# Monthly Archives: August 2013

## From Rulings to Augmentations

This is part III of a post on the relationship between augmentations and rulings. If you missed parts I and II, you can find them here and here.

Now we’ll work on going from rulings to augmentations. Fuchs does this using what he calls “splashes” in diagrams, but I find it easier to see this using Sabloff’s method of defining an augmentation for the dipped diagram as one can then get an augmentation for the original diagram.

Given a ruling for the original diagram in plat position, we will define an augmentation for the dipped diagram. First, augment $c_k$ if the ruling is switched at $c_k$, augment $a_{rs}$, the crossing of strands $r$ and $s$ in the $a$-lattice, if strands $r$ and $s$ are paired between $c_k$ and $c_{k+1}$, (this is what we called Property (R) in part II), and augment $b_{rs}$, the crossing of strands $r$ and $s$ in the $b$-lattice, if one of the following holds:

• ruling looks like (a) at the previous crossing $c_k$ and strands $r$ and $s$ are crossing strands,
• ruling looks like (b) or (c) at the previous crossing $c_k$ and strands $r$ and $s$ are crossing or companion strands,
• ruling looks like (e) or (f) at the previous crossing $c_k$ and strands $r$ and $s$ are companion strands.

Recall the various crossing configurations.

(From Sabloff’s paper.)

Let’s check for a couple of these cases that this gives an augmentation of the dipped diagram. In other words, check that for each crossing in the dipped diagram there are an even number of totally augmented disks in the diagram with positive corner at that crossing.

First, check the left end of the diagram. Since, in the ruling strands $2k$ and $2k-1$ are paired at the left, we know the crossing in the first $a$-lattice of strands $2k$ and $2k-1$ is augmented for $1\leq k\leq m$.

We then see that we have the totally augmented disks depicted.
So $\epsilon'\circ\partial=0$ on this portion of the diagram.

Most of the crossings in the dips I will leave for you to check, but we will check the dip after a crossing of configuration (c). Thus the ruling is switched at that crossing. Suppose strands $i$ and $i+1$ cross at the crossing $c_k$  and that strand $i$ is paired with $L$ and strand $i+1$ is paired with $K$.
Since the ruling is switched at the crossing, we know the crossing is augmented. We also see that the following other crossings are augmented as well, from the pairing of the strands in the ruling.
To check whether $\epsilon'\partial=0$ on the crossings in the dip after the original crossing, look for totally augmented disks.

Clearly there aren’t any totally augmented disks with positive corner at $c_k$, so $\epsilon'\partial c_k=0$.

We see that there are two totally augmented disks contributing to $\epsilon'\circ\partial$ of the crossing $b_1$ in the $b$-lattice of strands $K$ and $i$ and so $\epsilon'\partial(b_1)=1+1=0$. (Recall that we are working mod 2.)
We see that there are two totally augmented disks contributing to $\epsilon'\circ\partial$ of the crossing $b_2$ in the $b$-lattice of strands $L$ and $i$ and so $\epsilon'\partial(b_1)=1+1=0$.
Similarly, we have disks for crossings $b_3$ and $b_4$ in the $b$-lattice of, respectively, strands $K$ and $i+1$ and strands $L$ and $i+1$.
None of the remaining crossings in the $a$– or $b$-lattice have totally augmented disks, so we have checked that on this region of the dipped diagram, $\epsilon'$ is an augmentation.

Now, let’s look at the right end of the dipped diagram. Since we have the ruling of the original diagram, we know that at the right end of the diagram, strands $2k$ and $2k-1$ are paired in the ruling for $1\leq k\leq m$. Following our algorithm, this means that the crossings in the $a$-lattice of strands $2k$ and $2k-1$ are augmented for $1\leq k\leq m$.
Thus we have the following totally augmented disks for $q_k$.
Again, we see that there are two totally augmented disks with positive corner at $q_k$, so $\epsilon'\circ\partial(q_k)=1+1=0$.

Thus, with some checking of the remaining cases, we have shown that the augmentation of the dipped diagram we defined, is in fact an augmentation and so, given a way to define an augmentation of the dipped diagram of a knot from a ruling of the knot.

Filed under Uncategorized

## From Augmentations to Rulings

This is part II of a post on the relationship between augmentations and rulings. Here is part I if you missed it.

Given a front diagram of a Legendrian knot in plat position, we will use Sabloff’s method of dips to show that if there exists an augmentation of the Chekanov-Eliashberg DGA, then there exists a normal ruling of the diagram. We will extend the augmentation $\epsilon$ of the diagram to an augmentation $\epsilon'$ of the dipped diagram satisfying Property (R) which will simultaneously build a normal ruling of the original front diagram.

We will extend $\epsilon$ to $\epsilon'$ by adding dips to the diagram from left to right and extending the augmentation and ruling as we go as follows: If a crossing $c_k$ is augmented in the original front diagram, then augment $c_k$ in the dipped diagram. In a dip, we decide whether a crossing in the $a$-lattice is augmented by following Property (R).

Property (R): At the dip between $c_k$ and $c_{k+1}$, a crossing in the $a$-lattice is augmented iff the two strands which cross there are paired in the ruling of the original front diagram between $c_k$ and $c_{k+1}$.

A crossing $b$ in the $b$-lattice between crossings $c_k$ and $c_{k+1}$ is augmented based on whether the strands which cross at $b$ are crossing or companion strands in the ruling of the original front diagram and which diagram below $c_k$ looks like .

(Figure from Sabloff’s paper.)

In particular, if $c_k$ has configuration ____ and is augmented, then $b$ is augmented if the strands crossing at $b$ are _____ strands. (Use the following to fill in this statement.)

(a), crossing
(b) or (c), crossing and companion
(d), none
(e) or (f), companion

Note that this definition relies on simultaneously building a ruling for the original front diagram. So, let us describe how to build the ruling. If we look at the left end of the ruling of a front diagram in plat position, the ruling will have strand $2k$ paired with strand $2k-1$ for all $1\leq k\leq m$. To extend the ruling over a crossing $c_k$, if $c_k$ is augmented and it looks like configuration (a), (b), or (c) just to the left of $c_k$, then switch the ruling at $c_k$. Otherwise, don’t switch the ruling.

One can check that this gives us an augmentation of the dipped diagram. For example, suppose $c_k$ is augmented and has configuration (c) just to the left. We will denote augmented crossings by a large dot. Suppose strands $i$ and $i+1$ cross at $c_k$ and that strand $i$ is paired with strand $L$ and strand $i+1$ is paired with strand $K$ between crossings $c_k$ and $c_{k+1}$.

Since $c_k$ is augmented and looks like configuration (c) to the left, our ruling is switched at $c_k$.

(I apologize for the crude drawings, but Inkscape wasn’t cooperating.)

Now let us use this diagram to compute $\epsilon'\circ\partial$. Recall that the only disks that contribute to $\epsilon'\circ\partial(c)$ are totally augmented disks, disks with a positive corner at $c$ and with all negative corners augmented by $\epsilon'$. Each such disk will contribute 1 to $\epsilon'\circ\partial(c)$, so as long as there are a even number of such disks, $\epsilon'\circ\partial(c)=0$.

There are no such disks for $c_k$, so $\epsilon'\circ\partial(c)=0$. A more interesting computation is the crossing $b_1$ in the $b$-lattice of strands $i$ and $K$. We see that there are two such disks.

So $\epsilon'\circ\partial(b_1)=0$.

We also have the following disks for the crossing $b_2$ in the $b$-lattice of strands $i+1$ and $L$. So $\epsilon'\circ\partial(b_2)=0$.

A crossing with more standard disks appearing, is the crossing $b_3$ in the $b$-lattice where strands $i+1$ and $K$ cross. It has the following disks.

There are no such disks for any crossing in the $a$-lattice, so $\epsilon'\circ\partial=0$ on the $a$-lattices. One can check that $\epsilon'\circ\partial=0$ for all remaining crossings.

While the dipped diagram makes differentials much easier, they tend to be a pain to deal with. Luckily, there’s a shortcut for finding the ruling associated to an augmentation of the original front diagram, without going through the dipped diagram:

We will modify our augmentation of the crossings of the plat position diagram while extending the ruling from left to right. Start the ruling at the left cusps like usual, and extend as follows. If a crossing $c_k$ looks like configuration (a), (b), or (c) to the left and the crossing is augmented, then switch the ruling at the crossing and update the augmentation on the crossings to the right based on the number of disks to the right which contribute $Qc_kR$ and involve the crossing strands.

Otherwise do not switch at the crossing. However, if a crossing $c_k$ looks like configuration (e), or (f) to the left and $c_k$ is augmented, then update the augmentation of the crossings to the right based on the number of disks to the right which contribute $Qc_kR$ and involve the companion strands.

This is easier to see in an example. Let’s look at the trefoil where all crossings of degree $0$ are augmented.

Start the ruling at the left.

The first crossing is augmented and the left of the crossing matches (a), so switch the ruling at $c_1$.

Now update the augmentation by looking for the disks $QaR$. We only find one.

So we get the diagram so far, with updated augmentation.

As $c_2$ is not augmented, the ruling is not switched and no updating of the augmentation is necessary.

The final crossing does not match (a), (b), or (c) to the left, so do not switch the ruling at $c_3$.

It turns out the augmentation where only $c_1$ and $c_2$ are augmented will give the same ruling. Ng and Sabloff’s paper The Correspondence Between Augmentations and Rulings for Legendrian Knots discusses how this is a many-to-one map and gives the number of $\rho$-graded augmentations going to the same $\rho$-graded ruling for $\rho\vert2r(K)$ and $\rho=0$ or $\rho$ odd. (You’ll have to read up on what a $\rho$-graded augmentation is in the paper.)

1 Comment

Filed under Uncategorized

## Setup for the Relationship Between Augmentations and Rulings

The Chekanov-Eliashberg DGA is an invariant which associates a DGA over the integers mod 2 to a Lagrangian projection of a Legendrian knot. The generators come from crossings and the differential comes from counting immersed polygons whose edges lie in the diagram of the knot and whose vertices lie at crossings. (For more on how it is defined, see Chekanov’s paper Differential Algebras of Legendrian Links.) One can then look at augmentations of the DGA which are algebra maps $\epsilon:\mathcal{A}\to\mathbb{Z}/2$ which satisfy $\epsilon\circ\partial=0$ and $\epsilon(1)=1$. If $\epsilon(c)=1$, we call $c$ augmented. If we go back to the diagram of our knot, we can define a “normal ruling” of the diagram. In the paper Chekanov-Eliashberg invariants of Legendrian knots: Existence of augmentations, Fuchs uses “splashed” diagrams to show that if a Legendrian knot has a normal ruling, then it has an augmentation. Then, in Augmentations and rulings of Legendrian knots, Sabloff uses “dipped” diagrams to show that if a knot has an augmentation, then it has a normal ruling. We will use Sabloff’s method of “dipped” diagrams to show both results. However, in this posting, I will only give the necessary notation and definitions. Sorry there are so many.

First off, a ruling of the front diagram of a knot consists of a one-to-one correspondence between the left cusps and the right cusps and, for each pair of corresponding cusps, two paths in the front diagram that join them. They must also satisfy the conditions that two paths in the ruling meet only at crossing or cusps and the interior of the two paths joining corresponding cusps are disjoint and so only meet at the cusps and so bound a topological disk. Thus, the paths in a ruling cover the front diagram. Near a crossing, we call the two ruling paths (one from each of two paired cusps) which are on strands that cross, crossing paths and the paths paired with these, companion paths. We say the ruling is switched at a crossing if one of the crossing paths lies entirely above the other near the crossing. If all of the switched crossings of a ruling are one of the following, we call the ruling normal.

(Figure from Sabloff’s paper.)

For example, the three normal rulings of the trefoil are

(Figure from Sabloff’s paper.)

Recall that the DGA for the trefoil

is $\mathbb{Z}/2\langle a_1,a_2,a_3,q_1,q_2\rangle$ with the only nontrivial differentials

$\partial q_1=1+a_1+a_3+a_3a_2a_1$
$\partial q_2=1+a_1+a_3+a_1a_2a_3$

So an example of an augmentation for the trefoil is $\epsilon(a_1)=1=\epsilon(a_2)$ and $\epsilon(a_3)=0$.

Unfortunately, we still need a few more definitions.

We will assume that the front diagram of our knot is in plat position, in other words, all the left cusps have the same x-coordinate, all the right cusps have the same x-coordinate, and none of the crossings have the same x-coordinate. The trefoil above is in plat position. We can make this assumption as a series of Legendrian versions of the Reidemeister II moves and planar isotopies will put any front diagram in plat position. Call the crossings in this diagram $c_1,\dots,c_n$.

Suppose we have $m$ right cusps and at any x-coordinate in the diagram, label the strands in the diagram from bottom to top by $1,\ldots,2m$.

A dip is constructed by a sequence of Reidemeister II moves, which looks as follows in the front projection and Lagrangian projection.

(Picture from Sabloff’s paper.)

The dipped diagram involves introducing a dip between each crossing in the plat position diagram and in between the left, respectively right, cusps and the first, respectively last, crossing. Each Reidemeister II move introduces two new crossings. In each dip, call the new crossings which appear on the left the $b$lattice and the right the $a$lattice. While dipped diagrams have many more crossings than the original knot diagram, the differential $\partial$ on $\mathbb{Z}$-differential graded algebra is generally much simpler. Note that in the dipped diagrams, the differential of crossings in a $b$-lattice involve at most $c_k$, crossings in the previous $a$-lattice, and crossings in the $a$-lattice, the differential of crossings in an $a$-lattice only involve crossings in the $b$-lattice, and the
differential of $c_k$ is $0$ for all $k$. This greatly reduces the totally augmented disks (disks contributing to the differential of the crossing where all negative corners are augmented) for which to look to compute whether we have an augmentation. (These are the only disks which contribute to $\epsilon\circ\partial$.)

This is most of the setup necessary to show both that augmentations imply rulings and rulings imply augmentations.

Filed under Uncategorized

## News from University of Minnesota 4-manifolds workshop

I am writing from the 4-manifolds conference at the University of Minnesota. Lots of different approaches to and constructions of 4-manifolds have been discussed and slides from some of the courses are on the conference website . Today, I want to focus on Chris Wendl’s mini-course on characterizing symplectic fillings of certain contact structures (supported by a spinal open book with a planar page) in terms of Lefschetz fibrations. Additionally, the different notions of symplectic filling (weak, strong, Liouville, Weinstein, or Stein – see my previous post and Chris’ comment for details on these variations) all coincide in this case (up to blowing up). The results are contained in a paper of Lisi, Van Horn-Morris and Wendl coming out soon. This generalizes Chris’ earlier paper on the analogous theorem for regular planar open book decompositions . The results seem really powerful if one can make sense of how to find good spinal open books supporting contact manifolds of interest.

Though there may be some similarly defined notions in high dimensions, these results are only for 4-manifolds with 3-manifold boundary.

The definition for a spinal open book is motivated by looking at the decomposition induced on the boundary of a Lefschetz fibration. A Lefschetz fibration on a 4-manifold $E$, is a smooth map to a surface $\Sigma$, $\pi: E\to \Sigma$ which has finitely critical points so that each critical point has a coordinate neighborhood so that $\pi$ in these coordinates looks like $(z_1,z_2)\mapsto z_1^2+z_2^2$ where $z_1,z_2$ are complex coordinates. The base surface $\Sigma$ and the generic fibers can have many boundary components, but $\Sigma$ should have at least one boundary component if we want to study Lefschetz fibrations of manifolds with boundary. It is useful, particularly when defining spinal open book decompositions, to choose $E$ to be a manifold with corners so that you can think of the fibers as sitting vertically and the sections to lie horizontally. Then the boundary decomposes into a horizontal piece and a vertical piece which meet at a codimension 2 corner. In the case with corners it is more accurate to use the term bordered Lefschetz fibration and one needs to be a little careful when talking about smooth maps/structures near the corners.

In this schematic picture the red horizontal boundary forms an $S^1$ bundle over three disjoint copies of the base surface $B$ (one for each boundary component of the fiber). This fibration makes up the “spine” of the induced spinal open book decomposition. [Note since $B$ has boundary this fibration is always trivial in this dimension.] The blue vertical boundary forms a $F \sqcup F$ fibration over $S^1$ (one copy of the fiber surface $F$ for each boundary component of the base $B$). This fibration is called the paper of the spinal open book decomposition. Each connected component of a fiber in this piece is called a page.

A special case of this is when the base surface is a disk. Then the induced spinal open book on the boundary is a genuine open book decomposition. The spine part is an $S^1$ bundle over $\sqcup_k D^2$ (solid tori forming the binding), and the paper is a surface bundle over the circle with connected fibers (since the number of connected components in a fiber corresponds to the number of boundary components in the base surface $D^2$) so each fiber forms the page of an open book decomposition.

In general a spinal open book is a decomposition of a 3-manifold into the spine: any $S^1$ fibration over any surface $\Sigma$ with boundary (not necessarily connected), and the paper: any surface (with boundary and possibly disconnected) bundle over $S^1$ so that the intersection of the spine and the paper is a union of 2-dimensional tori. Many of these spinal open book decompositions do not arise as the boundary of bordered Lefschetz fibrations. However there are some slightly more complicated ways spinal open book decompositions can be found on the boundary of a bordered Lefschetz fibration than the above picture indicates. One thing that can happen is that the base surface of the spine may be a nontrivial cover of the base of the Lefschetz fibration. Maybe the schematic for this looks something like this:

The spine (red part) of the spinal open book decomposition can be a circle bundle over a nontrivial (in this picture double) cover of the base surface of the Lefschetz fibration. One can rule out these sort of Lefschetz fibrations by restricting attention to spinal open book decompositions that are “simple”: for any page, different boundary components of that page touch different components of the spine. In the simple case, a spinal open book whose spine is a circle fibration over a disconnected surface $\Sigma$ must have all connected components of $\Sigma$ be diffeomorphic surfaces. This rules out the existence of Lefschetz fillings for lots of spinal open book decompositions.

The main results of the mini-course were:

Theorem 1 (Decompositions -> Contact/Symplectic Geometry): Spinal open books determine a unique isotopy class of supported contact structures. Bordered Lefschetz fibrations determine a nonempty contractible space of symplectic forms on the 4-manifold which are weak/strong/Stein** fillings of the contact boundary.

**One must assume the bordered Lefschetz fibration is allowable (vanishing cycles are homologically essential) to get this result for Stein fillings.

Theorem 2 (Contact/Symplectic Geometry -> Decompositions): Suppose we have a contact manifold supported by a spinal open book which has a page of genus zero, and a weak symplectic filling of that contact manifold $(W,\omega)$ such that the restriction of $\omega$ to the spinal part of the boundary is exact. Then $(W,\omega)$ is supported by a bordered Lefschetz fibration (determined up to isotopy by the deformation class of $\omega$) which induces the given spinal open book decomposition on its boundary.

I will write the compatibility conditions for the contact/symplectic structure with the spinal open book/Lefschetz decomposition, but omit the discussion of the proof of theorem 1 for now.

A contact form $\alpha$ is compatible with a spinal open book decomposition if
(1) $d\alpha$ restricted to the interior of each page is a positive area form
(2) On the spine $\pi_{\Sigma}: M_{\Sigma}\to \Sigma$, the $S^1$ fibers are Reeb orbits for $\alpha$.

There is a version of this compatibility notion when you are thinking of the spinal open book decomposition as the boundary of a 4-manifold with corners. In this case you look at germs of 1-forms thought of as the restriction of a smooth 1-form on the 4-manifold to the (not quite smooth) boundary. Require this to be a contact form in the complement of the corner parts between the spine and the paper plus conditions (1) and (2).

The basic compatibility conditions between a bordered Lefschetz fibration of $W$ and a symplectic form $\omega$ which ensure $(W,\omega)$ is a weak symplectic filling are:

(1) The restriction of $\omega$ to each fiber is positive away from critical points
(2) Near critical points $\omega$ tames an almost complex structure $J$ which is defined in advance near each critical point so that the fibers are $J$ holomorphic.
(3) Near the horizontal boundary (spine part) $\omega=d\lambda$ where $\lambda$ is a contact form on the spine compatible with the spinal open book. [This additional partial exactness criterion is special to spinal open book decompositions since it is trivially satisfied for regular open books because the solid tori spines do not support any second cohomology.]

For strong fillings, require $\omega = d\lambda$ near the entire boundary.

The surprising result is theorem 2, the spinal generalization of Wendl’s earlier theorem that characterizes symplectic fillings of contact manifolds supported by planar open books as Lefschetz fillings of that particular open book decomposition. The idea of the proof is to symplectically complete $(W,\omega)$ (attach half of the symplectization to the end), and then fill that completion with a (singular) foliation by $J$-holomorphic curves.

Symplectic Completions

Given a contact manifold $(M,\xi=\ker(\alpha))$ one can form the symplectization $(\mathbb{R}\times M,\omega =d(e^t\alpha))$ where $t$ is the $\mathbb{R}$ coordinate. In this case the vector field $V=\partial_t$ is Liouville:

$\mathcal{L}_V\omega = d\iota_V(d(e^t\alpha))=d(\iota_V(e^t dt\wedge \alpha +e^t\alpha))=d(e^t\alpha)=\omega$

If a symplectic manifold $(W,\omega)$ has convex boundary $(M,\xi)$, we can match up a collared neighborhood $N$ of $\partial W=M$ with a piece of this symplectization and form a non-compact symplectic manifold $(\widehat{W},\widehat{\omega})= (W,\omega)\cup_{N} ([0,\infty)\times M,d(e^t\alpha))$. Similarly for concave boundary one can attach $((-\infty,0]\times M,d(e^t\alpha)$ to a collared neighborhood of the boundary. The advantage of such completions is that the flow of the Liouville vector field no longer runs off the boundary (it’s complete).

Chris discussed a weaker structure than a contact structure one can have near the boundary of a symplectic manifold so that there is still a natural symplectic completion. This is a stable Hamiltonian structure.

A symplectic manifold $(W,\omega)$ has stable boundary if there is a collared neighborhood of $\partial W=M$ identified as $(-\varepsilon,0]\times M$ such that $\ker(\omega|_{\{t\}\times M})$ is independent of $t\in (-\varepsilon,0]$. This is true if and only if the collared neighborhood is symplectomorphic to $((-\varepsilon,0]\times M, d(e^t\lambda)+\Omega)$ such that:
(1) $d\Omega=0$
(2) $\lambda \wedge \Omega>0$
(3) $\ker(\Omega)\subset \ker(d\lambda)$

In this case $(\lambda,\omega=d(e^t\lambda)+\Omega)$ is called a stable Hamiltonian structure (SHS) on the boundary of $(W,\omega)$. When $\omega|_M = d\alpha$ for a contact form $\alpha$, the pair $(\alpha,\omega)$ is a SHS where $\Omega = \omega|_M$. This is the convex boundary case, where $\partial_t$ gives a Liouville vector field near the boundary. For a general stable Hamiltonian boundary, $\partial_t$ is no longer a Liouville vector field but there is still a natural symplectic structure matching up the boundary to $(\mathbb{R}\times M, d(e^t\lambda)+\Omega)$.

The advantage of working with completions using stable Hamiltonian structures is that one can work with weak fillings instead of strong fillings and still get the same results as long as the symplectic structure is exact near the spine part of the spinal open book.

Foliating by J-holomorphic curves

We assume for the rest of this post that we have a weak symplectic filling $(W,\omega)$ such that
(1) It weakly fills a contact manifold supported by a planar spinal open book decomposition
(2) On the spine part $M_{\Sigma}$ of $\partial W=M$, $\omega$ is exact.

Start by building J-holomorphic curves in the product part of the completion using a stable Hamiltonian structure that is nice with respect to the spinal open book decomposition of $M$.

Since $\omega|_{M_{\Sigma}}$ is exact, if $\pi_P: M_P\to S^1$ is the paper fibration, $\omega$ is cohomologous to $d\pi_P\wedge \eta$ for a closed 1-form $\eta$ supported in $M_P$.

On the spine, $M_{\Sigma}\cong S^1\times \Sigma$ and a compatible contact form is one whose Reeb vector field points in the $S^1$ direction. We can construct such a contact form as follows. Choose a complex structure $j$ on $\Sigma$ and a plurisubharmonic function $\phi: \Sigma \to \mathbb{R}$ which is constant on boundary components of $\Sigma$, and setting $\sigma = -d\phi\circ j$ (such a plurisubharmonic function exists because $\Sigma$ has boundary). Then let $\alpha = d\theta+\sigma$ where $\theta$ is the $S^1$ coordinate on $M_{\Sigma}$.

Proposition: Given the above assumptions, it is possible to deform $(W,\omega)$ near $\partial W$ such that there is a SHS on the boundary $(\lambda, \Omega)$ satisfying:
(1) In the spine away from the paper, $M_{\Sigma}$, the stabilizing 1-form $\lambda = d\theta +\sigma$ and $\Omega=d\lambda$.
(2) In the paper, $\pi_P: M_P\to S^1$, the stabilizing 1-form $\lambda=d\pi_P$ and $\Omega = kd\alpha+d\pi_P\wedge \eta$ for some large $k$ where $\alpha$ is a “Giroux form” i.e. a contact form (or the cornered version) compatible with the spinal open book (so $d\alpha$ is positive on the pages).

Using this Stable Hamiltonian Structure to define the completion, there exists an almost complex structure $J$ satisfying:
(1) $\mathbb{R}$-invariant on the $[0,\infty)\times M$ piece
(2) Each page $\times \{0\}$ is $J$ invariant
(3) For each gradient flowline $\gamma$ of the plurisubharmonic function $\phi:\Sigma \to \mathbb{R}$ which defined the Liouville 1-form $\sigma$, there is a lift of $S^1\times \gamma\subset M_{\Sigma}$ in $[0,\infty)\times M$ to a $J$-holomorphic half-cylinder $u_{\gamma}:[0,\infty)\times S^1\to [0,\infty)\times M$.

Here is a picture of some example gradient flowlines (in blue and red) on $\Sigma$, which should exit the boundary of $\Sigma$ and land on a page (indicated by the green lines).

(4) Each page $\times \{0\}$ matches up to a union of the half cylinders $u_{\gamma}$ to form a finite energy embedded $J$-holomorphic curve.

In fact there is a family of such $J$-holomorphic curves, of index $2-2g(P)$ for a page $P$. Therefore when we have at least one page of genus zero, there is a 2-dimensional family of $J$-holomorphic curves which start filling up the completion $\widehat{W}$. The space of points hit by such $J$-holomorphic curves is open, and closed if you include all possible degenerations of the curves. The generic curves are genus zero (a genus zero page with some cylinders attached). These can degenerate in two ways:

Either the vanishing cycle splits the curve into different components each with some bounday or it bubbles off a closed sphere. It is possible to show that if it bubbles off a closed sphere, this sphere has self-intersection number -1 which implies $\widehat{W}$ is not minimal. So if we assume we have a minimal weak symplectic filling, the degenerations must split off pieces with boundary and these types of degenerations can only happen finitely many times (since there are finitely many boundary components). In the end, this implies that the space of points hit by these (possibly degenerate) J-holomorphic curves is both open and closed so (assuming our filling was connected) it is the whole of $\widehat{W}$.

To get a Lefschetz fibration, one must restrict to a compact subdomain of the (noncompact) symplectic completion. This can be done in the case that the spinal open book decomposition is simple.