# Kylerec – Weinstein fillings

Continuing on with the Kylerec posts… (see the first one here as well as notes to follow along with here).

This post is a synthesis of the following talks:

• Day 1 Talk 2 – François-Simon Fauteux-Chapleau’s talk on Weinstein handles and contact surgery
• Day 1 Talk 3 – Orsola Capovilla-Searle’s talk on Kirby calculus for Stein manifolds
• Day 1 Talk 4 – Alvin Jin’s talk on Lefschetz fibrations and open books
• Day 2 Talk 1 – Bahar Acu’s talk on mapping class factorizations and Lefschetz fibration fillings
• Day 3 Talk 2 – Sarah McConnell’s talk on applications of Wendl’s theorem to fillings
• Day 5 Talk 1 – Ziva Myer’s talk on flexible and loose Legendrians

### Weinstein surgery theory

I assume the reader is familiar with smooth surgery theory. Recall the following definition.

Definition: A Weinstein cobordism consists of a quadruple $(W,\omega,V,\phi)$, where

• $(W,\omega)$ is a compact symplectic manifold with boundary
• $V$ is a Liouville vector field for $(W,\omega)$, meaning $\mathcal{L}_V\omega = \omega$, which is also transverse to the boundary $\partial W$
• $\phi \colon W \rightarrow \mathbb{R}$ is a Morse function
• $V$ is gradient-like for $\phi$, meaning there is some constant $\delta$ with $d\phi(V) \geq \delta(|d\phi|^2 + |V|^2)$ with respect to a given Riemannian metric.

In this case, the boundary decomposes as $\partial W = \partial^+ W \sqcup \partial^-W$, where $V$ points out of $\partial^+ W$ and into $\partial^- W$. Note that the 1-form $\lambda = \iota_V \omega$ satisfies $d\lambda = \omega$, and is sometimes called the Liouville 1-form, since it encodes the same data as $V$. Also note that a Weinstein cobordism with $\partial^- W = \emptyset$ is what we called a Weinstein filling.

The gradient-like condition is meant to give $V$ some directionality (since $d\phi(V) > 0$) and ensure that the critical points of $V$ are non-degenerate. One typically doesn’t think of the precise choice of pair $(V,\phi)$ as very important, but rather the data up to some notion of homotopy. For example, one can always perturb the Morse function so that each of $\partial^- W$ and $\partial^+ W$ is a regular $\phi$-level set, regardless of the number of components, and so we might as well assume this from the start. The equivalence hinted at here is called Weinstein homotopy, by which we perturb the pair $(V,\phi)$, possibly through birth-death type singularities.

Lemma: The descending manifolds in a Weinstein cobordism, i.e. the set of points which flow along $V$ to a given critical point in infinite time, are isotropic submanifolds.

Proof: Standard Morse theory implies these submanifolds are smooth. Let $\phi_V^t$ be the flow along $V$ at time $t$, and suppose we choose some $q \in D_p^-$ where $D_p^-$ is some descending manifold for a given critical point $p$. Suppose $v \in T_qD_p^-$ is a vector in the tangent space. Then since $\mathcal{L}_V\lambda = d\iota_V\lambda + \iota_V d\lambda = d\iota_V^2\omega + \iota_V\omega = 0 + \lambda = \lambda$, we have that

$e^t\lambda_q(v) = ((\phi_V^t)^*\lambda)(v) = \lambda(d\phi_V^t(v))$

As $t \rightarrow \infty$, the right hand side goes to zero since $\phi_V^t(q') \rightarrow p$ for all $q'$ in a curve $\gamma$ along $D_p^-$ with tangent vector $v$ at $q$. Hence, $\lim_{t\rightarrow \infty} e^t \lambda_q(v) = 0$, from which it follows that $\lambda_q(v) = 0$. Hence, $\lambda|_{D_p^-} = 0$, and so also $\omega|_{D_p^-} = d\lambda_{D_p^-} = 0$.

Corollary: All critical points in a Weinstein cobordism $(W^{2n},\omega,V,\phi)$ are of index at most $n$. Smoothly, any such manifold can be built up by surgery starting from a neighborhood of $\partial^-W$ and attaching handles of index at most $n$.

One would like to be a bit more precise about how the surgery interacts with the symplectic geometry. As a first step, along a regular level set $W_c := \phi^{-1}(c)$, the symplectic condition on $\omega$ implies that $\lambda|_{W_c}$ is a contact form. The proof of the lemma above further implies that $D_p^- \cap W_c$ gives an isotropic submanifold of $W_c$ with respect to $\lambda|_{W_c}$.

So we can think, at least smoothly, that our Weinstein cobordism is built up, starting from $\partial^- W$, by attaching handles with isotropic cores and attaching spheres along isotropics in level sets of $\phi$ (which are contact submanifolds). But there’s a little more that we know about neighborhoods of isotropics. In a symplectic manifold, the neighborhood of an isotropic $M \subset (W,\omega)$ is completely determined up to symplectomorphism by its symplectic normal bundle, $(TM)^{\omega}/TM$, as a symplectic vector bundle (with symplectic structure induced by $\omega$ on the fibers).  A similar statement holds for isotropic submanifolds in contact manifolds, but now with their neighborhoods determined up to contactomorphism by the conformal symplectic normal bundle $(TM)^{d\alpha}/TM$, where $\alpha$ is a contact form so that $d\alpha$ is symplectic on $\xi$. Furthermore, if we fix $\alpha$, then the symplectic vector bundle structure determined by $d\alpha$ on the nose determines the neighborhood up to exact contactomorphism. Patching these two things together, one finds:

Theorem [Weinstein, before the term “Weinstein handle” was coined]: Weinstein handle attachment is completely specified (up to Weinstein homotopy) by matching the symplectic framing data determined by $\lambda$ along the isotropic attaching spheres.

One therefore thinks of $\partial^+ W$ as being built up from $\partial^- W$ by contact surgery along isotropic submanifolds with given framing information compatible with the underlying symplectic topology.

Consider a Weinstein cobordism of dimension $2n$. Then the handles of index $k \in \{0,1,\ldots,n-1\}$ are called subcritical handles, whereas the handles of index $k = n$ are called critical handles. When $k = n$, the aformenetioned symplectic normal bundles are trivial automatically, and so one specifies critical handle attachment simply by drawing a Legendrian sphere on $\partial^- W$.

Recall that the proof of the h-cobordism theorem requires some ability to cancel (and create) pairs of handles with index differing by 1 whose ascending and descending manifolds intersect in a 1-dimensional manifold, to move around attaching spheres, and to move critical values around. The last of these we can always do, so we can attach the handles in order of their index. It turns out that when $2n > 4$, we can recreate all parts of the proof of the h-cobordism theorem for subcritical Weinstein cobordisms. In some sense, subcritical Weinstein domains have no symplectic geometry in them – they are encoded by algebro-topological information, and so this gives some flexibility phenomena.

It turns out that some critical handles behave the same way. The key obstruction to the aforementioned flexibility is that sometimes the data of an attaching Legendrian does not boil down to purely toplogical information. However, Emmy Murphy defined a class of Legendrians, called loose Legendrians, for which there is such a so-called h-principle. The Weinstein h-cobordism theorem works for Weinstein cobordisms which can be built (up to Weinstein homotopy) out of subcritical and loose critical handle attachments. We call such Weinstein cobordisms flexible.

We often care about the case when $2n = 4$. In this case, it is pretty easy to describe a connected Weinstein domain (or its contact boundary). One can first order the handles by index, and then cancel 0-handles with 1-handles until we are in the situation where there is precisely one 0-handle and possibly many 1- and 2-handles. The boundary of the 0-handle is just a standard contact $S^3$, and 1-handle attachment is trivially described by picking pairs of points in $S^3$ (the bundle data boils down to showing $\pi_0(\text{Sp}(2,\mathbb{R})) = 0$). So it suffices to draw Legendrians on $S^3$ with $k$ pairs of points identified, which is just $\#^k (S^1 \times S^2)$. Any Legendrian $L$ has a canonical framing of its normal bundle given by the twisting of the Reeb chord around the Legendrian. Eliashberg showed that adding a left twist to this framing gives the smooth framing which determines the corresponding smooth surgery data.

Gompf showed that in this case $2n = 4$, one can draw standard Kirby calculus type surgery diagrams. We think of all of these 1-handle attachments and Legendrians as missing a point in $S^3$, so that we can draw our diagrams in $(\mathbb{R}^3, \ker dz - ydx)$. The front projection is the projection to the coordinates $(x,z)$, so that $y$ is determined by $dz/dx$. It might not be obvious how to draw a smooth knot in this projection since the curve can’t have infinite slope, but we are allowed semi-cubical cusps, corresponding to $(x,y,z) = (t^2,3t/2,t^3)$. Note that transverse crossings are also allowed, since the $y$-coordinates are distinct. One usually draws the front projection of a Legendrian without showing which strand lies over the other, but we include this extra information in the next figure, where we imagine the $y$-axis as pointing into the page.

A Legendrian trefoil knot

Gompf’s standard form for these Legendrians looks like the following, where the pairs of balls in each row corresponds to where the 1-handles are attached, and the Legendrian strands simply go through the handles as though they were wormholes.

An example of a Gompf surgery diagram. There are three 1-handles (in blue, red, and green) and two 2-handles with attaching spheres given by the Legendrian tangle above. All of the information can be made to live inside of the purple rectangle (i.e. without going horizontally or vertically outside of where the 1-handles are attached).

### Weinstein fillings, Lefschetz fibrations, and open book decompositions

Definition:Lefschetz fibration is a smooth map $\pi \colon W^4 \rightarrow \Sigma^2$ with finitely many critical points with distinct critical values such that locally around the critical points, $\pi$ looks like a complex Morse function (i.e. $(z_1,z_2) \mapsto z_1^2+ z_2^2$ in local coordinates). When $\Sigma$ has boundary, we assume the critical values of $\pi$ are all in the interior of $\Sigma$.

We shall typically be concerned with the case where $\Sigma = \mathbb{D}$ (although see this post by Laura Starkston which slightly generalizes some of what is discussed here).

A schematic for a Lefschetz fibration over the disk

In the case where $\Sigma = \mathbb{D}$, we see that the boundary decomposes as $\partial W = \partial^v W \cup \partial^h W$, where the superscripts are meant to indicate vertical and horizontal. That is, $\partial^v W = \pi^{-1}(\partial \mathbb{D})$, while $\partial^hW = \sqcup_{p \in \mathbb{D}} \partial \pi^{-1}(p)$. If we write $F$ for a regular fiber of $\pi$, then $\partial^h W = \partial F \times \mathbb{D}$. Meanwhile, we see that $\partial^v W$ is just a fibration over $S^1$ with fiber $F$, and hence can be described by some monodromy map $\phi \colon F \rightarrow F$ fixing the boundary, so that $\partial^v W = F \times [0,1]/{\sim}$ where $(\phi(x),0) \sim (x,1)$ (the mapping torus of $\phi$).

The structure on the boundary, in which we have a fibration over $S^1$ with fiber $F$ glued together with $\partial F \times \mathbb{D}$ in the natural way, is called an open book decomposition. It is given completely by the pair $(F,\phi)$. We think of each fiber over $S^1$ as a page, and the subset $F \times \{0\}$ as the binding, analogous to what one would get if one took their favorite book and matched the covers so that the pages radiate outwards. So Lefschetz fibrations yield open books on the boundary. To be a little more precise, one should extend each page so that the boundary of each page is actually the binding.

Some pages near the binding of an open book. I guess the name “Rolodex” wasn’t as catchy as “open book.” (Image from Wikipedia)

Now suppose $0 \in \mathbb{D}$ is a regular value (which can always be arranged up to small perturbation of $\pi$). Then $\pi^{-1}(\epsilon \mathbb{D}) \cong F \times \mathbb{D}$. One can ask what happens when we extend to $\pi^{-1}(U)$, where $\epsilon \mathbb{D} \subset U$ and there is exactly one critical value $p$ on $U \setminus \epsilon\mathbb{D}$.

Since we have a nice fibration away from critical points, we see that paths in $\mathbb{D}$ yield monodromy maps (up to isotopy preserving boundary) on the fibers. We can choose a connection on the fibration if we wish to make this a map on fibers, not just a map up to isotopy. If we take a path $\gamma$ from 0 to $p$ which intersects $\partial \epsilon \mathbb{D}$ once and otherwise avoids critical values then for whatever connection we chose, we can see what points flow to the critical point over $p$. Over each regular fiber, this is just a circle, and the union of all of them together with the critical point yields a disk. The path $\gamma$ is called a vanishing path, and each circle on the regular fiber is called a vanishing cycle (one really should think of it as a homology cycle, but for concreteness, one can think of it as a curve). The disk consisting of the union of vanishing cycles above a path is called a thimble.

The green circles in the regular fibers above the purple vanishing path are the vanishing cycles. Their union is the thimble.

It is then not hard to see that $\pi^{-1}(U)$ is obtained from $\pi^{-1}(\epsilon \mathbb{D})$ by 2-handle attachment, where the attaching curve is just the vanishing cycle above $\gamma \cap \partial \epsilon \mathbb{D}$ and the core of the handle is the thimble. Furthermore, one can check by a local computation that the monodromy map in a loop around $p$ is just given by a Dehn twist (positive or negative, depending on orientations) around the vanishing cycle. Hence, one can write out the open book determined by the Lefschetz fibration explicitly – it is just the product of the Dehn twists on the vanishing cycles, performed in an order determined by a sequence of vanishing paths.

Notice that for a given regular value on $\partial \mathbb{D}$, one can choose a different basis of vanishing paths, and this yields a possibly different factorization for the monodromy. Such changing of the basis is generated by so-called Hurwitz moves, as drawn below.

A Hurwitz move swapping the $i$th and $(i+1)$st critical points. Note that the corresponding vanishing cycles for the critical point corresponding to $\gamma_i$ and $\gamma_{i+1}'$ are actually different, but the overall monodromy on the open book at the boundary is the same.

Hence, understanding Lefschetz fibrations over the disk essentially corresponds to understanding factorizations of mapping class group elements into Dehn twists.

Now, this whole story can be repeated in the symplectic context, as follows.

Definition:symplectic Lefschetz fibration is a Lefschetz fibration with $(W,\omega)$ a symplectic manifold such that each fiber is symplectic submanifold away from the critical points, while at the critical points the coordinates in which $\pi$ locally looks like a complex Morse function can be taken to be holomorphic for some compatible almost complex structure $J$.

In this case, one can take the connection to be the symplectic connection given the symplectic orthogonal complement to the vertical directions. In this way, the thimbles produced will actually be Lagrangian disks, which suggests one can think of these as the descending disks for a Weinstein domain filling the boundary. In addition, the monodromy maps are now compositions of positive Dehn twists only, since the symplectic condition gives the proper orientations. In other words, our Lefschetz fibration is itself positive. If the vanishing cycles of a Lefschetz fibration are homologically nontrivial, we shall call it allowable.

With a little more work, we can obtain the following theorem of Loi and Piergallini (although an alternative proof by Akbulut and Özbağci is more in line with the exposition presented here):

Theorem: Any positive allowable Lefschetz fibration (PALF) yields a Weinstein domain, and any Weinstein domain comes from a PALF in this way.

Furthermore, one obtains a little bit more compatibility at the boundary.

Definition: An open book decomposition on a manifold $M$ is said to support a cooriented contact structure $\xi$ if there is some contact form $\alpha$ for $\xi$ such that the binding is a contact submanifold, $d\alpha$ is a symplectic form on the pages, and the boundary orientation of the page (with respect to $d\alpha$) matches the orientation of the binding with respect to $\alpha$.

One checks that the open book on the boundary of a PALF does indeed support the contact structure determined by being the boundary of a Weinstein domain.

Our surgery theory for these Lefschetz fibration builds the fiber up by subcritical surgery, and the 2-handle attachments correspond to the critical points of the fibration. One can always produce, for any Weinstein manifold, a cancelling pair consisting of a 1-handle and a 2-handle. The way that this affects the open book is by positive stabilization, meaning that one adds a 1-handle to the page, but kills it by adding an extra Dehn twist to the monodromy through a circle which passes through the handle.

The following theorem implies that all 3-dimensional contact geometry can actually be encoded (somewhat non-trivially) in the study of open books up to positive stabilization, and hence the study of Weinstein fillings reduces to studying positive factorizations of given elements of the mapping class group of a surface with boundary (up to this not-so-easy-to-work-with notion of positive stabilization).

Theorem [Giroux correspondence]: There is a one-to-one correspondence between contact structures on a closed 3-manifold up to isotopy with open books up to positive stabilization.

### Applications to Weinstein fillings

To summarize the previous section, an explicit surgery decomposition of a Weinstein filling yields a PALF which in turn gives an open book structure supporting the contact boundary of the Weinstein filling with monodromy factored into positive Dehn twists. Conversely, given a supporting open book for a contact structure with monodromy factored into positive Dehn twists, one obtains a Weinstein filling.

One common question we ask is whether a single contact manifold has multiple Weinstein fillings. From the above construction, one possible way to attack this problem is to look for distinct positive factorizations of a given element in a mapping class group.

Theorem [Auroux]: There is an element in the mapping class group of the surface $\Sigma_{1,1}$ (of genus 1 and with one boundary component) with two distinct factorizations into positive Dehn twists such that the Weinstein fillings are distinguished by their first homology.

Remark: In this setting, the first homology is just given by $H_1(F)/V$ where $V$ is the span of the vanishing cycles. The only real trick of Auroux is therefore to find a good candidate for the above theorem to hold, and just compute.

Generalizing a bit more:

Theorem [Baykur – Van Horn-Morris]: There exists an element in the mapping class group of $\Sigma_{1,3}$ (of genus 1 with three boundary components) which admits infinitely many positive factorizations such that the corresponding Weinstein fillings are all distinguished from each other by their first homology.

Finally, as one last application, I want to consider a result of Plamenvskaya and Van Horn-Morris, but I need to define the contact structures in question to begin. Honda’s classification of tight contact structures on the lens spaces $L(p,1)$ can be formulated in Gompf’s surgery diagrams by the following diagrams, coming from a single 2-handle attachment to standard $S^3$. We denote the corresponding contact structures by $\xi_1,\xi_2,\ldots, \xi_{p-2}$.

The surgery diagram for the contact structure $\xi_k$.

Of these, the universal covers of $\xi_1$ and $\xi_{p-2}$ are also tight, where as the others’ universal covers are overtwisted. We say $\xi_2, \ldots, \xi_{p-3}$ are virtually overtwisted.

Theorem [PV]: Each virtually overtwisted $(L(p,1), \xi_k)$ has a unique Weinstein filling (up to symplectic deformation) and a unique minimal weak filling.

Proof sketch: Let us first discuss the Weinstein part. There are a few nontrivial theorems which go into this, which we won’t discuss, but essentially we have the following sequence of results. The open book given by the surgery diagrams above induce open books with genus 0 pages. When we discussed Wendl’s theorem in part 2 of the J-holomorphic curve posts, one thing we mentioned was that one can apply his techniques when there is a planar open book (meaning pages have genus 0). He proves that if a contact manifold has a given supporting planar open book, then every Weinstein filling is diffeomorphic to one compatible with that specified planar open book. Hence, it suffices to study Lefschetz fibrations compatible with the one just described, which in turn becomes studying factorizations of an element in the mapping class group of $\mathbb{D}_n$, the disk with $n$ holes. A nontrivial result of Margalit and McCammond gives that every such presentation must be in a certain form, from which one can use smooth Kirby calculus to conclude that the surgery diagram must come from $-p$-surgery on some knot. Finally, an appeal to work of Kronheimer, Mrowka, Ozsváth, and Szabó using Seiberg-Witten Floer homology (also called monopole Floer homology) yields that this knot must have been an unknot, and since the framing is $-p$, this determines the canonical framing of the knot, which in turn implies we could only have had one of our original surgery diagrams.

Finally, to obtain the weak part, one can use work of Ohta and Ono to boost a weak filling up to a strong filling, from which Wendl’s theorem implies that any minimal weak filling is symplectic deformation equivalent to a Weinstein filling.