Monthly Archives: May 2013

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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