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.
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 of a link and a fibration , whose fibers are the interior of compact surface with boundary B. Note that flowing once around the base gives a homeomorphism that is the identity near the boundary. So, we could instead define an abstract open book, which is a pair of a compact surface with nonempty boundary and a monodromy map . This defines a closed, oriented 3-manifold
with an obvious open book decomposition.
The same manifold admits many open book decompositions. For instance, take an abstract open book and attach a 2-dimensional 1-handle to any way you please. Now, choose any essential simple closed curve that intersects the cocore of the 1-handle exactly once. Then the abstract open books define the same 3-manifold as , where denotes performing a positive Dehn twist along . These are referred to as positive/negative stabilizations of the open book.
Undoing this procedure is called a destabilization. Let be an arc in with boundary points in and suppose that only intersect on the boundary. Let be the simple closed curve obtained by concatenating . Then one of the mapping classes fixes up to isotopy and we can choose it to be the identity near . Thus,it restricts to a well-defined monodromy on the surface given by cutting along and a new open book. Choosing 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 if there is a contact form for such that (1) B is a positively transverse knot, and (2) is a positive area form on all of the pages .
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 , 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
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 and a mapping class 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 embedded on a page 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 is an abstract open book for .
Proof. (Intuitive) Think of the surgery torus sitting in the cylinder . Now, take a piece of string (thought of as sitting in a page below the solid torus and transverse to ) and pull it through the surgery torus.
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 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 is supported by the open book (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 , there is a supporting open book such that L lies on some page.
This means that Legendrian surgery can be encoded by a tuple 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 that is (1) invariant under positive (de)stabilizations, (2) invariant under composition with positive Dehn twists along any essential simple closed curve in , and is (3) equivalent to the tightness of the supported contact structure.
One way to study mapping classes is by understanding the relation between some collection of arcs or curves in the surface and their images under a nice representative of the mapping class.
Take a compact surface , an embedded arc with boundary points on the boundary of and a mapping class . We can also choose a representive of the mapping class that is efficient with respect to , meaning the do not form any trivial bigons on the surface that can be isotoped away. Orient so that their concatenation is an oriented loop. We say that $latex\alpha$ is right-veering if the orientation given by at
some both boundary point agrees with orientation on and that 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)]
Theorem: (Honda-Kazez-Matic) A contact structure is overtwisted if and only if there is an open book supporting with a
left-veering non-right-veering arc.
So, to know that a contact structure is overtwisted, all we need to know is that there is at least one open book decomposition supporting 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 be an overtwisted contact structure [Edit: I’ve rewritten the following paragraph -PLC]
and denote the standard overtwisted contact structure . Then since the contact structures are homotopic, they must be isotopic. [Edit: As Marco points out in the comments below, the overtwisted contact structure on supported by an open book with an annular page and monodromy a single Dehn twist is not homotopic to the standard tight contact structure . In order to get a contact structure on homotopic to , we need to connect sum with another contact structure on so that the algebraic topology works out correctly: and are homotopic.] Now, 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 homotopic to the standard tight contact structure on . Since for all contact structures, the connect sum is isomorphic to , this implies that if is overtwisted then are homotopic, hence isotopic by Eliashberg.
There is another familiar overtwisted contact structure 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 such that and are isomorphic. Thus is isomorphic to the double connect sum . Connect sum is equivalent to a Murasugi sum of the open books, which for 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 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: . 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 are convex, because the vertical vector field is contact. Note that there are dividing curves when for some nonnegative integer k, which is when .
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.
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 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.
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.
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.
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.
So, we have a bypass for sitting in one of the handlebodies , attached along and a second bypass sitting in the other handlebody attached along . When the arc is
left-veering non-right-veering, we can perform bypass rotation and find a new bypass sitting in , attached along . Thus, glue up to an overtwisted disk. .
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:
left-veering non-right-veering arc 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 , 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.
(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 to the left. Again, the Dehn twist drags across the 1-handle and makes a left-veering arc into a right-veering arc.
Note that right/left-veering-ness is localized to the boundary of . 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]