# Complex vs convex Morse functions and geodesic open books

Pierre Dehornoy  Univ. Grenoble Alpes, CNRS, Institut Fourier, F-38000
Grenoble, France<EMAIL_ADDRESS>and  Burak Ozbagci
Department of Mathematics
Koç University
Istanbul, Turkey<EMAIL_ADDRESS>
###### Abstract.

Suppose that $\Sigma$ is a closed and oriented surface equipped with a
Riemannian metric. In the literature, there are four seemingly distinct
constructions of open books on the unit (co)tangent bundle of $S$, having
complex, symplectic, contact, and dynamical flavors, respectively. Each one of
these constructions is based on either an admissible divide or an ordered
Morse function on $\Sigma$. We show that the resulting open books are pairwise
isomorphic provided that the ordered Morse function is adapted to the
admissible divide on $\Sigma$. Moreover, we observe that if $\Sigma$ has
positive genus, then none of these open books are planar and furthermore, we
determine the only cases when they have genus one pages.

##  1\. Introduction

Let $\Sigma$ be a closed and oriented surface equipped with a Riemannian
metric. The bundle of cooriented lines tangent to $\Sigma$ (aka the space of
cooriented contact elements), which we denote by $V(\Sigma)$ in this paper, is
defined as the set of pairs $(q,L)$ where $q\in\Sigma$ and $L$ is a cooriented
line in $T_{q}\Sigma$. One can identify $V(\Sigma)$ with the unit tangent
bundle $ST\Sigma$ as well as with the unit cotangent bundle $ST^{*}\Sigma$. As
an oriented $3$-manifold, $ST^{*}\Sigma$ (resp. $ST\Sigma$) is diffeomorphic
to the circle bundle over $\Sigma$ with Euler number $-\chi(\Sigma)$ (resp.
$\chi(\Sigma)$). We orient the circle bundle $V(\Sigma)$ so that it is fiber
and orientation preserving diffeomorphic to $ST^{*}\Sigma$.

Let $DT^{*}\Sigma$ denote the disk cotangent bundle, which is a disk bundle
over $\Sigma$ with Euler number $-\chi(\Sigma)$, whose boundary is
$ST^{*}\Sigma$. Let $\lambda$ denote the Liouville $1$-form on $T^{*}\Sigma$.
Under the identification of $ST^{*}\Sigma$ with $V(\Sigma)$, the canonical
contact structure $\xi=\ker\lambda|_{ST^{*}\Sigma}$ on $ST^{*}\Sigma$
coincides with the canonical contact structure on $V(\Sigma)$, which we again
denote by $\xi$ throughout the article.

A _divide_ $P\subset\Sigma$ is a generic immersion of the disjoint union of
finitely many copies of the unit circle, which is said to be _admissible_ if
it is connected, each component of $\Sigma\setminus P$ is simply connected and
$\Sigma\setminus P$ admits a black-and-white coloring so that two sides of an
edge in $P$ have opposite colors, see Figure 1. We say that an ordered Morse
function $f:\Sigma\to\mathbb{R}$ (meaning that the higher the index of a
critical point the greater its value) is adapted to a given admissible divide
$P\subset\Sigma$ if $P=f^{-1}(0)$, each double point of $P$ corresponds to a
critical point of $f$ of index $1$, and each black (resp. white) region of
$\Sigma\setminus P$ contains exactly one index $2$ (resp. $0$) critical point
of $f$. Note that for each admissible divide $P\subset\Sigma$, there exists an
ordered Morse function $f:\Sigma\to\mathbb{R}$ adapted to $P$. We also say
that an admissible divide $P\subset\Sigma$ is convex with respect to the given
metric on $\Sigma$, if it is a set of geodesics on $\Sigma$, every geodesic on
$\Sigma$ meets $P$ in bounded time, and every region of $\Sigma\setminus P$
can be foliated by concentric closed curves with non-vanishing curvature.

######  Theorem 1.1.

Suppose that $P$ is an admissible divide on a closed and oriented surface
$\Sigma$ equipped with a Riemannian metric. If $f:\Sigma\to\mathbb{R}$ is an
ordered Morse function adapted to $P$, then the open books described in (1) -
(4) below, are pairwise isomorphic under the natural identifications of the
$3$-manifolds $ST\Sigma$, $ST^{*}\Sigma$, and $V(\Sigma)$.

(1) (A’Campo [2] $\&$ Ishikawa [20])  The Morse function
$f:\Sigma\to\mathbb{R}$ can be extended to a complex Morse function
$f_{\mathbb{C}}:T\Sigma\to\mathbb{C}$, which restricts to a smooth
achiral111There is orientation error in [20, Lemma 2.6]: The Lefschetz
fibration $DT\Sigma\to D^{2}$ Ishikawa constructs must be achiral or
equivalently, it must be considered on the cotangent (rather than tangent)
disk bundle $DT^{*}\Sigma$. Lefschetz fibration $DT\Sigma\to D^{2}$. There is
an induced open book on the boundary $ST\Sigma$.

(2) (Johns [21]) The Morse function $f:\Sigma\to\mathbb{R}$ can be extended to
an exact symplectic Lefschetz fibration $f_{\omega}:E\to D^{2}$, where $E$ is
conformally exact symplectomorphic to $(DT^{*}\Sigma,\omega=d\lambda)$. The
induced open book on the boundary $ST^{*}\Sigma$ supports $\xi$.

(3) (Giroux [17]) The Morse function $f:\Sigma\to\mathbb{R}$ can be modified
to an ordered222The fact that $f_{\xi}$ is ordered, provided that $f$ is
ordered, was observed by Massot [25] along with some other details of Giroux’s
construction [17, Example 4.9] restricted to dimension three. $\xi$-convex (or
contact) Morse function $f_{\xi}:V(\Sigma)\to\mathbb{R}$, which induces an
open book for $V(\Sigma)$ that supports $\xi$.

(4) (Birkhoff [5] & Fried [15]) If $P$ is a convex, then the lift of $P$ to
$ST\Sigma$ is the binding of a ”geodesic” open book, that is an open book
whose pages are negative333We use the term _negative_ since the natural
orientation given by the geodesic flow of the binding of this open book is
opposite to that induced from a natural orientation of the page. Birkhoff
cross sections of the geodesic flow.

######  Remark 1.2.

The open books in items (1)-(3) are independent of the given metric on the
surface $\Sigma$, up to isomorphism. Moreover, when $\Sigma$ is of genus at
least two and the metric is hyperbolic, the geodesic flow is independent of
the metric, up to homeomorphism, by a theorem of Gromov [19] (see Section 5
for details) and hence the geodesic open books in item (4) are uniquely
determined up to isomorphism. More generally, if $\Sigma$ is of any genus and
is equipped with two different metrics, the resulting geodesic open books in
item (4) are isomorphic as long as $P$ is convex with respect to both metrics.

Note that the open books in items (2) and (3) support the canonical contact
structure $\xi$ on $ST^{*}\Sigma\cong V(\Sigma)$ by construction, and the
“duals” of the geodesic open books in item (4), when considered for the
cogeodesic flow on $ST^{*}\Sigma$, also support $\xi$, since the cogeodesic
flow agrees with the Reeb flow of the Liouville $1$-form $\lambda$ (see, for
example, [16, Theorem 1.5.2]). As we explain in Remark 2.1 below, Ishikawa’s
Lefschetz fibrations should be considered on $DT^{*}\Sigma$ (rather than
$DT\Sigma$) and thus they induce open books for $ST^{*}\Sigma$. Therefore, the
A’Campo-Ishikawa open books in item (1) support $\xi$ as well when they are
viewed for $ST^{*}\Sigma$, as a consequence of Theorem 1.1.

######  Corollary 1.3.

For each admissible divide $P$ on a closed and oriented surface $\Sigma$, the
A’Campo-Ishikawa open book (when viewed for $ST^{*}\Sigma$) supports the
canonical contact structure $\xi$.

Our method of proof also implies that for each connected divide on $D^{2}$,
the open book that A’Campo constructs for $\mathbb{S}^{3}$ supports the
standard contact structure. Notice that A’Campo already proved [1, Section 4]
that the binding of his open book is transverse to the standard contact
structure, which is a necessary condition.

Provided that the genus of $\Sigma$ is at least one, none of the open books
mentioned in Theorem 1.1 are planar, by construction. This is consistent with
the fact that the support genus of $(ST^{*}\Sigma,\xi)$ is equal to one for a
surface $\Sigma$ of positive genus, which follows by combining an obstruction
to planarity due to Etnyre [13], with constructions of Massot [25] when the
genus of $\Sigma$ is at least two and Van Horn-Morris [29] for a surface
$\Sigma$ of genus one (see [27] for further details). The only case for which
the open book in Theorem 1.1 is planar is the one where the divide $P$ is a
single embedded circle on $S^{2}$. The corresponding open book adapted to
$(ST^{*}S^{2}=\mathbb{R}P^{3},\xi)$ is planar, whose page is an annulus and
monodromy the square of the Dehn twist along the core circle. Here, we
characterize those open books in Theorem 1.1 with genus one pages.

######  Theorem 1.4.

Up to homeomorphism, there are exactly three admissible divides on a closed
and oriented surface $\Sigma$ of genus at least one, as depicted in Figure 8,
that yield genus one open books obtained by any of the methods listed in
Theorem 1.1. These open books have $4g,4g+2,$ and $4g+4$ binding components,
respectively.

As another consequence of Theorem 1.1, one can draw the following conclusion.
The monodromies of the open books in item (3), which are not immediately clear
from their construction, can be computed via Theorem 1.1, since the
monodromies of the open books in items (1) and (2) are given explicitly as a
product of Dehn twists [1, 21] and the monodromies in items (1) and (4) are
given as a product of two involutions [3, 12].

Moreover, Theorem 1.1 answers Questions (1) and (3) affirmatively listed at
the end of [27], while Theorem 1.4 and the discussion in Section 7 give a
negative answer for Question (4).

###  1.1. Pairwise identification of the $3$-manifolds in Theorem 1.1

The identification of $V(\Sigma)$ with $ST\Sigma$ is obtained by taking the
unit tangent vector positively normal to the given cooriented line. The
identification of $V(\Sigma)$ with $ST^{*}\Sigma$ is obtained by further
composing the identification above with the natural bundle map $ST\Sigma\to
ST^{*}\Sigma$ induced by the Riemannian metric on $\Sigma$. Equivalently, a
more direct identification of $V(\Sigma)$ with $ST^{*}\Sigma$ is obtained by
taking for each cooriented line $L$ in $T_{q}\Sigma$, the unit covector (a
linear map $T_{q}\Sigma\to\mathbb{R}$) in $T_{q}^{*}\Sigma$, whose kernel is
$L$ with its coorientation. In other words, the bundle of cooriented lines
tangent to $\Sigma$ has a natural identification with the bundle of rays in
$T^{*}\Sigma$, which therefore can be denoted by $ST^{*}\Sigma$ (see, for
example, [26]), but for the purposes of this paper, we opted to denote the
bundle of cooriented lines by $V(\Sigma)$, to distinguish it from the unit
cotangent bundle, although they are orientation-preserving diffeomorphic to
each other.

###  1.2. Open book constructions in chronological order

Our goal here is to bring together seemingly unrelated work in the literature
over a span of more than one hundred years, going all the way back to
Birkhoff’s article [5] on dynamical systems, where he constructs a “surface of
section” for the geodesic flow on the unit tangent bundle $ST\Sigma$. The
existence of such a surface reduces the study of some dynamical properties of
the flow to understanding a self-homeomorphism of this surface. Birkhoff’s
approach was later popularized by Fried [15], who utilized it to study
arbitrary transitive Anosov flows on closed $3$-manifolds. With the
terminology used in this paper, the surface of section $S$ is a page of an
open book for $ST\Sigma$, so that the interior of $S$ is transverse to the
geodesic flow, while the binding $\partial S$ is a collection of periodic
orbits.

The notion of a divide on $D^{2}$, and the link of a divide in
$\mathbb{S}^{3}$, was introduced by A’Campo [1, 2], who was interested in
computing geometric monodromies of isolated plane curve singularities. In
particular, A’Campo proved that the link of any isolated plane curve
singularity appears as the link of a divide, the link of a connected divide is
fibered, and that this fibration is a model for the Milnor fibration of the
singularity. Note that, for each connected divide, A’Campo’s construction
gives nothing but an open book for $\mathbb{S}^{3}$ whose binding is the link
of the given divide. The work of A’Campo was later generalized by Ishikawa
[20], who showed that the link of an admissible divide444Birkhoff used the
term _primary set of curves_ for what Ishikawa (following A’Campo) called an
admissible divide. on an arbitrary surface $\Sigma$ (not just $D^{2}$) is
fibered in $ST\Sigma$ and moreover $DT\Sigma$ admits an achiral Lefschetz
fibration over $D^{2}$, which induces the same open book on the boundary.

In an other direction, at the turn of this century, a major breakthrough in
contact topology was achieved by Giroux [18], who proved that every closed
contact manifold is convex, which is equivalent to the fact that every contact
manifold admits an adapted open book with Weinstein pages. By combining with
the earlier work of Giroux [17], it follows in particular that $V(\Sigma)\cong
ST^{*}\Sigma$ admits an open book [25] that supports its canonical contact
structure $\xi$.

Finally, building on the fundamental work of Seidel [28], Johns [21]
constructed an exact symplectic Lefschetz fibration on the symplectic disk
cotangent bundle $(DT^{*}\Sigma,\omega)$, which in turn, restricts to an open
book for $ST^{*}\Sigma$ supporting $\xi$. The difference from Ishikawa’s
Lefschetz fibration is that Johns’ fibration is not just a smooth fibration
but adapted to the canonical symplectic structure $\omega=d\lambda$, meaning
that the regular fibers are symplectic submanifolds.

In this article, we relate the four different constructions of open books
described in the last four paragraphs, where the first one involves the
dynamics of the geodesic flow, the second one is motivated by the study of
isolated plane curve singularities, the third one illustrates the power of
convexity in contact topology and the last one has a distinctive symplectic
flavor.

######  Remark 1.5.

It is not true that every open book adapted to $(ST^{*}\Sigma,\xi)$ arises
from an admissible divide or an ordered Morse function on $\Sigma$, using any
one of the (eventually equivalent) constructions in Theorem 1.1. This is
simply because each open book appearing in Theorem 1.1 has an even number of
binding components and there is no reason for this to hold for an arbitrary
open book adapted to $(ST^{*}\Sigma,\xi)$. For example, a genus one open book
adapted to $(ST^{*}T^{2}=T^{3},\xi)$ with three binding components, was
constructed by Van Horn-Morris [29] (see also [11, Section 3] for similar
examples obtained by the first author). Note that one can always get an
adapted open book with an odd number of binding components, by positively
stabilizing an adapted open book with an even number of binding components,
but this is not the case for the aforementioned example of Van Horn-Morris.

Outline of the paper: First, we review in some details the four distinct
constructions above due to A’Campo $\&$ Ishikawa, Johns, Giroux, and Birkhoff
$\&$ Fried in Sections 2 to 5, respectively. Then we describe a proof of
Theorem 1.1 in Section 6, while we give a proof of Theorem 1.4 in Section 7.

##  2\. Complexification of a Morse function on a surface

In this section, we briefly review Ishikawa’s construction [20] of Lefschetz
fibrations on the disk (co)tangent bundle of a closed and oriented surface
$\Sigma$, which is a generalization of the work of A’Campo [1, 2]. These
Lefschetz fibrations induce natural open books for the unit (co)tangent bundle
by restricting to the boundary.

For any admissible divide $P\subset\Sigma$, there is an ordered Morse function
$f:\Sigma\to\mathbb{R}$ adapted to $P$, satisfying the following conditions

$\bullet$ $P=f^{-1}(0)$,

$\bullet$ each double point of $P$ corresponds to a critical point of $f$ of
index $1$, and

$\bullet$ each black (resp. white) region of $\Sigma\setminus P$ contains
exactly one index $2$ (resp. $0$) critical point of $f$.

Figure 1. An admissible divide $P$ (in red) on a genus 2 surface, a black-and-
white coloring of the complement $\Sigma\setminus P$, and some level sets of
an ordered Morse function adapted to $P$. The divide $P$ consists of six
curves $(c(P)=6)$ and includes seven double points $(v(P)=7)$.

The Morse function $f:\Sigma\to\mathbb{R}$ can be extended to an _almost
complexified_ Morse function $f_{\mathbb{C}}:T\Sigma\to\mathbb{C}$ defined as

(2.1) $f_{\mathbb{C}}(x,u)=f(x)+i\eta
df_{x}(u)-\dfrac{1}{2}\eta^{2}\chi(x)d^{2}f_{x}(u,u)$

for $x\in\Sigma$ and $u\in T_{x}\Sigma$. Here the linear map
$df_{x}:T_{x}\Sigma\to\mathbb{R}$ denotes the differential, and the bilinear
map $d^{2}f_{x}:T_{x}\Sigma\times T_{x}\Sigma\to\mathbb{R}$ denotes the
Hessian of $f$ at the point $x$. The smooth map $\chi:\Sigma\to\mathbb{R}$ is
a suitable cut-off function which vanishes outside of a sufficiently small
neighborhood of the critical points, and $\eta$ is a positive real number,
whose role will be clarified below.

The map $f_{\mathbb{C}}$ is almost complexified in the sense that the Morse
singularities of $f$ becomes complex Morse singularities of $f_{\mathbb{C}}$.
More precisely, the Morse singularities of $f$ of the form
$-x_{1}^{2}-x_{2}^{2},-x_{1}^{2}+x_{2}^{2},x_{1}^{2}+x_{2}^{2}$ in local
coordinates on $\Sigma$ becomes singularities of $f_{\mathbb{C}}$ of the form
$-z_{1}^{2}-z_{2}^{2},-z_{1}^{2}+z_{2}^{2},z_{1}^{2}+z_{2}^{2}$, respectively,
in local complex coordinates on $T^{*}\Sigma$, which are all equivalent to the
complex Morse singularity $z_{1}^{2}+z_{2}^{2}$, under a complex change of
coordinates. Moreover, by choosing $\eta$ sufficiently small, one can
guarantee that there are no other critical points of $f_{\mathbb{C}}$. It
follows that a point in $T\Sigma$ is a critical point of $f_{\mathbb{C}}$ if
and only if it belongs to $\Sigma$ and it is a critical point of $f$. The map
$f_{\mathbb{C}}:T\Sigma\to\mathbb{C}$ descents to an achiral (see Remark 2.1)
Lefschetz fibration $DT\Sigma\to D^{2}$. The singular fiber over $0\in D^{2}$
contains a singularity for each double point of $P$. Corresponding to each
index $2$ or index $0$ critical point of $f$, there is a singular fiber
containing a unique singularity. Moreover, the regular fiber and the vanishing
cycles of the Lefschetz fibration $DT\Sigma\to D^{2}$, are described
explicitly in [20] by a method due to A’Campo [2].

######  Remark 2.1.

There is an orientation issue in [20, Lemma 2.6]. Following the notation in
[20], suppose that $x_{1},x_{2}$ are local coordinates on the surface $\Sigma$
and $u_{1},u_{2}$ are the corresponding coordinates in the tangent fibers, so
that $\\{x_{1},x_{2},u_{1},u_{2}\\}$ is an oriented chart for $T\Sigma$. But
the orientation of this chart is opposite to the complex orientation that is
required in the definition of a Lefschetz fibration, since Ishikawa uses the
complex chart

$(z_{1},z_{2})=(x_{1}+iu_{1},x_{2}+iu_{2})$

for $T\Sigma$ in the proof of [20, Lemma 2.6]. Except the orientation issue,
however, everything else works in Lemma 2.6 and the subsequent discussion
leading to Proposition 3.1 and Lemma 3.2 in [20]. In other words, the
Lefschetz fibration Ishikawa constructs is achiral on $DT\Sigma$ and the
corresponding Lefschetz fibration $DT^{*}\Sigma\to D^{2}$, obtained by
reversing the orientation, induces an open book on the boundary
$ST^{*}\Sigma$. It follows, by Theorem 1.1, that this open book supports the
contact canonical structure on $ST^{*}\Sigma$, which is not clear from
Ishikawa’s work [20].

$S_{0}$$S_{\frac{\pi}{4}}$$S_{\frac{\pi}{2}}$$S_{\frac{3\pi}{4}}$$S_{\pi}$$S_{\frac{5\pi}{4}}$$S_{\frac{3\pi}{2}}$$S_{\frac{7\pi}{4}}$
Figure 2. Some of the pages $S_{\theta}$ of the A’Campo-Ishikawa open book for
$ST\Sigma$, viewed as sets of vectors tangent to $\Sigma$.

######  Remark 2.2.

When restricted to $ST\Sigma$, the complexification map $f_{\mathbb{C}}$ in
(2.1) vanishes on the link $L(P)$ of the divide $P$, which is the set of unit
vectors tangent to $P$ (cf. [2, 20]). Moreover, provided that $\eta$ is
sufficiently small, this map restricts to a fibration $ST\Sigma\setminus
L(P)\to S^{1}$, obtained by dividing the image by its norm, and therefore
inducing an open book for $ST\Sigma$ with binding $L(P)$, hereafter called
_A’Campo-Ishikawa_ open book. If the unit circle $S^{1}$ is parametrized by
$\theta$, then we denote the closure of the inverse image of $\theta$ under
the fibration map as $S_{\theta}$, the $\theta$-page.

As depicted in Figure 2, the interior of $S_{0}$ (resp. $S_{\pi}$) consists of
the unit vectors tangent to the level sets of $f$ in black (resp. white)
regions, while the interior of $S_{\pi/2}$ (resp. $S_{3\pi/2}$) consists of
the unit vectors along the divide $P$ pointing into black (resp. white)
regions of $\Sigma\setminus P$, except around the double points of the divide
where the term involving the Hessian in Equation (2.1) smooths the surface and
separate the surfaces associated to different values of $\theta$. Around the
double points of $P$, where the function $\chi$ does not vanish, the surfaces
$S_{\theta}$ are depicted in Figure 3.

Figure 3. The pages $S_{\theta}$ of the A’Campo-Ishikawa open book for
$ST\Sigma$ around the fiber of a double point of a divide $P$, for
$\theta=\frac{k\pi}{4}$ with $k=0,\dots,7$.

The topological type of the pages of the A’Campo-Ishikawa open book can be
easily determined as follows. Denote by $v(P)$ the number of double points of
a divide $P$ and by $c(P)$ the number of circles that form $P$ (see Figure 1,
for an example).

######  Proposition 2.3.

For any admissible divide $P$, the A’Campo-Ishikawa open book has $2c(P)$
binding components and its page genus is given by $1+v(P)-c(P)$.

###### Proof.

Since all the pages have the same genus, we focus on one page, say
$S_{\frac{\pi}{2}}$, in particular. First notice the boundary of
$S_{\frac{\pi}{2}}$ is the link $L(P)$ of the divide, which has two components
for every circle in $P$. Now consider $P\subset\Sigma$ as a graph whose
vertices are the double points of $P$ and whose edges are the segments of $P$
connecting these double points. Then the surface $S_{\frac{\pi}{2}}\subset
ST\Sigma$ is made of one rectangle in the fiber of every segment of $P$. The
horizontal sides of these rectangles are in the link $L(P)$, hence in the
boundary of $S_{\frac{\pi}{2}}$. The vertical sides can be divided into two
segments, which are glued with the rectangle associated to the next and
previous edge of $P$ respectively, when rotating around the corresponding
vertex of $P$, as depicted in Figure 3. One checks that every rectangle
contributes by $-1$ to the Euler characteristics of $S_{\frac{\pi}{2}}$. Since
$P$ has twice more edges than vertices, we deduce that
$\chi(S_{\frac{\pi}{2}})=-2v(P)$. The number of boundary components of
$S_{\frac{\pi}{2}}$ is $2c(P)$. Hence the genus of $S_{\frac{\pi}{2}}$ is
$1+v(P)-c(P)$. ∎

##  3\. An exact symplectic Lefschetz fibration on the cotangent bundle

Let $\lambda$ denote the Liouville $1$-form on the cotangent bundle
$T^{*}\Sigma$. It is well-known that the symplectic disk cotangent bundle
$(DT^{*}\Sigma,\omega=d\lambda)$ is a Weinstein domain and in fact a Stein
filling of its contact boundary $(ST^{*}\Sigma,\xi=\ker\lambda)$, see, for
example, [8]. Therefore, according to [4, 22], the Stein filling
$DT^{*}\Sigma$ admits a smooth Lefchetz fibration over $D^{2}$.

Here we briefly review Johns’ construction [21] of an exact symplectic
Lefschetz fibration on $(DT^{*}\Sigma,\omega=d\lambda)$, using methods other
than those described in [4, 22]. Apparently inspired by the work A’Campo on
the complexification of smooth Morse functions (but not referring to
Ishikawa’s work [20]), Johns starts with an educated guess for the fiber and
the vanishing cycles of a smooth Lefschetz fibration $E\to D^{2}$. By a
standard method (see, for example, [28]) the total space $E$ can be equipped
with an exact symplectic form so that the regular fibers are symplectic
submanifolds of $E$ and the vanishing cycles are Lagrangian submanifolds of a
regular fiber. The crux of the matter is that the construction of the fiber
depends on the choice of a smooth Morse function $f:\Sigma\to\mathbb{R}$ and
for each critical point of $f$, there is a vanishing cycle of the fibration.
More precisely, the fiber is constructed by a smooth plumbing of some annuli,
each one of which represents a neighborhood of a vanishing cycle corresponding
to a critical point of $f$ of index zero or one. The plumbing is dictated by
the way that the $1$-handles are attached to the $0$-handles in the handle
decomposition of $\Sigma$ given by $f$. Moreover, the vanishing cycles
corresponding to the critical points of $f$ of index two are obtained from the
vanishing cycles corresponding to the critical points of $f$ of index zero and
one, by a simultaneous “Lagrangian surgery” where they meet on the fiber.

To summarize, based on the handle decomposition of the surface $\Sigma$ given
by $f$, Johns describes how to construct (the diffeomorphism type of) the
regular fiber of an exact symplectic Lefschetz fibration $f_{\omega}:E\to
D^{2}$, and the Lagrangian vanishing cycles explicitly on this fiber. The map
$f_{\omega}$ can be viewed as, roughly speaking, a symplectic convexification
of the smooth Morse function $f$ with respect to $\omega$, as opposed to a
contact convexification due to Giroux, which we will discuss in the next
section.

The hardest part of Johns’ article is his proof that the exact symplectic
$4$-manifold $E$ with convex boundary is conformally exact symplectomorphic to
$(DT^{*}\Sigma,\omega)$, after smoothing the corners of $E$, which he
accomplishes by following the steps (i)-(iii) below.

(i) There is an exact Lagrangian embedding $\Sigma\subset E$.

(ii) Critical points of $f_{\omega}$ are in $\Sigma$,
$f_{\omega}(\Sigma)=[a,b]\subset\mathbb{R}$, and $f_{\omega}|_{\Sigma}=f$ (up
to reparametrizing $\Sigma$ and $\mathbb{R}$ by diffeomorphisms).

(iii) $E$ is conformally exact symplectomorphic to $(DT^{*}\Sigma,\omega)$.

Recall that a conformal exact symplectomorphism between exact symplectic
manifolds $(E_{i},\omega_{i},\theta_{i})$ (where $\omega_{i}=d\theta_{i}$ is a
symplectic form on $E_{i}$), for $i=1,2$, is a map

$\psi:(E_{1},\omega_{1},\theta_{1})\to(E_{2},\omega_{2},\theta_{2})$

such that $\psi^{*}\theta_{2}=K\theta_{1}+dh$, for some $K>0$ and
$h:E_{1}\to\mathbb{R}$. In particular, $\psi^{*}\omega_{2}=K\omega_{1}$, which
indeed implies that $\psi$ is a conformal symplectomorphism.

The main step in the proof is (i), where Johns uses a Milnor style handle
decomposition of $\Sigma$ given by the Morse function $f$, to embed $\Sigma$
into $E$ “handle by handle” as an exact Lagrangian submanifold. The statement
(ii) follows by the construction of the embedding $\Sigma\subset E$. The last
step (iii) is achieved by describing a retraction of $E$, using a Liouville
type flow, onto a small Weinstein neighborhood of $\Sigma$ in $E$,
symplectomorphic to $DT^{*}\Sigma$.

##  4\. Convex Morse functions on the bundle of hyperplanes

Our goal in this section is to review Giroux’s construction [17] of an open
book adapted to the contact $3$-manifold $(V(\Sigma),\xi)$, where $\xi$ is the
canonical contact structure on the bundle $V(\Sigma)$ of cooriented lines
tangent to $\Sigma$. We begin with some general facts about convex contact
manifolds, and then give some details of Giroux’s proof of the convexity of
the canonical contact structure on the bundle of cooriented hyperplanes $V(M)$
for an arbitrary smooth manifold $M^{n+1}$ in Section 4.1, before we turn our
attention to the case $n=1$ (i.e, $M=\Sigma$, a surface) in Section 4.2.

######  Definition 4.1.

[17] A contact structure $\zeta$ is called convex if it is invariant under the
flow of a vector field $X$ which is gradient-like for a Morse function. Such a
function is called $\zeta$-convex (or contact) Morse, and $X$ is called a
contact vector field. The characteristic hypersurface $C_{X}$ is defined as
the set of points where $X$ is tangent to $\zeta$.

In his groundbreaking work, Giroux proved that every contact structure is
convex, which implies that every contact manifold admits an adapted open book
by Theorem 4.2 below.

######  Theorem 4.2 (Giroux).

Let $(V,\zeta)$ be a closed contact manifold of dimension $2n+1$. Suppose that
$X$ is a contact vector field on $V$, which is gradient-like for an ordered
Morse function $F:V\to\mathbb{R}$. Let $L$ be a regular level set of $F$ above
the critical values of index $n$ and below the critical values of index $n+1$.
Then there is an open book adapted to $(V,\zeta)$ whose binding $K$ is the
transverse intersection of $L$ and the characteristic hypersurface $C_{X}$.
Moreover, the link $K=L\cap C_{X}$ cuts $L\cup C_{X}$ into four pages of this
open book.

Conversely, any open book for $V$ supporting $\xi$ comes from this
construction.

###  4.1. Convexity of the canonical contact structure on the bundle of
hyperplanes

Let $V(M)$ denote the bundle of cooriented hyperplanes tangent to a closed
manifold $M^{n+1}$ and $\xi$ denote the canonical contact structure on $V(M)$.
Note that $V(M)$ can be identified with the unit cotangent bundle $ST^{*}M$
and $\xi$ is given by the kernel of the Liouville $1$-form $\lambda$, under
this identification (see, for example, [16, page 32]). In the following, we
review Giroux’s construction [17, Example 4.8] of a $\xi$-convex Morse
function on $V(M)$.

Any diffeomorphism $\psi:M\to M$ lifts to a contactomorphism
$\widetilde{\psi}$ of $(V(M),\xi)$ defined by

$\widetilde{\psi}(x,H)=(\psi(x),d\psi_{x}(H)).$

It follows that any vector field $X$ on $M$ lifts to a contact vector field
$\widehat{X}$ on $(V(M),\xi)$ by lifting the flow of $X$.

Let $f:M\to\mathbb{R}$ be a Morse function and $X$ be a gradient-like vector
field for $f$ such that at each critical point of $p$ of $f$, the eigenvalues
of $D_{p}X$ are real and simple. Then there is a contact vector field
$\widehat{X}$ for $(V(M),\xi)$ obtained as the lift of $X$ as explained above.
To show that $(V(M),\xi)$ is convex, it suffices to construct a Morse function
$f_{\xi}:V(M)\to\mathbb{R}$ such that $\widehat{X}$ is gradient-like for
$f_{\xi}$. For any critical point $p$ of $f$, let $V_{p}(M):=\pi^{-1}(p)$
denote the sphere fiber, where $\pi:V(M)\to M$ is the bundle projection. Since
$\widehat{X}$ projects to $X$, it is vertical above the critical points of
$f$, i.e., it is tangent to $V_{p}$. The restriction of $\widehat{X}$ to
$V_{p}(M)$ is the gradient of some Morse function
$g_{p}:V_{p}(M)\to\mathbb{R}$, having exactly $2n+2$ critical points,
corresponding to the $n+1$ hyperplanes generated by $n$ eigendirections of
$D_{p}X$.

We set

(4.2) $f_{\xi}:=f+\sum\limits_{p\in Crit(f)}\chi_{p}g_{p},$

where $\chi_{p}$ is a suitably defined cut-off function which vanishes outside
of a sufficiently small neighborhood of the critical points.

Figure 4. A gradient-like vector field for an ordered Morse function
associated to a divide on a surface.

###  4.2. Canonical contact structure on the bundle of cooriented lines

We restrict our attention to the case $n=1$, namely we take $M$ to be a closed
and oriented surface $\Sigma$. Suppose that $f:\Sigma\to\mathbb{R}$ is an
ordered Morse function and $X$ is a gradient-like vector field for $f$ on
$\Sigma$ (see Figure 4 for an example). The vector field $X$ lifts to a vector
field $\widehat{X}$ on $V(\Sigma)$, which is gradient-like for the modified
lift $f_{\xi}:V(\Sigma)\to\mathbb{R}$. As observed by Massot [25], the
$\xi$-convex Morse function $f_{\xi}:V(\Sigma)\to\mathbb{R}$ defined by
Equation (4.2) is ordered provided that $f:\Sigma\to\mathbb{R}$ is an ordered
Morse function555Massot [25] uses a self-indexed Morse function $f$, but the
conclusion holds true if self-indexed is replaced by ordered. Here we assume
that $f$ is ordered and $f^{-1}(0)$ includes all index $1$ critical points..
In particular the singularities of $f$ lift to quadruples of singularities of
$f_{\xi}$ in $V(\Sigma)$, as for example, we depicted on the left in Figure 5
for an index $1$ singularity that lifts to two index $1$ and two index $2$
singularities.

Figure 5. Given a vector field $X$ on a surface $\Sigma$ and a hyperbolic
fixed point $p$, the differential of the flow shrinks tangent lines toward the
expanding eigendirection. The lift $\widehat{X}$ in the bundle $V(\Sigma)$ has
four critical points in the fiber above $p$, which correspond to the
cooriented lines tangent to the eigendirections as depicted on the left. When
$X$ is a (pseudo-) gradient of an ordered Morse function $f$ on $\Sigma$, one
can lift $f$ to $V(\Sigma)$ and modify it using Equation (4.2) into an ordered
Morse function $f_{\xi}$, so that the critical points of $f_{\xi}$ are the
critical points of $\widehat{X}$ and the levels of $f_{\xi}$ away from the
fibers of the critical points of $f$ are the lifts of the levels of $f$. In
the center we depicted two critical level sets (blue and yellow) and the
intermediate regular level set (green) $f_{\xi}^{-1}(0)$. On the right, we
depicted the level set $f_{\xi}^{-1}(0)$ (green) and the characteristic
surface $C_{\widehat{X}}$ (yellow).

Now one can apply Theorem 4.2 to find an open book, which we call a _Giroux_
open book, that supports the canonical contact structure $\xi$ on $V(\Sigma)$
using the $\xi$-convex ordered Morse function
$f_{\xi}:V(\Sigma)\to\mathbb{R}$. On the right in Figure 5, we depicted the
characteristic surface $C_{\widehat{X}}$ (yellow) and the regular level
$f_{\xi}^{-1}(0)$ (green) around the fiber of a double point of the divide.
They intersect along the link consisting of the cooriented lines based at $P$
and tangent to $\xi$.

Suppose that $P$ is an admissible divide on $\Sigma$ and let
$f:\Sigma\to\mathbb{R}$ be an ordered Morse function adapted to $P$. Recall
that $c(P)$ is the number of circles in $P$ and $v(P)$ is the number of double
points of $P$. By our assumption, $v(P)$ is equal to the number of index $1$
critical points of $f$.

######  Proposition 4.3.

[26] For any admissible divide $P$, the Giroux open book has $2c(P)$ binding
components and its page genus is given by $1+v(P)-c(P)$.

###### Proof.

The indices of the critical points of $f_{\xi}:V(\Sigma)\to\mathbb{R}$ can be
computed from the indices of the critical points of $f:\Sigma\to\mathbb{R}$,
which in turn, determines the topology of the characteristic surface
$C_{\widehat{X}}$. This is because $f_{\xi}|_{C_{\widehat{X}}}$ is a Morse
function whose critical points are exactly the critical points of $f_{\xi}$
and moreover an index $i$ critical point of $f_{\xi}$ gives a critical point
of $f_{\xi}|_{C_{\widehat{X}}}$, whose index is $i$ if $i\leq 1$, and $i-1$,
otherwise. Since $C_{\widehat{X}}$ is the union of the two pages of an open
book supporting $\xi$, the topology of the page is determined by the number of
connected components of the binding, which is equal to twice the number of
circles in $P=f^{-1}(0)$. Since $v(P)$ is equal to the number of critical
points of index one of the Morse function $f$, by our assumption, the Euler
characteristic of the page is given by $-2v(P)$, and thus the genus of the
page is given by $1+v(P)-c(P)$. ∎

It follows that the topology of the pages of the Giroux open book agrees with
that of the A’Campo-Ishikawa open book (see Proposition 2.3).

##  5\. Geodesic flow and Birkhoff sections

In this section we recall Birkhoff’s classical construction [5] of a negative
Birkhoff cross section for the geodesic flow from a convex divide (which he
called a primary set of curves) on a surface. The geodesic flow $\Phi$ is the
flow on $ST\Sigma$, whose orbits are the lifts of the geodesics on $\Sigma$.
More precisely, if $g$ is a geodesic with unit speed on the Riemann surface
$\Sigma$, then the orbit of $\Phi$ going through $(g(0),\dot{g}(0))$ is given
by

$\Phi^{t}(g(0),\dot{g}(0))=(g(t),\dot{g}(t)).$

Although the geodesic flow depends on the choice of a Riemannian metric on
$\Sigma$, Gromov [19] showed that the geodesic flows corresponding to two
negatively curved metrics on a surface are topologically conjugated, which
means that there is a homeomorphism between the unit tangent bundles of
$\Sigma$ with respect to these metrics, taking the oriented orbits of one onto
the other. We conclude that as far as the topological (rather than dynamical)
properties are concerned, there is essentially a unique geodesic flow on a
negatively curved surface.

######  Definition 5.1.

Suppose that $X$ is a non-singular vector field on a closed and oriented
$3$-manifold $M$. A Birkhoff cross section666also called Poincaré-Birkhoff
section, or global cross section for $(M,X)$ is a compact orientable surface
$S$ with boundary such that $S$ is embedded in $M$, $X$ is transverse to the
interior of $S$, the boundary $\partial S$ is tangent to $X$, and every orbit
of $X$ intersects $S$ after a bounded time (see Figure 6 left).

It follows that $\partial S$ is the union of finitely many periodic orbits of
$X$, and near its boundary, a Birkhoff cross section $S$ looks like a
helicoidal staircase (see Figure 6 right). Notice that $S$ is cooriented by
$X$ since the interior of $S$ is transverse to $X$. Therefore, there is an
induced orientation on $S$ (and on its boundary $\partial S$), since $M$ is
oriented. On the other hand, $\partial S$ has a natural orientation as a
collection of periodic orbits of the vector field $X$. A Birkhoff cross
section $S$ is said to be a positive (resp. negative) if for each component of
$\partial S$, the natural orientation given by $X$ coincides with (resp. is
opposite of) its orientation inherited as the boundary of $S$.

Figure 6. Birkhoff cross sections: on the left the general picture, and on the
right how it looks around boundary components, assuming the flow is locally
vertical, for a negative and a positive Birkhoff cross section, respectively.

Notice that a positive Birkhoff cross section $S$ for $(M,X)$ is a page of an
open book for $M$ whose oriented binding is $\partial S$. On the other hand,
if $S$ is a negative Birkhoff cross section for $(M,X)$, then $\overline{S}$
is a positive Birkhoff cross section for $(\overline{M},X)$ and hence
$\overline{S}$ a is a page of an open book for $\overline{M}$ whose oriented
binding is $\overline{\partial{S}}$.

######  Definition 5.2.

We say that an admissible divide $P$ on a surface $\Sigma$ equipped with a
Riemannian metric is convex if every curve in $P$ is a closed geodesic, every
geodesic on $\Sigma$ intersects $P$ in bounded time, and every region of
$\Sigma\setminus P$ can be foliated by concentric curves with non-vanishing
curvature. In this context, a foliation of the regions of $\Sigma\setminus P$
by concentric curves is called a convex foliation.

Note that every admissible divide on a surface with constant curvature is
convex. An admissible divide may fail to be convex if, for example, there is a
closed geodesic staying in one region of $\Sigma\setminus P$, as depicted in
Figure 7.

Figure 7. A divide (red) that is not convex: the geodesic (blue) does not
intersect the divide.

######  Theorem 5.3 ([5, 15]).

Assume that $P$ is a convex divide and $\mathcal{F}$ is convex foliation of
$\Sigma\setminus P$. The set of pairs $(p,v)$, where $p$ belongs to some black
region of $\Sigma\setminus P$ and $v$ is a unit vector tangent to
$\mathcal{F}$ is a Birkhoff cross section for the geodesic flow on $ST\Sigma$.

The proof of Theorem 5.3 is straightforward. Denoting by $S^{\bullet}$ the set
of tangent vectors mentioned in Theorem 5.3, it is easily seen that
$S^{\bullet}$ is a surface whose boundary is the link $L(P)$ of the divide
$P$. Hence the boundary $\partial S^{\bullet}$ is indeed tangent to the
geodesic flow. Since the foliation $\mathcal{F}$ is convex, the contact
between leaves of this foliation and geodesic is only of order 1 (because of
the assumption on the curvature), hence of order 0 in the tangent bundle. This
means that the interior of $S^{\bullet}$ is transverse to the geodesic flow.
Finally since every geodesic on $\Sigma$ intersects $P$ in bounded time, every
orbit of the geodesic flow intersects $S^{\bullet}$ in bounded time.

######  Remark 5.4.

Theorem 5.3 provides one Birkhoff cross section canonically associated to a
given divide $P$, which corresponds to the page $S_{0}$ depicted in Figure 2.
Another natural Birkhoff cross section is the set of pairs $(p,v)$, where $p$
belongs to some white region of $\Sigma\setminus P$ and $v$ is a unit vector
tangent to $\mathcal{F}$. It corresponds to the page $S_{\pi}$, also depicted
in Figure 2.

######  Remark 5.5.

Isotopy classes of negative Birkhoff cross sections whose boundary is
symmetric were classified by Cossarini-Dehornoy [9]. Marty proved that a flow
cannot admit at the same time positive and negative Birkhoff sections [24],
hence there are no positive Birkhoff cross sections for geodesic flows.

##  6\. Proof of Theorem 1.1

The proof of Theorem 1.1 follows by combining Proposition 6.2, Proposition
6.3, and Lemma 6.8 below.

######  Definition 6.1.

Suppose that $P$ is an admissible divide on a closed and oriented surface
$\Sigma$ which is equipped with a Riemannian metric and let
$f:\Sigma\to\mathbb{R}$ be an ordered Morse function adapted to $P$. Then

  * •

$\overline{\pi}_{1}:DT\Sigma\to D^{2}$ denotes Ishikawa’s achiral Lefschetz
fibration and $\overline{\mathop{\rm OB}\nolimits}_{1}$ the induced open book
for $ST\Sigma$. Similarly, $\pi_{1}:DT^{*}\Sigma\to D^{2}$ denotes the
corresponding Lefschetz fibration, obtained by reversing the orientation of
$\overline{\pi}_{1}$, and $\mathop{\rm OB}\nolimits_{1}$ the induced open book
for $ST^{*}\Sigma$ (see Remark 2.1).

  * •

$\pi_{2}:DT^{*}\Sigma\to D^{2}$ denotes Johns’ Lefschetz fibration and
$\mathop{\rm OB}\nolimits_{2}$ the induced open book for $ST^{*}\Sigma$, that
supports the canonical contact structure $\xi$. (Note that $f_{\omega}:E\to
D^{2}$ described in Section 3 induces a Lefschetz fibration $DT^{*}\Sigma\to
D^{2}$, which we denote by $\pi_{2}$ here.)

  * •

$\mathop{\rm OB}\nolimits_{3}$ denotes Giroux’s open book for $V(\Sigma)$ that
supports the canonical contact structure $\xi$.

  * •

$\overline{\mathop{\rm OB}\nolimits}_{4}$ denotes the geodesic open book for
$ST\Sigma$ induced by the geodesic flow, whose pages are negative Birkhoff
cross sections, provided that $P$ is convex with respect to the metric on
$\Sigma$, and $\mathop{\rm OB}\nolimits_{4}$ denotes the corresponding open
book for $ST^{*}\Sigma$, whose pages are positive Birkhoff cross sections.

We will show below that the open books $\mathop{\rm OB}\nolimits_{1}$,
$\mathop{\rm OB}\nolimits_{2}$ and $\mathop{\rm OB}\nolimits_{3}$ are pairwise
isomorphic for any surface $\Sigma$ and $\mathop{\rm OB}\nolimits_{4}$ is
isomorphic to any one of them when $\Sigma$ is a Riemannian surface such that
$P$ is convex (see Definition 5.2), which is the content of Theorem 1.1. As a
corollary, it follows that they all support the canonical contact structure
$\xi$ on $ST^{*}\Sigma\cong V(\Sigma)$.

Given a Morse function $f$ which satisfies the hypothesis of Theorem 1.1, for
$i\in\\{0,1,2\\}$, denote by $m_{i}$ the number of critical points of index
$i$.

######  Proposition 6.2.

The Lefschetz fibrations $\pi_{1}$ and $\pi_{2}$ on $DT^{*}\Sigma$ are
isomorphic.

###### Proof.

We claim that $\pi_{1}$ and $\pi_{2}$ are isomorphic, meaning that there is a
diffeomorphism from the regular fiber of $\pi_{1}$ to that of $\pi_{2}$ taking
the ordered vanishing cycles of $\pi_{1}$ to that of $\pi_{2}$. In [27], we
proved this result only for one particular Morse function, which gives the
minimum genus Lefschetz fibration on $DT^{*}\Sigma$. Here we prove it for all
Morse functions satisfying the assumptions of Theorem 1.1. Notice that an
isomorphism between the Lefschetz fibrations $\pi_{1}$ and $\pi_{2}$ induces
an isomorphism of the open books $\mathop{\rm OB}\nolimits_{1}$ and
$\mathop{\rm OB}\nolimits_{2}$.

The regular fiber $F_{1}$ and the ordered vanishing cycles

$\alpha_{1},\ldots,\alpha_{m_{0}},\beta_{1},\ldots,\beta_{m_{1}},\gamma_{1},\ldots,\gamma_{m_{2}}$

of $\pi_{1}$ are described explicitly in [2, 20] based on the divide
$P=f^{-1}(0)$. The fiber $F_{1}$ is obtained by starting with $m_{0}$ disjoint
roundabouts (certain embedded annuli), one for each double point of $P$, and
then connecting them with twisted bands, one for each edge in $P$ connecting
any two double points. The curves $\beta_{1},\ldots,\beta_{m_{1}}$ are the
core circles of the roundabouts, the curves $\alpha_{1},\ldots,\alpha_{m_{0}}$
appear as the boundaries of the white regions, while
$\gamma_{1},\ldots,\gamma_{m_{2}}$ appear as the boundaries of the black
regions in $\Sigma\setminus P$.

On the other hand, the regular fiber $F_{2}$ and the ordered vanishing cycles

$a_{1},\ldots,a_{m_{0}},b_{1},\ldots,b_{m_{1}},c_{1},\ldots,c_{m_{2}}$

of $\pi_{2}$ are described explicitly in [21] based on the handle
decomposition of $\Sigma$ given by the Morse function $f$. By our assumptions
on $f$, the surface $\Sigma$ is obtained by gluing $m_{1}$ $1$-handles to the
disjoint union of $m_{0}$ disks and then gluing $m_{2}$ disks to close off the
resulting boundary. The fiber $F_{2}$ is obtained by starting with $m_{0}$
disjoint annuli $A_{0},\ldots,A_{m_{0}}$ (one for each $0$-handle) and
plumbing $m_{1}$ disjoint annuli $B_{0},\ldots,B_{m_{1}}$ (one for each
$1$-handle), onto this union, according to the handle decomposition of
$\Sigma$. More precisely, a $B$-annulus is plumbed for each $1$-handle,
exactly corresponding to the feet of that $1$-handle in the handle
decomposition of $\Sigma$. Notice that the vanishing cycle $a_{i}$ is the core
circle of $A_{i}$, and the vanishing cycle $b_{j}$ is the core circle of
$B_{j}$, while the vanishing cycles $c_{1},\ldots,c_{m_{2}}$ are obtained by a
(left) Lagrangian surgery on the set of curves
$a_{1},\ldots,a_{m_{0}},b_{1},\ldots,b_{m_{1}}$ at the points where they
intersect.

To see that $F_{1}$ is diffeomorphic to $F_{2}$, we view the construction of
$F_{1}$ slightly differently. Rather than starting with the roundabouts as we
described above, we consider the disjoint union of an annulus neighborhood of
each of the curves $\alpha_{1},\ldots,\alpha_{m_{0}}$ and view the roundabouts
as a plumbing onto this disjoint union. This was already discussed in [27] for
twisted $1$-handles and a similar proof holds for the untwisted $1$-handles as
well, which implies that $F_{1}$ is obtained in the same way as $F_{2}$ from
this point of view. Moreover, this also proves the isomorphism of the ordered
vanishing cycles, since first of all $\alpha_{i}$ can be identified with
$a_{i}$ and $\beta_{j}$ can be identified with $b_{j}$. Recall that
$c_{1},\ldots,c_{m_{2}}$ are obtained by a surgery on
$a_{1},\ldots,a_{m_{0}},b_{1},\ldots,b_{m_{1}}$ at the points where they
intersect. Similarly, one can check that the curves
$\gamma_{1},\ldots,\gamma_{m_{2}}$ are obtained from
$\alpha_{1},\ldots,\alpha_{m_{0}},\beta_{1},\ldots,\beta_{m_{1}}$ at the
points where they meet, so that the set $\gamma_{1},\ldots,\gamma_{m_{2}}$ can
be identified with the set $c_{1},\ldots,c_{m_{2}}$ and hence the ordered
vanishing cycles of $\pi_{1}$ are mapped to that of $\pi_{2}$ under the above
diffeomorphism of $F_{1}$ and $F_{2}$, which finishes the proof of the
isomorphism of $\pi_{1}$ and $\pi_{2}$. ∎

######  Proposition 6.3.

The open book $\mathop{\rm OB}\nolimits_{1}$ for $ST^{*}\Sigma$ and the open
book $\mathop{\rm OB}\nolimits_{3}$ for $V(\Sigma)$ are isomorphic.

###### Proof.

This is essentially contained in the first author’s PhD thesis [10, Chapter
1], where tangent bundles were considered rather than cotangent bundles. Here
we work with $V(\Sigma)\cong ST^{*}\Sigma$ which is more natural from the
contact geometric point of view. We would like to show that $\mathop{\rm
OB}\nolimits_{1}$ for $ST^{*}\Sigma$ is orientation-preserving isomorphic to
$\mathop{\rm OB}\nolimits_{3}$ for $V(\Sigma)$. Instead, we first set up an
isomorphism between $\overline{\mathop{\rm OB}\nolimits}_{1}$ and $\mathop{\rm
OB}\nolimits_{3}$, and then derive the conclusion by reversing the orientation
on the pages of $\overline{\mathop{\rm OB}\nolimits}_{1}$, as explained in
Remark 6.9 below.

######  Lemma 6.4.

Under the identification of $ST\Sigma$ with $V(\Sigma)$, the binding of
$\overline{\mathop{\rm OB}\nolimits}_{1}$ corresponds to the binding of
$\mathop{\rm OB}\nolimits_{3}$.

###### Proof.

The binding of $\overline{\mathop{\rm OB}\nolimits}_{1}$ (see Remark 2.2 and
Figures 2 and 3) is the set of unit vectors tangent to the divide $P$. Under
the identification of $ST\Sigma$ with $V(\Sigma)$, each unit tangent vector
represents a cooriented line, by taking its normal. Now we show that the set
of these cooriented lines normal to $P$ is the binding of $\mathop{\rm
OB}\nolimits_{3}$ for $V(\Sigma)$. Recall that the characteristic surface
$C_{\widehat{X}}$ (see Section 4.2) consists of those cooriented lines
containing the gradient-like vector field $X$ for $f$, which can be assumed to
be the gradient $\nabla f$ away from the critical points of $f$. Notice that
we fixed $f$ from the beginning of the proof, but we are free to choose the
gradient-like vector field $X$, as long as it is in a certain form near the
critical points of $f$ as we explained in Section 4.1.

The binding of $\mathop{\rm OB}\nolimits_{3}$ is given by the intersection of
$f_{\xi}^{-1}(0)$ with $C_{\widehat{X}}$. First of all, we observe that
$f_{\xi}^{-1}(0)$ does not intersect the fibers above the Morse charts around
index $2$ or index $0$ critical points. Away from the index $1$ critical
points of $f$, the surface $f_{\xi}^{-1}(0)$ consists of all the circle fibers
above the divide $P=f^{-1}(0)$. Since $X=\nabla f$ away from the critical
points, there are two points in each such circle fiber that belongs to
$C_{\widehat{X}}$, corresponding to the normal directions to $P$. So, we
conclude that, possibly except for a neighborhood of index $1$ critical
points, the binding of $\mathop{\rm OB}\nolimits_{3}$ consists of cooriented
normal lines along $P$. Nevertheless, one can check that this is also the case
on a Morse chart around each index $1$ critical point of $f$ either by looking
at Figure 5 right, or by the following argument.

Using local coordinates $x_{0},x_{1}$ for a Morse chart centered at an index
$1$ critical point, and denoting the corresponding cotangent fiber coordinates
by $y_{0}=\cos\theta$ and $y_{1}=\sin\theta$, we have (cf. [25]):

$f(x_{0},x_{1})=-x_{0}^{2}+x_{1}^{2}$ (6.3) $X=-x_{0}\frac{\partial}{\partial
x_{0}}+x_{1}\frac{\partial}{\partial x_{1}}$ $\xi=\cos\theta dx_{0}-\sin\theta
dx_{1}$ (6.4)
$f_{\xi}(x,y)=-x^{2}_{0}+x_{1}^{2}+\widetilde{\eta}\chi(x)(y_{0}^{2}-y_{1}^{2})=-x^{2}_{0}+x_{1}^{2}+\widetilde{\eta}\chi(x)\cos
2\theta$ $\widehat{X}=-x_{0}\frac{\partial}{\partial
x_{0}}+x_{1}\frac{\partial}{\partial x_{1}}-\sin
2\theta\frac{\partial}{\partial\theta}$

where $\widetilde{\eta}>0$ is a sufficiently small real number. Notice that

$\xi(\widehat{X})=0\iff-x_{0}\cos\theta=x_{1}\sin\theta.$

It follows that the binding $f_{\xi}^{-1}(0)\cap C_{\widehat{X}}$ consists of
the following four disjoint line segments

$\theta=\pi/4$ and $-x_{0}=x_{1}$

$\theta=3\pi/4$ and $x_{0}=x_{1}$

$\theta=5\pi/4$ and $-x_{0}=x_{1}$

$\theta=7\pi/4$ and $x_{0}=x_{1}$

in the trivialized circle bundle over this Morse chart. Therefore, since
$P=f^{-1}(0)$ is given by the lines $x_{0}=\pm x_{1}$ in this chart, we
conclude that the binding consists of cooriented normal lines along $P$, on
this chart as well, which finishes the proof of the lemma. ∎

Next we show that the four distinct pages of $\overline{\mathop{\rm
OB}\nolimits}_{1}$ coincides with that of $\mathop{\rm OB}\nolimits_{3}$,
under the identification of $ST\Sigma$ with $V(\Sigma)$. The four pages we
have in mind are the pages whose images are purely real or purely imaginary
under the fibration map defining $\overline{\mathop{\rm OB}\nolimits}_{1}$
given in Remark 2.2, namely the pages $S_{0}$, $S_{\pi/2}$, $S_{\pi}$ and
$S_{3\pi/2}$ of $\overline{\mathop{\rm OB}\nolimits}_{1}$.

######  Lemma 6.5.

Under the identification of $ST\Sigma$ with $V(\Sigma)$, the union $S_{0}\cup
S_{\pi}$ corresponds to the characteristic surface $C_{\widehat{X}}$, which is
the union of two pages in $\mathop{\rm OB}\nolimits_{3}$.

###### Proof.

The interior of the page $S_{0}$ (resp. $S_{\pi}$) of $\overline{\mathop{\rm
OB}\nolimits}_{1}$ consists of the unit vectors tangent to the level sets of
$f$ in black (resp. white) regions of $\Sigma\setminus P$ (see Figure 2). In
particular, the whole circle fiber above a critical point of index $2$ (resp.
index $0$) is included in $S_{0}$ (resp. $S_{\pi}$). We claim that the set of
corresponding conormal lines of all the level sets in black and white regions
belongs to the characteristic surface $C_{\widehat{X}}\subset V(\Sigma)$, up
to isotopy. This is clear outside of a neighborhood of the critical points,
since the vector field $X$ is assumed to be equal to $\nabla f$, which is
indeed normal to the level sets of $f$. In addition, $X$ is normal to the
level sets in a neighborhood of an index $1$ critical point by (6.3), see
Figure 5.

On the other hand, over a critical point of index $2$ or index $0$, the whole
fiber in $V(\Sigma)$ is included in $C_{\widehat{X}}$. In addition, by
choosing the vector field $X$ to be sufficiently close to $\nabla f$ on a
Morse chart around an index $2$ (resp. index $0$) critical point, we can
guarantee that the piece of $S_{0}$ (resp. $S_{\pi}$) is isotopic to the piece
of $C_{\widehat{X}}$ above that chart. In other words, we consider an isotopic
fibration map defining the open book $\overline{\mathop{\rm
OB}\nolimits}_{1}$, where the isotopy only takes place above the Morse charts
around index $2$ or index $0$ critical points. Therefore, this isotopy does
not affect the discussion in the proof of Lemma 6.4, since the binding in each
open book does not intersect the fibers above the Morse charts around the
critical points of index $2$ or index $0$. Hence, we finish the proof of the
lemma, by adding in the bindings of both open books. ∎

######  Lemma 6.6.

Under the identification of $ST\Sigma$ with $V(\Sigma)$, the union
$S_{\pi/2}\cup S_{3\pi/2}$ corresponds to the level set $f_{\xi}^{-1}(0)$,
which is the union of two pages in $\mathop{\rm OB}\nolimits_{3}$.

###### Proof.

The interior of the page $S_{\pi/2}$ (resp. $S_{3\pi/2}$) of
$\overline{\mathop{\rm OB}\nolimits}_{1}$ consists of unit vectors along the
divide $P$ pointing into black (resp. white) regions of $\Sigma\setminus P$.
Notice that $f_{\xi}^{-1}(0)$ does not intersect the fibers above the Morse
charts around index $2$ or index $0$ critical points. Moreover, away from the
index $1$ critical points of $f$, the zero-level set of $f_{\xi}$ consists of
all the circle fibers above the divide $P$ and thus all cooriented lines are
in $f_{\xi}^{-1}(0)$, where the binding of $\mathop{\rm OB}\nolimits_{3}$
separates the two pages. To summarize, away from the index $1$ critical
points, the union $S_{\pi/2}\cup S_{3\pi/2}$ corresponds to the level set
$f_{\xi}^{-1}(0)$.

Near each index $1$ critical point, we understand both $f_{\xi}^{-1}(0)$ and
the union $S_{\pi/2}\cup S_{3\pi/2}$ explicitly in local coordinates (again,
see Figure 5 right). According to (6.4), the surface $f_{\xi}^{-1}(0)$ is
given locally by

$\\{(x_{0},x_{1},y_{0},y_{1})\;|\;-x_{0}^{2}+x_{1}^{2}+\widetilde{\eta}\chi(x)(y_{0}^{2}-y_{1}^{2})=0\\}$

while according to Remark 2.2, the union $S_{\pi/2}\cup S_{3\pi/2}$ is given
locally by

$\\{(x_{0},x_{1},u_{0},u_{1})\;|\;-x_{0}^{2}+x_{1}^{2}-\dfrac{1}{2}\eta^{2}\chi(x)(u_{1}^{2}-u_{0}^{2})=0\\}.$

So, we see that these surfaces coincide near each index one critical point, by
taking $\widetilde{\eta}=\eta^{2}/2$ and using identical cut-off functions
$\chi:\Sigma\to\mathbb{R}$. ∎

######  Lemma 6.7.

Suppose that $\mathop{\rm OB}\nolimits$ (resp. $\widetilde{\mathop{\rm
OB}\nolimits}$) is an open book for a closed and oriented $3$-manifold $M$
(resp. $\widetilde{M}$). Let $S$ (resp. $\widetilde{S}$) be a fixed oriented
page of $\mathop{\rm OB}\nolimits$ (resp. $\widetilde{\mathop{\rm
OB}\nolimits}$). If there is an orientation preserving diffeomorphism
$\varphi:M\to\widetilde{M}$ such that $\varphi(S)=\widetilde{S}$, then
$\mathop{\rm OB}\nolimits$ is isomorphic to $\widetilde{\mathop{\rm
OB}\nolimits}$.

###### Proof.

The result follows by observing that $M\setminus S$ is diffeomorphic to a
trivial $S$-bundle over $I$, while $\widetilde{M}\setminus\widetilde{S}$ is
diffeomorphic to a trivial $\widetilde{S}$-bundle over $I$. ∎

We conclude the proof of Proposition 6.3 by Lemma 6.5 and Lemma 6.6 that the
four pages of $\overline{\mathop{\rm OB}\nolimits}_{1}$ coincides with the
four pages of $\mathop{\rm OB}\nolimits_{3}$ under the aforementioned
diffeomorphism between $ST\Sigma$ and $V(\Sigma)$. This is more than enough to
conclude that $\mathop{\rm OB}\nolimits_{1}$ is isomorphic to $\mathop{\rm
OB}\nolimits_{3}$ by Lemma 6.7, after reversing the orientation of the pages
of $\overline{\mathop{\rm OB}\nolimits}_{1}$ first as explained in Remark 6.9.
∎

######  Lemma 6.8.

The open books $\mathop{\rm OB}\nolimits_{1}$ and $\mathop{\rm
OB}\nolimits_{4}$ for $ST^{*}\Sigma$ are isomorphic, provided that $P$ is
convex with respect to the given metric on $\Sigma$.

###### Proof.

The binding of both open books correspond to the link $L(P)$ consisting of
those vectors tangent to $P$, hence they coincide. The geometric description
in Remark 2.2 of the page $S_{0}$ of the open book $\overline{\mathop{\rm
OB}\nolimits}_{1}$ of A’Campo & Ishikawa coincides with the page of the open
book $\overline{\mathop{\rm OB}\nolimits}_{4}$ of Birkhoff & Fried given by
Theorem 5.3, based on a set of geodesics satisfying the assumptions of a
divide (see the top left of Figure 2). Indeed they both correspond to the set
of vectors tangent to the level sets of $f$ in the black regions of
$\Sigma\setminus P$. By reversing the orientations, we get the desired
isomorphism of $\mathop{\rm OB}\nolimits_{1}$ and $\mathop{\rm
OB}\nolimits_{4}$ using Lemma 6.7. ∎

In fact, one can show that these two open books have _four_ “common” pages,
namely the pages $S_{0},S_{\pi/2},S_{\pi}$, and $S_{3\pi/2}$ from $\mathop{\rm
OB}\nolimits_{1}$.

######  Remark 6.9.

Let $S$ be an oriented surface with boundary and $h$ an element of the mapping
class group $\mathrm{MCG}(S,\partial S)$, which is a product of positive Dehn
twists. Note that the sign of the Dehn twist depends on the orientation of
$S$. Let $M$ be the oriented $3$-manifold which is the total space of an
abstract an open book $(S,h)$ with page $S$ and monodromy $h$. Let $\eta$ be
the positive Stein fillable contact structure supported by $(S,h)$. The total
space of the open book $(\overline{S},h)$ is $\overline{M}$, where $h$ is now
a product of _negative_ Dehn twists in
$\mathrm{MCG}(\overline{S},\partial\overline{S})$. Moreover, $\overline{\eta}$
is the positive contact structure on $\overline{M}$ supported by
$(\overline{S},h)$, which is overtwisted. On the other hand, $\eta$ is a
negative contact structure on $\overline{M}$.

######  Remark 6.10.

The manifolds $ST^{*}\Sigma$ and $DT^{*}\Sigma$ are orientable, even if
$\Sigma$ is a nonorientable surface. While Johns’ construction of a symplectic
Lefschetz fibration on $DT^{*}\Sigma$ works for any surface $\Sigma$, Ishikawa
only deals with oriented surfaces. On the other hand, both constructions work
for an oriented surface with nonempty boundary and the proof of Proposition
6.2 extends to cover that case.

##  7\. Lefschetz fibrations and open books of minimal genus

A particular admissible divide on a closed and oriented surface $\Sigma$ of
genus $g>1$ was given by Fried [15], following Birkhoff [5]. This divide
consists of $2g+2$ simple closed curves, with $2g+2$ double points, as we
depicted at the top right of Figure 8, for $g=3$. Fried noticed that, when
$\Sigma$ is hyperbolic, the corresponding Birkhoff cross section has genus
one, which can also be recovered from Proposition 2.3. The associated
Lefschetz fibration $DT^{*}\Sigma\to D^{2}$, whose fiber is of genus one
(minimal possible) with $4g+4$ boundary components, was explicitly constructed
by the second author in [27]. Here we simply observe that this construction
does not give the minimal number of boundary components of a fiber among all
genus one Lefschetz fibrations, answering Question (4) at the end of [27]
negatively.

######  Proposition 7.1.

777This statement was independently proven by C. Bonatti (unpublished, private
communication).

For any closed and oriented surface $\Sigma$ of genus $g\geq 1$, there is an
explicit genus one Lefschetz fibration $DT^{*}\Sigma\to D^{2}$ whose fiber has
$4g$ boundary components. As a consequence, there is an explicit genus one
open book adapted to $(ST^{*}\Sigma,\xi)$, whose binding has $4g$ components.

Figure 8. Three admissible divides that yield genus one Lefschetz fibrations
on $DT^{*}\Sigma$. The top row corresponds to the Birkhoff-Fried divide which
consists of $2g+2$ curves that we depicted on the left for a torus and on a
genus $3$ surface on the right. The middle row corresponds to Brunella’s
improvement with $2g+1$ curves. It is obtained by smoothing one crossing, thus
connecting the two white regions. The bottom row corresponds to an additional
improvement with $2g$ curves only, obtained by smoothing another crossing and
connecting the two black regions.

###### Proof.

The set of $2g$ simple closed curves on $\Sigma$ depicted at the bottom row of
Figure 8 yields an admissible divide on $\Sigma$. Since each of these curves
intersect exactly two other curves, the regular fiber of the Lefschetz
fibration $DT^{*}\Sigma\to D^{2}$ obtained by using the methods of either
Ishikawa or Johns is of genus one, with $4g$ boundary components. Moreover,
the monodromy of this Lefschetz fibration can be explicitly computed as a
product of Dehn twists. It follows that, there is an explicit genus one open
book adapted to $(ST^{*}\Sigma,\xi)$, whose binding has $4g$ components.
Furthermore, by Theorem 1.1, this open book is isomorphic to the Giroux open
book as well as the geodesic open book. ∎

An open book for a closed $3$-manifold induces a Heegaard splitting, where the
Heegaard surface is the union of two pages along the binding. If the page of
the open book is a surface of genus $h$ with $k$ boundary components, then the
Heegaard surface is of genus $2h+k-1$. According to Boileau-Zieschang [6], for
any closed and oriented surface $\Sigma$ of genus $g$, the Heegaard genus of
$ST^{*}\Sigma$ is $2g+1$. Therefore, the binding of any genus one open book
for $ST^{*}\Sigma$ must have at least $2g$ connected components, which implies
that the fiber of any genus one Lefschetz fibration $DT^{*}\Sigma\to D^{2}$
must have at least $2g$ boundary components. In other words, for an oriented
surface $\Sigma$ of genus at least two, the binding number $\mathop{\rm
bn}\nolimits(ST^{*}\Sigma,\xi)$ [14] satisfies the following inequality

$2g\leq\mathop{\rm bn}\nolimits(ST^{*}\Sigma,\xi)\leq 4g.$

This simple observation indicates that there is some room for improvement for
the result stated in Proposition 7.1, using methods other than discussed in
this paper. For example, a genus one open book adapted to
$(ST^{*}T^{2}=T^{3},\xi)$ with three binding components, was constructed by
Van Horn-Morris [29] (see also [11, Section 3] for similar examples obtained
by the first author).

Next we observe that Proposition 7.1 can not be improved utilizing the
constructions described in this article. Notice that Theorem 7.2 below implies
Theorem 1.4 from the introduction.

######  Theorem 7.2.

Up to homeomorphism, there are exactly three admissible divides (depicted in
Figure 8) on a genus $g\geq 1$ surface $\Sigma$ that yield genus one Lefschetz
fibrations $DT^{*}\Sigma\to D^{2}$.

###### Proof.

Let $P$ be an admissible divide on $\Sigma$. Recall that $c(P)$ denotes the
number of curves that form $P$ and $v(P)$ the number of double points in $P$.
By Proposition 2.3, the genus of the associated Lefschetz fibration
$DT^{*}\Sigma\to D^{2}$ is $1+v(P)-c(P)$. Hence this genus is equal to $1$ if
and only if $c(P)=v(P)$. Since $P$ is admissible, its complement can be black-
and-white colored, which means that every curve in $P$ has an even number of
double points (counting self-intersection points twice). Since $P$ is
connected, every component of $P$ has at least two double points. By counting
all double points twice (one for every curve that traverses the point), one
obtains $2v(P)\geq 2c(P)$, with equality if and only if every curve in $P$ has
exactly two double points on it. One can check that this is indeed the case
for all the divides depicted in Figure 8.

Assume now that $P$ is an admissible divide so that every curve in $P$ has
exactly two double points on it. Consider the dual graph $\Gamma(P)$ whose
vertices are the curves in $P$ and whose edges are the double points. Note
that $\Gamma(P)$ is a graph with loops and multiple edges. Since $P$ is
admissible, $\Gamma(P)$ is connected, for otherwise the complement
$\Sigma\setminus P$ would have a non-trivial topology. Our assumption on the
number of double points implies that the vertices of $\Gamma(P)$ all have
degree $2$. Hence $\Gamma(P)$ is a cycle, and therefore $P$ is cyclic chain of
circles.

If one removes a small open disk in each region of $\Sigma\setminus P$, one
obtains a surface with boundary, which deformation-retracts on a neighborhood
$\Sigma_{P}$ of $P$. The surface with boundary $\Sigma_{P}$ is then a cyclic
chain of circular ribbons, say $r_{1},\dots,r_{k}$, where $k=c(P)$. One can
recover $\Sigma_{P}$ by gluing one by one all the ribbons, starting with
$r_{1}$, then gluing $r_{2}$, then $r_{3}$, and so on, inductively. Finally
one adds $r_{k}$ and glues back $r_{k}$ to $r_{1}$. Since $\Sigma_{P}$ is
orientable, every ribbon is a standard annulus, and there is no freedom in the
homeomorphism type of the gluing, except at the end when gluing $r_{k}$ back
to $r_{1}$. Indeed, as shown in Figure 9, the open chain $r_{1}\cup\dots\cup
r_{k}$ has four boundary components, which are color-coded. When gluing
$r_{k}$ back to $r_{1}$, one has to chose which boundary component is glued
with which one.

Figure 9. Proof of Theorem 7.2 is illustrated for even $k$ on the left and for
odd $k$ on the right. When reconstructing a neighborhood $\Sigma_{P}$ of $P$
in $\Sigma$, one has a chain of ribbons $r_{1},\dots,r_{k}$. The surface
$\Sigma_{P}$ with boundary is obtained by connecting the two open bands in
both figures. When $k$ is even, there are two possibilities: either one glues
each (colored) boundary with itself, or one glues blue with green, green with
blue, red with brown, and brown with red. In the first case $\Sigma_{P}$ has
four boundary components of length $k$, while in the second it has two
boundary components of length $2k$. When $k$ is odd, there are also two
possibilities: either one glues blue with blue, brown with brown, red with
green, and green with red, or one glues blue with brown, brown with blue, red
with red, and green with green. In either case $\Sigma_{P}$ has three boundary
components, two of length $k$ and one of length $2k$.

If $k$ is even, there are exactly two possible cases that lead to two
different divides. In the first case, one glues each boundary component with
itself and therefore $\Sigma_{P}$ has four boundary components. When filling
these boundary components with disks, thus recovering $\Sigma$, one checks
that $\Sigma$ is made of four $k$-gons glued along their edges. It has genus
$\frac{k-2}{2}$, so that $k=2g+2$ and the divide $P$ corresponds to the
Birkhoff-Fried divide [15] on $\Sigma$ (as depicted at the top row of Figure
8). In the second case, one glues each boundary component with another one so
that $\Sigma_{P}$ has two boundary components. When recovering $\Sigma$, it is
made of two $2k$-gons glued along their edges. It has genus $\frac{k}{2}$, so
that $k=2g$ and the divide $P$ is the one depicted at the bottom row of Figure
8.

Finally when $k$ is odd, there is only one possibility up to homeomorphism:
two boundary components have to be glued to themselves, and the two others are
glued one with another. Thus $\Sigma$ is made of two $k$-gons and one $2k$-gon
glued along their edges. It has genus $\frac{k-1}{2}$, so that $k=2g+1$ and
the divide $P$ corresponds to the Brunella divide depicted at the middle row
of Figure 8.∎

Cossarini and Dehornoy [9] generalized the work of Birkhoff [5], Fried [15]
and Brunella [7], to construct and classify negative Birkhoff cross sections
of the geodesic flow associated to arbitrary Eulerian coorientations on a
given finite set of geodesics on a hyperbolic surface $\Sigma$. Moreover,
Marty [23] computed the monodromy of the corresponding open books for
$ST\Sigma$ as a product of Dehn twists along curves explicitly described on a
page. The following questions naturally arise as a result of this discussion:
Do these open books (when viewed for $ST^{*}\Sigma$) support the canonical
contact structure $\xi$ as well? Are these monodromy factorizations pairwise
Hurwitz equivalent? Is the total space of the Lefschetz fibration filling each
of these open books diffeomorphic to $DT^{*}\Sigma$?

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