a bypass for an trivial arc of attachment.

bypass_rotation

So, if we see the local picture above and know that a bypass exists for the right arc, we know a bypass exists for the left arc. Note that this is (not) true if we attach a bypass on the left first and then the right; the right arc is not forced to be a trivial arc of attachment.

bypass__not_rotation

So, we have a bypass D_1 for \Sigma sitting in one of the handlebodies H_1, attached along \alpha' and a second bypass D_2 sitting in the other handlebody H_2 attached along \phi(\alpha'). When the arc \alpha is left-veering  non-right-veering, we can perform bypass rotation and find a new bypass D_1' sitting in H_1, attached along \phi(\alpha'). Thus, D_1', D_2 glue up to an overtwisted disk. \Box.

However, it is (not) true that every OB decomposition for an overtwisted contact structure has a left-veering non-right-veering arc. By adding some positive stabilizations, we can hide the overtwistedness of the contact structure. Consider the following examples:

(1)Take a left-veering non-right-veering arc \alpha and consider a local picture of one boundary point of this arc that veers left. We can positively stabilize along the green arc in a neighborhood of this point and since the arc intersects the image \phi(\alpha), it gets modified by the Dehn twist and appears to be right-veering now. If we did that at both endpoints, the arc would now be right-veering instead of left-veering.

stabilize_left-veering

(2) In a second example, consider a local picture of an entire left-veering arc (left-veering at both ends). Positively stabilize along the green arc, which is a pushoff of \alpha to the left. Again, the Dehn twist drags \phi(\alpha) across the 1-handle and makes a left-veering arc into a right-veering arc.

stabilize_left-veering-full

Note that right/left-veering-ness is localized to the boundary of \Sigma. It is completely blind to what happens in the interior of a page. By cleverly applying some positive stabilizations, we can push the overtwistedness or left-veering-ness further and further into the surface.

Intuitively, Andy Wand’s approach is to undo this procedure and bring the negative twisting back to the boundary, where we can apply Honda-Kazez-Matic’s result. He has a characterization of some phenoma in the interior that, after some well-chosen positive (de)stabilizations, result in a left-veering non-right-veering arc at the boundary.

[*In fact, Andy Wand’s method only uses stabilizations, not both stabilizations and destabilizations]

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The Legendrian surgery conjecture

Update: Andy Wand has posted a preprint of his proof on the arXiv.  Obviously, definitions and content there supercede what was originally written in this series of blog posts.

Update: (12 May 2014) Some minor technical details have been corrected (Thanks to Patrick Massot for pointing these out)

The Legendrian surgery conjecture is one of the biggest problems in contact topology. The question is, does (-1) Legendrian surgery on Legendrian knots in a contact 3-manifold preserve tightness? Andy Wand has an approach to proving this by characterizing the tightness of some (M, \xi) in terms of the monodromy of an open book supporting the contact structure \xi. He gave a seminar talk last month at Georgia Tech that I attended and there is also a video of a similar talk at the Simons Center. He hasn’t posted a written proof yet, an hour-long lecture is not sufficient to cover all the details and I don’t completely understand every part yet, so I can’t verify the proof for myself. However, I do understand some suggestive examples that motivate his approach and I want to blog that. I’ll start by reviewing some background in contact topology, motivating the conjecture and describing Honda-Kazez-Matic’s work on non-right-veering monodromies before moving on to Andy’s work.

Let’s start from the beginning. An overtwisted disk in a contact 3-manifold in an embedded disk D^2 such that the contact structure \xi is tangent to the disk along its boundary. For example, consider the unit disk in the contact structure \xi = \text{ker}(\cos \pi r dz + \sin \pi r d \theta). A contact structure is overtwisted if it contains an overtwisted disk and is tight otherwise.

There is nice classification of overtwisted contact structures, due to Eliashberg.

Theorem. (Eliashberg) Let M be a closed, oriented 3-manifold. Then there is 1-1 correspondence between overtwisted contact structures on M, up to isotopy, and homotopy classes of 2-plane fields on M.

Therefore, the classification of overtwisted contact structures is governed by algebraic topology and is not really interesting from a geometric perspective. Understanding and classifying tight contact structures on 3-manifolds is a big remaining topic in contact topology.

Motivation for Legendrian surgery conjecture

One way to construct contact 3-manifolds is by performing Dehn surgery but in a manner that respects the geometry. Recall that p/q-Dehn surgery on a knot K \subset M consists of removing a tubular neighborhood S^1 \times D^2 of the knot, and then regluing via the diffeomorphism \phi: T^2 \rightarrow T^2 that sends the meridian \mu = pt \times \partial D^2 to p \mu + q \lambda, where \lambda = S^1 \times pt is a longitude (equivalently a framed pushoff of the knot). This operation is well-defined up to isotopy but depends upon the (framing), the explicit identification of \nu(K) \simeq S^1 \times D^2. Denote the new manifold as (M_K(p/q))

In contact topology, a Legendrian knot has a framing, the Thurston-Bennequin  framing, that is well-defined up to Legendrian isotopy and neighborhoods of all Legendrian knots are contactomorphic. This framing is given by any vector field in M along K that is positively transverse to the contact planes, such as the Reeb vector field for any contact form. When there is a well-defined null-homologous framing, the Thurston-Bennequin framing is often written the integer giving the difference between the null-homologous framing and the TB framing. To properly keep track of surgery framings, coefficients in parentheses will indicate framing relative to the TB framing and coefficients without parentheses will indicate framing relative to the null-homologous framing.

When the surgery coefficient is (1/k) for some integer k, the contact structure on the complement M \setminus \eta(L) extends uniquely across the surgery torus. When this coefficient is (-1), this operation is called Legendrian surgery.

In 3-manifold topology, the Lickorish-Wallace theorem states that every closed, oriented 3-manifold can be obtained by integral surgery on a link in S^3. There is an analogous result in contact topology:

Theorem (Ding-Geiges) Every closed contact 3-manifold (M, \xi) is given by (\pm 1)-surgery on a Legendrian link in (S^3, \xi_{std}).

Here (\pm 1) means that we perform (-1) surgery on some link components and (+1) surgery on the remaining components.

For example, the standard contact structure on S^1 \times S^2 is given by (+1) surgery on the tb=-1 unknot. The standard overtwisted contact structure on S^3, which is contactomorphic to the one described above in the definition of overtwisted disks, is given by (+1) surgery on the Hopf link of tb=-1 unknots. The former is tight, the latter is overtwisted and the standard contact structure on S^3 is tight (a result due to Bennequin), so (+1) surgery neither preserves tightness nor necessarily creates overtwistedness.

Integral knot surgery can be interpreted in low-dimensional topology as a cobordism from M to M_K(n) given by attaching a 4-dimensional 2-handle to M along K with framing n. This also has a geometric counterpart.  A Stein cobordism between contact manifold (M, \xi), (M', \xi') is a cobordism X that is a complex manifold, equipped with a J-convex function f such that M, M’ are regular level sets of f.

Lemma. There is a Stein cobordism from (M,\xi) to (M_L(-1), \xi_L(-1)) given by attaching a handle along each component in L with framing (-1).

Theorem. A cobordism from (M, \xi) to (M', \xi') is Stein if and only if it can be obtained by Legendrian surgery on a link in (M, \xi) [edit: plus 1-handles].

For details, see Cieliebak and Eliashberg’s book.

A contact structure (M, \xi) is Stein fillable if there exists a Stein bordism with boundary (M,\xi). Stein fillability is the strongest notion of fillability (see Laura’s post) and all fillable contact 3-manifolds are tight.

Proposition. Legendrian surgery preserves fillability.

So, Legendrian surgery preserves a large subclass of tight contact structures. Yet there are tight contact 3-manifolds that are not fillable. For example, (+1) surgery on the max TB right-handed trefoil in S^3. What about them?

In addition, Legendrian surgery seems to make things more tight. Consider the following lemma:

Lemma. There is a Stein cobordism from every contact 3-manifold to a Stein fillable contact structure.
Proof. (-1) and (+1) surgeries cancel each other out, topologically and geometrically. By Ding-Geiges, there exists a link L = L_- \cup L_+ \subset (S^3, \xi_{std}) such that (\pm 1) surgery on L_{\pm} gives (M,\xi). Now performing (-1) surgery on L_+ removes all of the (+1) surgeries, leaving a contact manifold obtained by (-1) surgery on a Legendrian link in the Stein fillable (S^3, \xi_{std}), hence it is Stein fillable.

This leads us to the following conjecture:

Conjecture. The contact structure given by Legendrian surgery on a Legendrian knot in a tight contact structure is tight.

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Morse Homotopy and A-infinity; Part 2

Last time, I describe the first step in Fukaya’s proof that Morse homology has an A-\infty structure, defined in terms of gradient flow trees.  This time I’ll describe how the higher relations (m_3, m_4, \dots) arise from Morse theory.

To describe m_3, we need to look for all trees connecting x,y,z to some w.  At this level there are some complications.  First, not all trees we need to consider will be isomorphic (at least if we fix a cyclic ordering of the exterior vertices).  For the m_2, every tree was a Y.  But at higher levels, we can have nonisomorphic trees, such as the following:

trees_m3Unlabeled

So we need to make sure we look for all possible trees.  As we get to the higher $A – \infty$ maps the space of all possible trees gets complicated.

Secondly, we need to start keeping track of the length of interior edges.  Each parametrizes a flow line, so we need to know exactly how long the partial flow line is that we want to parametrize.  This hasn’t been a problem because up to now, every flow line we considered had at least one noncompact end, because one end was asymptotic to a critical point, so its length was infinite.

We can solve both these problems by realizing that we can form a moduli space of italics(metric trees), an approach originally due to Stasheff.  Luckily, if we are careful, the moduli space is just some affine space \mathbb{R}^k for some dimension k.

Embed the tree in the unit disk with the exterior vertices (the 1-valent vertices which map to critical points) cyclically ordered along the boundary.  Furthermore, assume that no vertex has valence 2, as this corresponds to a broken flow line, which we will consider separately later on.  The [exterior] edges will be those connected to the boundary and [interior] edges will be any other edge.  Assign a positive real number to each interior edge (the exterior edges are assumed to have length \infty).  Then the lengths of the interior edges, of which there can be at most k-3, where k is the number of exterior vertices, identify parametrize the moduli space and identify it with \mathbb{R}^{k-3}.  Call this space \mathcal{T}_k

For example, suppose k = 4, which is the relevant space for the m_3.  The tree on the left corresponds to -m \in \mathbb{R} and the tree on the right corresponds to m \in \mathbb{R}.

trees_m3_Moduli

The space of metric trees is not compact but it can also be compactified, in the sense that as the length of the interior edge goes off to \pm \infty, the tree breaks into two metric trees.  For k = 4, this is just the union of a two 3-leaf Y trees.

trees_m3_ModuliCompactified

For higher k, it will break into two trees with j and k-j+2 exterior vertices, respectively.  Again, we see the principle that a moduli space can be compactified using the product of lower-dimensional moduli spaces of the same type of object.

Choose four Morse functions f_1,f_2,f_3,f_4 such that their differences $f_i-f_j$ are collectively generic.  Let \widetilde{\mathcal{M}}(x,y,z;w) denote the moduli of metric trees with 4 exterior vertices parametrizing flow lines of the difference functions f_i-f_j in the following way:  Each tree can be thought of as embedding in the unit disk and thus separates the disk into 4 regions.  Cyclically label each region with a function $f_i$.  Then an edge of a tree parametrizes a flow line of f_i-f_j if it separates the regions labeled by f_i and f_j.  The trees can be oriented so  that every interior vertex has exactly one outgoing edge and exactly one exterior vertex has an incoming edge.  Assume that the oriented flows respect this orientation on edges.

m3_Labeled

The m_3 map is defined as follows.

m_3: C(f_1,f_2) \otimes C(f_2,f_3) \otimes C(f_3,f_4) \rightarrow C(f_1,f_4)
m_3(x,y,x) = \sum_{w|[index]} |\widetilde{\mathcal{M}}(x,y,z;w)| w

In other words, count all rigid trees connecting x,y,z to w.

We’d now like to establish the A-\infty relation
m_3(d(x),y,z) + m_3(x, d(y),z) + m_3(x,y,d(z)) + m_2(m_2(x,y),z) + m_2(x,m_2(y,z)) + d(m_3(x,y,z))= 0

As with the m_2, each term here describes one way a 1-dimensional tree could degenerate.  The first three correspond to an incoming flow line breaking:

brokenTree_top
The last term corresponds to the outgoing flow line breaking:
brokenTree_bottom
And the terms involving m_2 correspond to an interior flow line breaking:
brokenTree_interior
Again, the relation follows because each possible combination of broken trees can be glued to form the boundary of a 1-dimensional moduli space.  Moreover, each 1-dimensional tree must break/degenerate in one of the above ways.  Either an exterior edge breaks, which corresponds to the familiar compactification of Morse flow lines, or an interior breaks, which corresponds to the compactification of the moduli of metric trees.

Fukaya notes that this m_3 relation descends to Massey products on cohomology but I won’t go into that here.

The higher A-\infty maps arise in the same way.  The map m_k is defined by counting rigid trees with k incoming and 1 outgoing edge.  The A-\infty relation follows from the fact that each 1-dimensional tree breaks into two rigid trees.

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Morse Homotopy and A-infinity; Part 1

I found it useful to thoroughly go through and understand why the differential in Legendrian Contact Homology squares to 0 and so did some others, so I’m going to continue and discuss where  higher A-infinity relations come from.  A-infinity things seem intimidating because of all the little details necessary to define them.  I hope it helps to have a geometric interpretation.  It also helps to open your mind to universal algebra and operads, but I won’t go into that here.

Recall the definition of an A-infinity algebra.  Let A be a graded k-vector space.  Then there exists an infinite family of maps \{m_k \}

m_k: A^{\otimes k} \rightarrow A

that satisfy the A-infinity relations, which are a nightmare to state.  I’ll describe them pictorially as follows.  Each m_k can be represented by a box with k strands entering on the top and 1 strand leaving on the bottom, i.e. there are k inputs and 1 output.  (Some people use trees to visualize this as well).

m_k map

Then, we some over all possible ways to combine two of these maps into a map that has l inputs and 1 output:

a_infinity relation

and require that this sum is 0, (if for simplicity we ignore signs and assume k = \mathbb{F}_2).  So, we sum over all i,j such that i + j -1 = l and we can move the m_i map left and right so that it takes as inputs any adjacent i-tuple.

The simplest condition is that m_1 \circ m_1 = 0, since there is only one way to combine two maps in a manner that takes 1 input and yields 1 output.

m_1 relation

As a consequence, this means that the m_1 map is a differential.

The new two conditions are m_2 ( d(x), y) + m_2 ( x,d(y)) + d( m_2 (x,y)) = 0.

m_2 relation

and m_2(m_2(x,y),z) + m_2(x,m_2(y,z)) + d(m_3(x,y,x)) + m_3(d(x),y,z) + m_3(x,d(y),z) + m_3(x,y,d(z)) = 0

m_3 relation

Using Morse theory, Fukaya proved that the cohomology ring of a real analytic manifold is actually an A-infinity algebra.  In the case of proving d^2=0, we used the fact that each term in d^2 corresponds to a union of two flowlines, called a broken flow.

Fukaya studied gradient flow trees.  Let T be the tree in figure 1, with 4 vertices and 3 edges in a Y pattern.  Now, pick 3 Morse functions f_1,f_2,f_3 such that their difference functions f_i - f_j are generic.  This means that the (un)stable manifolds of all difference functions intersect transversely.  To define the m_2 map, we are going to look at the moduli space of flow trees corresponding to Y.  That is, each edge will parametrize a flow line of some f_i-f_j.  Let \widetilde{\mathcal{M}}(x,y;z) denote the moduli space of gradient flow trees from x,y to z for x a critical point of f_1 - f_2, y a critical point of f_2 - f_3 and z a critical point of f_1 - f_3,:

m_2 source tree

One way to get our hands on this space of trees is to take W_u(x) \cap W_u(y) \cap W_s(z).  For each point in this space, there is a unique triple of oriented flow lines connecting it to x,y and z.  Together, these form a tree of the form we are looking for.  The dimension of the moduli space is I(x) + I(y) - I(z) - n, which can easily be checked because this is assumed to be a transverse intersection.

Then the m_2 map can be defined as

m_2:C(f_1 - f_2) \otimes C(f_2 - f_3) \rightarrow C(f_1 - f_3)
m_2(x,y) = \sum_{z: I(z) = I(x) + I(y) - n} |\widetilde{\mathcal{M}}(x,y;z)| z

We need to show that this satisfies the A-$\infty$ relation
d(m_2(x,y)) + m_2(d(x),y) + m_2(x,d(y)) = 0

In Morse theory, all moduli spaces, of flow lines and trees and of all dimensions, can be compactified.  We just need to know how trees/flows degenerate as they head off to the open end.  But as always, the principle here is that it can only degenerate into a union of trees you already know about.

For instance, take the Y.  Since everything is finite dimensional, any open end of the moduli space must come from an open end of the moduli of the individual flow lines.  So degeneration for trees looks exactly like degeneration for flow lines.  Any of the three edges could break into pieces.

m_2 broken trees

So, now we have a union of two trees, a segment and another Y.  But the segment is just a flow line from x to w, and so algebraically shows up in the differential.  And the Y is another tree corresponding to an m_2 map.

Let’s work the other way.  The each of term of m_2(d(x),y) corresponds to a pair of a rigid flow line from x to some $w$ and a rigid tree connecting w,y to z.  Similarly, each term of m_2(y,d(y)) corresponds to a pair of a rigid flow line from y to some w and a rigid tree connecting x,w to z.  Finally, each term of d(m_2(x,y)) corresponds to a pair of a rigid tree connecting x,y to some w and a rigid flow line from w to z.

As with the differential, these pairs can be glued together into a 1-dimensional tree and each pair corresponds to the endpoint of some 1-dimensional component of the moduli space of trees from x,y to z.  Since this 1-dimensional space can be compactified in such a way that if we look at the boundary of all 1-dimensional moduli of trees, we get pairs as in the figure above.

Thus, d(m_2(x,y)) + m_2(d(x),y) + m_2(x,d(y)) = 0 and we know that our chain complex is at least an A_2-algebra.

Now, we have been using 3 different Morse functions and then three other difference functions.  These critical points live in different chain complexes.  This is ok.  We already know that the chain homotopy type of the Morse complex is independent of the Morse function.  So the algebraic structure of the d map is the same in all three chain complexes.  So it’s OK to think of this relation as living on a single chain complex

Fukaya also shows how this m_2 map descends to the familiar cup product on cohomology.  First, recall the chain homotopy equivalence between Morse homology and singular homology.  In one direction, the descending manifold of an index i critical point, which is topologically a disk, is a singular i-chain, a continuous map of an i-dimensional simplex into the manifold M.  In the other direction, given a singular i-chain, its image will (generically) intersect the stable manifolds of an index i critical points in a finite number of points.  Summing over all such intersections gives the algebraic image of the chain in the Morse complex.  In more suggestive terms, one can think of flowing the image of the singular chain down by the descending gradient flow.  The singular chain will “hang” on some of the index i critical points and summing over these points gives the corresponding algebraic chain in the Morse complex.

Morse-Singular equivalence

In Morse cohomology (N.B the differential increases the grading, so we look at rigid, ascending flow lines), the cohomology complex is still generated by the critical points and their Poincare duals can be represented by the singular chains given by their ascending manifolds.  Thus, take two basis elements x \in H^i(M,\partial_{\text{Morse}}), y \in H^j(M,\partial_{\text{Morse}}).  These have Poincare duals X,Y which are disks of dimension n-i,n-j.  The Poincare dual of x \wedge y is X \cap Y, which has dimension n-i-j.  To determine which element this is in Morse cohomology, we flow this intersection upward by the gradient vector field and see which index i+j critical points it gets caught by.  In terms of gradient trajectories, we look for all the gradient flow lines from X \cap Y to some critical point z of index i + j.  There are a finite number of such flow lines, each of which corresponds to a unique tree connecting x,y to z.  In the chain complex, this is exactly the m_2 map and so passing to cohomology we recover the cup product from m_2.

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