Multisections with divides and Weinstein 4-manifolds

We will call the handlebodies which are standard neighborhoods of Legendrian graphs standard contact handlebodies. Note that a smooth handlebody of a given genus typically has multiple different “standard” contact structures, which are differentiated by the number of components of the dividing set on the boundary. The different options come from the fact that there are generally multiple different surfaces with boundary whose doubles are a genus $g$ surface (determined by the number of boundary components).

This notion of contact Heegaard splittings originated with Giroux [Giroux] and was also developed by Torisu [Tor00], and can be equivalently formulated as follows.

Though these equivalences are known, we review how to pass between them. If we start with $H_{1}$ and $H_{2}$ as standard neighborhoods of Legendrian graphs, we can see the page $F$ of the corresponding open book decomposition as a contact framed ribbon of the Legendrian. The contact planes are tangent to $F$ along the Legendrian, so in a standard neighborhood, $d\alpha$ is a positive area form when restricted to this page. Moreover, in the standard model, we can identify $H_{i}=F\times I/\sim$ (where $(x,t)\sim(x,t^{\prime})$ for $x\in\partial F$), such that $d\alpha$ is positive on each $F\times\{t\}$. In this way, we see that the first characterization gives rise to the second characterization. Conversely, given a supporting open book decomposition, we can Legendrian realize a spine on two pages. Restricting the contact structure defined on an abstract open book to half of the pages, we see that this is a standard neighborhood of this Legendrian spine. The dividing set on the boundary of a standard neighborhood of a Legendrian graph is precisely the intersection of the contact framed ribbon with $\Sigma=\partial H_{i}$. The meridian of each edge of the graph thus intersects the dividing set in exactly two points. Similarly, if $H_{1}$ and $H_{2}$ are two halves of an open book decomposition, the dividing set on $\Sigma=\partial H_{1}=\partial H_{2}$ is the binding of the open book $\partial F$. A collection of compressing disks for $F\times I/\sim$ is given by $a_{j}\times I/\sim$ where $\{a_{j}\}$ is a collection of arcs on $F$ which cut $F$ into a disk. Each $a_{j}\times I/\sim$ intersects the binding in exactly two points (the endpoints of $a_{j}$ in $\partial F$). If we have a Heegaard splitting with compression disks for each handlebody which each intersect the dividing set in two points, we can cut along these compressing disks to get a ball with the unique tight contact structure. Reversing the cuts amounts to attaching standard contact $1$-handles which gives a standard neighborhood of a Legendrian graph.

There is a fundamental challenge in obtaining compatibility between a multisection and a symplectic structure. Each interior $3$-dimensional handlebody in a multisection appears in the boundary of two $4$-dimensional sectors, but the boundary orientations are opposite to each other. Viewing a sector as a symplectic filling of its boundary induces a contact structure on the boundary which is a positive contact structure with respect to the boundary orientation. The sign of a contact structure (with respect to a fixed orientation) is an inherent property of the contact planes which measures the direction/handedness of the twisting of the contact planes. Note, this is not the same as the co-orientation of the contact structure which depends only on the contact form, not the contact structure. Therefore, we cannot have identical contact structures on a fixed manifold realize both positive and negative contact structures with respect to a fixed orientation. In general this suggests that we would need two different contact structures on each interior $H_{i}$ of a multisection. However, as we show in the following lemma, there are both positive and negative standard contact handlebodies which are orientation reversing contactomorphic to each other. Both the positive and negative contact structures are supported by the same half open book.

Motivated by the third characterization in Lemma 2.3, we define a diagrammatic version of contact Heegaard splittings.

Given a contact Heegaard diagram, we can reconstruct a contact manifold together with a contact Heegaard splitting. In particular, if $F$ is one of the halves of $\Sigma\setminus d$, then we endow each of the handlebodies the standard contact structures coming from $F\times I$. This induces an open book with page $F$ and binding $d$, which in turn induces a contact structure. Conversely, every contact Heegaard splitting has a contact Heegaard diagram by characterization (3) of Lemma 2.3.

This correspondence allows us to give a classification of genus-1 contact Heegaard splittings of $S^{3}$ as we will see in Section 5.1. In particular, by Euler characteristic considerations, genus-1 contact Heegaard splittings correspond to open book decompositions of $S^{3}$ with an annular page. The monodromy then consists or a left-handed or right handed Dehn twist about the core of this annulus. The right handed Dehn twist gives the tight contact structure on $S^{3}$ whereas the left handed Dehn twist gives an overtwisted structure. These lead to the Heegaard diagrams shown in Figure 2.

In Section 3, we will be looking at Heegaard splittings of $S^{3}$, or more generally $\#_{k_{1}}S^{1}\times S^{2}$ where one handlebody is a standard neighborhood of a Legendrian graph $\widetilde{\Lambda}$ described via a front projection, and the other handlebody is the complement. In order to verify that the complement is a standard contact handlebody, it will be useful to know exactly how to draw the dividing set on the boundary of the standard neighborhood of an explicitly embedded Legendrian graph in terms of the front projection.

We will mainly focus on trivalent Legendrian graphs. Higher valence vertices can be split into trivalent vertices by growing additional Legendrian edges via a Legendrian deformation which preserves the standard neighborhood and thus, the dividing set on its boundary. Given a front projection representing a Legendrian embedding in $\mathbb{R}^{3}$, we can draw the corresponding dividing set on the boundary of a standard neighborhood. Recall that the dividing set on the boundary of a standard contact handlebody is the boundary of the page of the compatible open book decomposition. A page of the open book is given by the contact framed ribbon of the Legendrian knot. Considering how the contact framing wraps around at left and right cusps and at left and right trivalent vertices, we obtain the local models for the dividing set as shown in Figure 3. The first five models cover the generic front projections of a trivalent Legendrian graph. The last model includes a Legendrian arc which degenerately projects to a single point, whose two end points are trivalent vertices. We can isotope this model to a generic front projection as in Figure 4, and thus derive its local model from the previous models. We include this last “compound” model for convenience as we will use it extensively in implementing our algorithm of Section 3.1.

A fundamental feature of multisections with divides is that they can be completely encoded on a surface. In this section we define these diagrams and show how the previous proof gives an algorithm for obtaining a bisection diagram with divides from a Kirby-Weinstein handlebody diagram.

The proof of Theorem 3.1 gives an algorithmic method to obtain a multisection diagram with divides as follows.

Starting from a Kirby-Weinstein handlebody diagram, construct the Legendrian graph $\widetilde{\Lambda}$ as in the proof. Use the models from Figure 3 to draw the dividing set $d$ on $\Sigma$ as the boundary of the neighborhood of $\widetilde{\Lambda}$ ($H_{2}$) in terms of the front projection.

We can describe cut systems $C_{i}$ $i\in\{1,2,3\}$ for the handlebodies $H_{1}$, $H_{2}$, and $H_{3}$ as follows. The cut system $C_{1}$ is given by taking the regions of the planar diagram together with a curve tracing through the vertical tunnels added between the 2-handles passing through each 1-handle, as in Figure 6–note the curve in this figure intersects the dividing set in two points. As the regions are diagrams of $tb=-1$ unknots, each of the curves given by the boundary of a region intersects the dividing set twice. This ensures that the pair $(\Sigma,C_{1})$ is a diagram of a standard contact handlebody.

The cut system $C_{2}$ is given by taking a meridian of each tunnel together with a meridian of each knot in the Kirby-Weinstein diagram. Using the local model at the top of Figure 3 and Figure 8 we see that these curves intersect the dividing set twice so that $(\Sigma,C_{2})$ is also a standard contact handlebody. Because $(\Sigma,C_{1},C_{2},d)$ represents a Heegaard splitting of the boundary of the $0$- and $1$-handles of the Kirby-Weinstein handlebody diagram, it is symplectically fillable and thus must support the tight contact structure on a connected sum of copies of $S^{1}\times S^{2}$.

To obtain $C_{3}$, we start by taking a Legendrian push off of each knot component $K_{i}$ in the diagram. Each such component intersects its chosen meridian in $C_{2}$ once and does not intersect the dividing set at all. Adding a left handed twist about the meridian to each component $K_{i}$ gives a curve $K_{i}^{\prime}$ whose surface framing is the contact framing $-1$ and which intersects the dividing set twice. Then replacing each meridian in $C_{2}$ with the corresponding $K_{i}^{\prime}$ gives a diagram $(\Sigma,d,C_{2},C_{3})$ representing a contact Heegaard splitting of a connected sum of copies of $S^{1}\times S^{2}$. Therefore $(\Sigma,d,C_{1},C_{2},C_{3})$ is a bisection diagram with divides of the given manifold.

This algorithm is carried out in Figures 9 and 10 for the result of attaching a Weinstein handle to the max $tb$ right handed trefoil and in Figures 11 and 12 for the Weinstein domain $T^{*}\mathbb{R}\mathbb{P}^{2}$, which is a disk bundle over $\mathbb{R}\mathbb{P}^{2}$.

Note that the cut systems $C_{2}$ and $C_{3}$ which are output from our algorithm have a very particular form. More specifically, each component $\gamma$ of $C_{3}$ either agrees with or is dual to a component $\beta$ of $C_{2}$. In the latter case, there exists a curve $V$ in $\Sigma$ which is disjoint from the dividing set, dual to the component $\beta$ with respect to $C_{2}$, such that $\gamma=\tau^{-1}_{\beta}(V)=\tau_{V}(\beta)$ where $\tau_{A}$ is a right-handed Dehn twist about $A$ with respect to the orientation on $\Sigma$ induced as the boundary of $H_{2}$. Let’s call two cut systems related in this way standard Weinstein cobordant. By isotoping each $V$ into the Legendrian core of the handlebody $H_{2}$, we find a Legendrian link in $H_{2}$ such that the $ct-1$ framing for this link is represented by the corresponding $\gamma$ curves in $C_{3}$, ($V$ represents the contact framing, and $\beta$ is a meridian of the surgery torus so $\tau^{-1}_{\beta}(V)$ represents the contact framing $-1$). Thus we see that if two cut systems with a dividing set are standard Weinstein cobordant, the corresponding sector can be endowed with the structure of a Weinstein cobordism from $H_{i}$ to $H_{i+1}$ obtained by attaching Weinstein $2$-handles to $H_{i}\times I$. We can always endow the first sector with the structure of a Weinstein $1$-handlebody (since by definition $(\Sigma,d,C_{1},C_{2})$ is a contact Heegaard splitting of $\#_{k_{1}}S^{1}\times S^{2}$ with the tight (fillable) contact structure). Therefore, we have an (overly strong) condition that ensures a multisection diagram with divides corresponds to a Weinstein $4$-manifold.

In Section 4, we will see another condition that ensures a multisection diagram with divides corresponds to a Weinstein $4$-manifold. It is an interesting question to ask whether there is a general characterization of all multisection diagrams with divides which correspond to Weinstein $4$-manifolds. In general, we only expect a multisection diagram with divides to correspond to a symplectic $4$-manifold, which may not admit any global Weinstein structure.

Fibration structures on symplectic manifolds have a long history of study, dating to Donaldson’s work in [Don99] where it was shown that every closed symplectic 4-manifold admits a Lefschetz pencil. Conversely, Gompf proved that every 4-manifold with a Lefschetz pencil admits a symplectic structure [GompfStipsicz]. The corresponding objects in the Weinstein category are positive allowable Lefschetz fibrations.

Following Loi and Piergallini [LoiPie01], every Weinstein domain admits a PALF. Conversely, every PALF supports a Weinstein structure [GompfStipsicz].

In this section, we will show how to use the PALF structure to obtain a decomposition of a Weinstein domain as a multisection with divides. As in Figure 13, we cut the disk $D^{2}$ into closed subdisks, $D_{1},\dots,D_{n}$, such that

• •

  Each $D_{i}$ contains a unique critical value of $\pi$,

• •

  $D_{i}\cap D_{i+1}$ is diffeomorphic to an interval for $i=1,\cdots,n-1$,

• •

  $\partial D^{2}\subset X_{1}\cup X_{n}$, and

• •

  $D_{1}\cap\cdots\cap D_{n}=\{(0,1),(0,-1)\}$.

Then let $X_{i}=\pi^{-1}(D_{i})$, $H_{1}=\pi^{-1}(\partial D^{2}\cap D_{1})$, $H_{i}=\pi^{-1}(D_{i-1}\cap D_{i})$ for $i=2,\dots,n$, $H_{n+1}=\pi^{-1}(\partial D^{2}\cap D_{n})$. Then $H_{i}=X_{i-1}\cap X_{i}$ and $X_{1}\cap\cdots\cap X_{n}=\partial H_{i}$ (after quotienting by the $D^{2}$ factor along points in $\partial F\times D^{2}$ which is a Weinstein homotopic domain).

A PALF can be encoded combinatorially through the fiber surface $F$ and the ordered set of vanishing cycles $(c_{1},\dots,c_{n})$. In this section we show how to use the combinatorial data of a PALF to obtain the combinatorial data of the multisection diagram with divides corresponding to the decomposition from Theorem 4.2.

The monodromy about a Lefschetz critical value with vanishing cycle $c$ is a right handed Dehn twist about $c$, which we denote by $\tau(c)$. Thus a PALF can equivalently be encoded by an ordered sequence of right handed Dehn twists about the vanishing cycles called a monodromy factorization.

The core surface of the multisection $\Sigma$ is diffeomorphic to the union of two copies of $F$ glued together along their boundary. The dividing set on $\Sigma$ is given by the boundary of $F$. More precisely, if $\Psi_{i}:F\times I\to H_{i}$ is the diffeomorphism defined by parallel transport, $\Sigma=\Psi_{i}(F\times\{0\}\cup F\times\{1\})=F\cup\Psi_{i}(F\times\{1\})$ (note we are suppressing the quotient of the $I$ direction at points in $\partial F$).

To understand a multisection diagram with divides, we want to fix an identification of $\Sigma$, and then draw cut systems for each handlebody $H_{i}$. We will use $\Psi_{1}$ to identify $\Sigma=\partial H_{1}$ as $F_{0}\cup F_{1}$. Then the restriction of $\Psi_{i}$ to $F_{0}\cup F_{1}$ gives a diffeomorphism from $\Sigma$ to $\partial H_{i}$.

Let $\{a_{1},\dots,a_{k}\}$ be a complete arc system for $F$ i.e. a collection of properly embedded arcs which cut $F$ into a disk. Then $\{\Psi_{i}(a_{1}\times I),\dots,\Psi_{i}(a_{k}\times I)\}$ gives a cut system of disks for $H_{i}$. We want to see the boundaries of these disks on our fixed identification of $\Sigma$. Namely, we want to describe the curves $\Psi_{i}^{-1}(a_{j}\times\{0\}\cup a_{j}\times\{1\})$ in $F_{0}\cup F_{1}$. Each $\Psi_{i}$ is the identity on $F_{0}$, and defines parallel transport along the arc from $(0,1)$ to $(0,-1)$ over which $H_{i}$ lies. Since the $\Psi_{i}$ define parallel transport, and we are using $\Psi_{1}$ to identify $\Sigma$ with $F_{0}\cup F_{1}$, we see that $\Psi_{i}(a_{j}\times\{1\})$ in $\Sigma$ is the image of $a_{j}$ under the monodromy around the curve which goes from $(0,-1)$ to $(0,1)$ along the $H_{i}$ curve and then goes from $(0,1)$ to $(0,-1)$ along the $H_{1}$ curve. This monodromy is $\tau(c_{1}),...,\tau(c_{i-1})$. Thus, the cut system for $H_{i}$ is obtained from the cut system for $H_{i-1}$ by applying the right handed Dehn twist $\tau(c_{i-1})$ where $c_{i-1}\subset F_{1}$. Any choice of complete arc system defines a cut system for $H_{1}$ by gluing together the same arcs on $F_{0}$ and $F_{1}$.

To summarize, given a PALF with fiber $F$ and ordered vanishing cycles $\{c_{1},\dots,c_{n}\}$, the corresponding multisection with divides is given by

• •

  $\Sigma=F_{0}\cup_{\partial}F_{1}$ where $F_{0}$ and $F_{1}$ are copies of $F$ ($F_{0}$ is oppositely oriented).

• •

  The dividing set $d$ is $\partial F=F_{0}\cap F_{1}$.

• •

  The cut system for $H_{1}$ is $\{a_{1}\cup a_{1},\dots,a_{g}\cup a_{g}\}$ where $\{a_{1},\dots,a_{g}\}$ is a complete arc system for $F$.

• •

  For $i>1$, the cut system for $H_{i}$ is obtained from the cut system for $H_{i-1}$ by applying a right handed Dehn twist about $c_{i-1}\subset F_{1}$.

Note that each cut curve intersects the dividing set in exactly two points (the end points of the arc $a_{j}$). (This is clearly true for $H_{1}$, and it remains true for each $H_{i}$ because the Dehn twists are applied in the interior of $F_{1}$ which is disjoint from the dividing set.)

The previous construction suggests that multisection can also be encoded by a monodromy factorization, and here, we show that this is indeed the case. Through some care, one can determine a particular monodromy factorization from a multisection, however for the present paper it will be sufficient to have a family of possible factorizations.

Fixing a surface $\Sigma$, a system of cut curves defines a handlebody with boundary $\Sigma$. Given a handlebody, we can choose any system of cut curves which bound disks in the handlebody to encode it. If we specify a diffeomorphism $\Phi$ of $\Sigma$, we can apply that diffeomorphism to a cut curve system, to produce a new cut curve system. If the original cut curve system defines a handlebody $H$, we let $H_{\Phi}$ denote the handlebody defined by the image of a cut-system for $H$ under the map $\Phi$. Note this is independent of the choice of cut-system for $H$, because two different cut systems $\{\alpha_{i}\}$ and $\{\alpha_{i}^{\prime}\}$ for $H$ will be related by some sequence of handle slides, and thus $\{\Phi(\alpha_{i})\}$ will be related to $\{\Phi(\alpha_{i}^{\prime})\}$ by a sequence of handle slides as well. We begin with a lemma which yields a sufficient condition for $H$ and $H_{\Phi}$ to form a Heegaard splitting of $\#_{g-1}S^{1}\times S^{2}$ when $\Phi$ is a Dehn twist.

Using the above lemma, we obtain a sufficient criteria for Dehn twists on cut systems to yield two sequential handlebodies in a multisection. Conversely, the following proposition shows that the sequential handlebodies in all multisections can be obtained in the fashion.

We call the product $\Pi_{i=1}^{n}\Pi_{j=1}^{g-k_{i}}\tau(c_{j}^{i})$ a monodromy factorization for $\mathfrak{M}$. By following through our construction in Theorem 4.2, we can track the monodromy of a PALF onto the monodromy of a multisection which immediately yields the following.

In this section we will demonstrate how a monodromy substitution affects a multisection with divides. We begin with the analogous construction for PALFs.

Monodromy substitution has been used extensively to produce new symplectic manifolds from existing ones. In particular, in [EndGur10], the authors show that the lantern relation, pictured in Figure 14, can be used to perform a rational blowdown on the configuration $C_{2}$ (see [GomSti99] Section 8.5 for an exposition on these operations). This was later generalized in [EMV11] to realize an infinite family of rational blowdowns as monodromy substitutions using daisy relations. In general, any monodromy substitution can be thought of as some symplectic cut-and-paste operation.

There is an analogous process of monodromy substitution on a multisection.

It follows immediately from Corollary 4.7 that we can find monodromy substitutions by doubling a PALF and a monodromy substitution of that PALF. Carrying this out for the lantern relation gives us a monodromy substitution on a multisection with divides yielding the operation outlined in Figure 15.

In this subsection we will provide a characterization of genus-1 multisections with divides. For examples of the diagrams for the unique 2- and 3-sections with divides, see Figure 16. Smooth genus-1 multisections are well characterized by their diagrams, which consist of sequences $(\alpha_{1},...\alpha_{n})$ with $|\alpha_{i}\cap\alpha_{i+1}|=1.$ In [IKLM], the authors show that smooth genus-1 $n$-sections with boundary correspond to linear plumbings of $(n-1)$ disk bundles over the sphere. Moreover, given the oriented sequence of cut curves $(\alpha_{1},\alpha_{2},...\alpha_{n})$ defining a genus-1 multisection diagram, the Euler number of the $i^{th}$ disk bundle is given by the algebraic intersection $\langle\alpha_{i-2},\alpha_{i}\rangle$.