# (Achiral) Lefschetz fibration embeddings of $4$–manifolds

Suhas Pandit  Indian Institute of Technology Madras, IIT PO.Chennai, 600036,
Tamilnadu, India<EMAIL_ADDRESS>and  Selvakumar A  Indian Statistical
Institute, Bangalore, Stat-Math Unit, 8th Mile, Mysore Road RVCE Post,
Bangalore 560059, India<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove Lefschetz fibration embeddings of achiral as well as
simplified broken (achiral) Lefschetz fibrations of compact, connected,
orientable $4$–manifolds over $D^{2}$ into the trivial Lefschetz fibration of
$\mathbb{C}P^{2}\times D^{2}$ over $D^{2}$. These results can be easily
extended to achiral as well as simplified broken (achiral) Lefschetz
fibrations over $\mathbb{C}P^{1}.$ From this, it follows that every closed,
connected, orientable $4$–manifold admits a smooth (simplified broken)
Lefschetz fibration embedding in $\mathbb{C}P^{2}\times\mathbb{C}P^{1}.$ We
provide a huge collection of bordered Lefschetz fibration which admit bordered
Lefschetz fibration embeddings into a trivial Lefschetz fibration
$\tilde{\pi}:D^{4}\times D^{2}\to D^{2}.$ We also show that every closed,
connected, orientable $4$-manifold $X$ admits a smooth embedding into
$S^{4}\times S^{2}$ as well as into $S^{4}\tilde{\times}S^{2}$. From this, we
get another proof of a theorem of Hirsch which states that every closed,
connected, orientable $4$–manifold smoothly embeds in $\mathbb{R}^{7}.$ We
also discuss Lefschetz fibration embedding of non-orientable $4$–manifolds
$X$, where $X$ does not admit $3$– and $4$–handles in the handle
decomposition, into the trivial Lefschetz fibration of $\mathbb{C}P^{2}\times
D^{2}$ over $D^{2}$.

###### Key words and phrases:

$4$–manifolds, embeddings, Lefschetz fibration

###### 1991 Mathematics Subject Classification:

Primary: 57R40, 57R45, 57R15, 57R65

##  1\. Introduction

In this note, we work with Lefschetz fibration $\pi:X^{4}\to D^{2}$, where
$X^{4}$ is a compact, connected, orientable $4$–manifold with non-empty
boundary. We shall show that given a Lefschetz fibration $\pi:X^{4}\to D^{2}$,
it admits a Lefschetz fibration embedding into a trivial Lefschetz fibration
$\mathbb{C}P^{2}\times D^{2}$ over $D^{2}$, see Section 3, Theorem 6.

In Section 4, we show that given an achiral Lefschetz fibration $\pi:X\to
D^{2}$ on a $4$–manifold $X$ with regular fiber, a closed, connected,
orientable surface $\Sigma_{g}$ of genus $g\geq 1$ such that the vanishing
cycle corresponding to each achiral singularity is a non-separating simple
closed curve on $\Sigma_{g}$, then there exists a Lefschetz fibration
embedding of $X$ into a trivial Lefschetz fibration
$\tilde{\pi}:\mathbb{C}P^{2}\times D^{2}\to D^{2}$, see Theorem 13.

Also, we show that given an achiral Lefschetz fibration $\pi:X\to D^{2}$ on a
$4$–manifold $X$ with regular fiber a closed, connected, orientable surface
$\Sigma_{g}$ of genus $g\geq 1$, there exists a Lefschetz fibration embedding
of $X$ into a trivial Lefschetz fibration
$\tilde{\pi}:(\mathbb{C}P^{2}\\#\mathbb{C}P^{2})\times D^{2}\to D^{2}$ as well
as into a trivial Lefschetz fibration
$\tilde{\pi}:(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}})\times D^{2}\to
D^{2}$, see Theorem 15.

We also discuss similar results for simplified broken achiral Lefschetz
fibration and bordered Lefschetz fibrations, refer Sections 5 and 6, see
Theorems 20 and 27. Also, in Section 6, we provide a huge collection of
bordered Lefschetz fibration which admit bordered Lefschetz fibration
embeddings into a trivial Lefschetz fibration $\tilde{\pi}:D^{4}\times
D^{2}\to D^{2},$ see Theorem 29 and 32.

Our results can be easily extended to the closed, connected, oriented
$4$–manifolds $X$ with Lefschetz fibration $\pi:X\to\mathbb{C}P^{1}$, see
Subsection 4.1, Theorem 16. Moreover, if $\pi:X^{4}\to\mathbb{C}P^{1}$ is a
simplified broken achiral Lefschetz fibration, where $X$ is a closed connected
orientable $4$–manifold such that each regular fiber in the higher genus piece
is a closed, connected, orientable genus $g>2$ surface $\Sigma_{g}$ and if the
vanishing cycle corresponding to each achiral singularity is non-separating on
$\Sigma_{g}$, then there exists a Lefschetz fibration embedding of
$\pi:X^{4}\to\mathbb{C}P^{1}$ into a trivial Lefschetz fibration
$\tilde{\pi}:\mathbb{C}P^{2}\times\mathbb{C}P^{1}\to\mathbb{C}P^{1}$, see
Section 5, Theorem 23.

It is known that every closed, connected, oriented $4$–manifold admits a
simplified broken Lefschetz fibration over $\mathbb{C}P^{1}$. From this, it
follows that every closed, connected, orientable $4$–manifold admits a smooth
embedding in $\mathbb{C}P^{2}\times\mathbb{C}P^{1}$, see Section 5, Corollary
24.

We also show that every closed, connected, orientable $4$-manifold $X$ admits
a smooth embedding into $S^{4}\times S^{2}$ as well as into
$S^{4}\tilde{\times}S^{2}$, see Section 6, Theorem 36. From this, we get
another proof of a theorem of Hirsch which states that every closed,
connected, orientable $4$–manifold smoothly embeds in $\mathbb{R}^{7}$, see
Corollary 37.

Finally in Section 7, we have Lefschetz fibration embedding results for non-
orientable manifolds $X$ over $D^{2}$ such that the handle decomposition of
$X$ does not admit $3$– and $4$–handles, see Theorems 39 and 40.

In [5], Ghanwat and Pancholi have proved that every closed, oriented smooth
$4$–manifold admits a smooth embedding in $\mathbb{C}P^{3}.$

##  2\. Preliminaries

In this section, we will recall the necessary notions for this article.

###  2.1. Mapping class group

######  Definition 1.

(The mapping class group of a manifold) The mapping class group of a manifold
$M$ is the group of isotopy classes of diffeomorphisms of $M$, where the group
operation is induced by the composition of diffeomorphisms. When $M$ is an
orientable manifold, we consider the diffeomorphisms and the isotopies
preserving the orientation of $M$. In case if $M$ has non-empty boundary
$\partial M$, we assume that all the diffeomorphisms and the isotopies are the
identity in a collar neighborhood of the boundary. We will denote the mapping
class group of $M$ by $\mathcal{M}CG(M)$.

Lickorish in [12] proved that every element of the mapping class group of a
closed orientable surface is a product of isotopy classes of the Dehn twists
along the simple closed curves on the surface. Let us recall the definition of
a Dehn twist.

######  Definition 2.

(Dehn twist)

A Dehn twist along a circle $S^{1}\times\\{\frac{3}{2}\\}$ in
$S^{1}\times[0,3]$ is a diffeomorphism of $S^{1}\times[0,3]$ given by
$(e^{i\theta},t)\to(e^{i(\theta\pm 2\pi f(t))},t),$ where $f:[0,3]\to[0,3]$ is
a smooth function that is identically $0$ on the interval $[0,1],$ on the
interval $[1,2]$ it increases monotonically from $0$ to $1$ and $f$ is
identically $1$ on the interval $[2,3]$. Note that the Dehn twist fixes a
collar of both the boundary components of $S^{1}\times[0,3]$, see Figure 1.
Hence, given an embedded circle $c$ in a surface $\Sigma,$ we can define the
Dehn twist of an annular neighborhood $N(c)=S^{1}\times[0,3]$ of
$c=S^{1}\times\\{\frac{3}{2}\\}$ in $\Sigma$ which is the identity when
restricted to the boundary of this annular neighborhood. Clearly, we can
extend this diffeomorphism by the identity in the complement of the annular
neighborhood to produce a diffeomorphism of $\Sigma$. This diffeomorphism is
called a Dehn twist $d_{c}$ along the embedded circle $c$ in the surface
$\Sigma$.

Figure 1. A Dehn twist along the centre curve $c$ of an annulus.

Thus, the right (respectively left) handed Dehn twist $d_{c}$ on $\Sigma$ is
given by removing an annular neighbourhood of $c$ and regluing it after giving
a full $2\pi$ twist to the right (respectively left). The right handed Dehn
twist $d_{c}$ is determined up to the isotopy class of $c$ and is independent
of the orientation on $c.$

Lickorish [12], proved that the mapping class group of a closed orientable
surface is generated by the Dehn twists along non-separating simple closed
curves on the surface as shown in Figure 2. For a compact orientable surface
with non-empty boundary the mapping class group is generated by the curves
given in Figure 3. We call the curves shown in Figure 2 and 3, the _Lickorish
generating curves_. In [10], Humphries provided a minimal set of generators
for mapping class group of orientable surfaces.

Figure 2.  The orientable closed surface of genus $g$ with the Lickorish
generating curves. Figure 3.  The orientable surface of genus $g$ with $n$
boundary components along with the Lickorish generating curves. Here,
$\sigma_{ij}$ is a curve which is a boundary component of a planar subsurface
of $F$ whose boundary components are precisely $\sigma_{i},\sigma_{j}$ and
$\sigma_{ij}$. Figure 4. The Szepietowski generating set for the mapping class
group of a closed non-orientable genus $g$ surface.

Lickorish in [13], also showed that to generate the mapping class group of a
non-orientable surface, the Dehn twists are not enough, in addition, we
require homeomorphisms which are known as $Y$–homeomorphisms. Consider a
Möbius band $M_{D}$ with interior $\mathring{D}$ of a disc $D$ removed. We
attach a Möbius band $M$ along $\partial D$ to get a Klein bottle $K$ with a
hole, which we call a $Y$–space. Now, move $M$ once along the core of $M_{D}$
to get a homeomorphism called the $Y$–homeomorphism of $K$ which fixes the
boundary of $K$. We can consider an embedding of $K$ in a non-orientable
surface $N$ and extend this homeomorphism by the identity on $N\setminus K$,
to a $Y$–homeomorphism of $N$ about $K$.

Recall that a closed non-orientable surface is a connected sum of the real
projective planes, $\mathbb{R}P^{2}$’s. The number of $\mathbb{R}P^{2}$’s
occurring in the connected sum is called the _genus_ of the non-orientable
surface.

In [8], Hirose showed that Szepietowski’s system of generators, given in [18],
for the mapping class group of a non-orientable surface is a minimal
generating set by Dehn twists and a $Y$–homeomorphism, see Figure 4.

###  2.2. Lefschetz fibration

Now, we discuss the notion of Lefschetz fibration.

######  Definition 3.

(Lefschetz fibration on $2n$–manifold) Let $X$ be a compact, connected,
orientable smooth $2n$–manifold with non-empty boundary. A proper smooth map
$\pi:X\to D^{2}$ is said to be a Lefschetz fibration if

  1. (1)

the map $\pi$ has finitely many critical (or singular) points $q_{i},$ all lie
in the interior of $X$ with critical (or singular) values $f(q_{i})=p_{i}$,
$1\leq i\leq n$ and $p_{i}\neq p_{j}$, for $i\neq j$,

  2. (2)

for each critical point $q_{i}$ and its corresponding critical value $p_{i}$,
there exist coordinate charts $U_{q_{i}}=\mathbb{C}^{n}$ around
$q_{i}=(0,...,0)$ and $V_{p_{i}}=\mathbb{C}$ around $p_{i}=0$ in $X$ and
$D^{2}$, respectively which agree with the orientations of $X$ and $D^{2}$,
such that the restriction of $\pi$ on $U_{q_{i}}$ is the map $\pi:U_{q_{i}}\to
V_{p_{i}}$ defined as $\pi(z_{1},...z_{2})=z_{1}^{2}+z_{2}^{2}+...+z_{n}^{2}.$
Such a singularity is called a Lefschetz singularity.

  3. (3)

for a regular value $p$, the fiber is a closed, connected, oriented co-
dimension $2$–submanifold of $X^{2n}.$

We call the fiber $\pi^{-1}(p_{i})$ corresponding to a singular point $p_{i}$
a singular fiber and the fiber $\pi^{-1}(p)$ corresponding to a regular point
$p$ a regular fiber.

###  2.3. Topology of Lefschetz fibration

Consider a Lefschetz fibration $\pi:X^{4}\to D^{2}$ on a $4$–manifold $X$ with
non-empty boundary. Suppose that $q_{1},q_{2},...,q_{n}$ and
$p_{1},p_{2},...,p_{n}$ are the critical points and the critical values of
$\pi$ respectively, such that $\pi(q_{i})=p_{i}$. Now, we have a corresponding
handle decomposition of $X$ as follows: Fix a regular value $p$ in the
interior of $D^{2}$ and an identification of the fiber $\pi^{-1}(p)$ with a
compact surface $\Sigma_{g}$ possibly with non-empty boundary. Consider an arc
$\alpha_{i}$ in the interior of $D^{2}$ connecting $p$ to the singular value
$p_{i}$ for each $i$ such that $\alpha_{i}\cap\alpha_{j}=\\{p\\},\ i\neq j.$
The labels $p_{i}$’s for the singular values and the arcs $\alpha_{i}$’s can
be chosen in a cyclic order with respect to a small circle around $p$ with the
anticlockwise orientation on the circle.

For each critical value $p_{i}$ and the regular value $p$, we choose small
open disc $D_{p_{i}}$ centered at $p_{i}$ and $D_{p}$ centered at $p$
respectively, such that the discs $D_{p_{i}}$, $D_{p}$ are disjoint from each
other in $D^{2}$, see Figure 5. In the complement of the critical values, the
map

$\pi|_{X\setminus(\cup_{i=1}^{n}\pi^{-1}(p_{i}))}:X\setminus(\cup_{i=1}^{n}\pi^{-1}(p_{i}))\to
D^{2}\setminus\\{p_{1},p_{2},...,p_{n}\\}$

is a locally trivial fiber bundle. Note that
$\pi^{-1}(D_{p})=D_{p}\times\Sigma_{g}$ as $D_{p}$ is contractible. Let
$\nu(\alpha_{i})$ be a regular neighbourhood of the arc $\alpha_{i}.$ Then,
$\pi^{-1}(D_{p}\cup\nu(\alpha_{1})\cup D_{p_{1}})$ is diffeomorphic to
$D^{2}\times\Sigma_{g}$ with a $2$–handle $H_{p_{1}}$ attached along a closed
curve $\gamma_{1}$ in a fiber $\\{pt\\}\times\Sigma_{g}\subset\partial
D^{2}\times\Sigma_{g}$ with the framing $-1$ with respect to the surface
framing. We describe the handle attachment as follows:

Choose the coordinate charts $U_{q_{1}}=\mathbb{C}\times\mathbb{C}$ around
$q_{1}=(0,0)$ and $V_{p_{i}}=\mathbb{C}$ around $p_{1}$ in $X$ and $D^{2}$
respectively, such that the restriction of $\pi$ on $U_{q_{1}}$ is the maps
$\pi(z_{1},z_{2})=z_{1}^{2}+z_{2}^{2}.$ Hence, near the critical point
$q_{1}$, the regular fiber is given by $z_{1}^{2}+z_{2}^{2}=t$, and after
multiplying $\pi$ by a unit complex number, we can assume that $t>0.$ Consider
the plane
$\mathbb{R}^{2}=\\{(z_{1},z_{2})\in\mathbb{C}^{2}:z_{1}=x_{1}+0i,z_{2}=x_{2}+0i\\}$
in $\mathbb{C}^{2}$. The intersection of the fibers with this plane are the
circles $x_{1}^{2}+x_{2}^{2}=t$ in $\mathbb{R}^{2}$. We assume $\partial
D_{1}=\Sigma_{g}^{1}\cap\mathbb{R}^{2}$ is the curve $\gamma_{1}$. Note that
each circle $x_{1}^{2}+x_{2}^{2}=t$ bounds a disc $D_{t}\subset\mathbb{R}^{2}$
and as $t$ approaches to $0$, the disc $D_{t}$ shrinks to a point in
$\mathbb{R}^{2}$. Note that $\partial D_{t}=\Sigma_{g}^{t}\cap\mathbb{R}^{2}$
and the singular fiber $\Sigma_{g}^{0}$ being created from $\Sigma_{g}^{t}$ by
the collapse of $D_{t}.$ We call the circle $\partial D_{1}=\gamma_{1}$ _the
vanishing cycle_ of the critical point $q_{1}.$ A regular neighbourhood
$\nu(\Sigma_{g}^{0})$ of the singular fiber is obtained from the neighbourhood
$\Sigma_{g}^{1}$ by adding a regular neighbourhood of $D_{1}.$ The later
neighbourhood is a $2$–handle $H_{p_{1}}$ attached to $\nu(\Sigma_{g}^{1})$.
For more details regarding framing of the handle attachments refer [6] and
[15].

Note that if $\Sigma_{g}$ is closed, $\pi^{-1}(D_{p}\cup\nu(\alpha_{1})\cup
D_{p_{1}})$ is diffeomorphic to $D^{2}\times\Sigma_{g}\cup H_{p_{1}}$ and
$\partial(\pi^{-1}(D_{p}\cup\nu(\alpha_{1})\cup
D_{p_{1}}))=\partial(D^{2}\times\Sigma_{g}\cup H_{p_{1}})$ is diffeomorphic to
a $\Sigma_{g}$–bundle over $S^{1}$. If we continue in the anticlockwise
direction around $p$ and include other critical values in the similar way, we
get

$\displaystyle
X_{0}=\pi^{-1}(D_{p}\cup(\cup_{i=1}^{n}\nu(\alpha_{i}))\cup(\cup_{i=1}^{n}D_{p_{i}}))$

which is diffeomorphic to
$(D^{2}\times\Sigma_{g})\cup(\cup_{i=1}^{n}H_{p_{i}}),$ where $H_{i}$ is a
$2$–handle attached along the vanishing cycle $\gamma_{i}$ in the distinct
fibers of $\partial D^{2}\times\Sigma_{g}\to\partial D^{2}$ with the framing
$-1$ with respect to the surface framing. Also, the boundary of $X_{0}$ is

$\displaystyle\partial
X_{0}=\partial((D^{2}\times\Sigma_{g})\cup(\cup_{i=1}^{n}H_{p_{i}})),$

which is a $\Sigma_{g}$–bundle over $S^{1}$ with the monodromy the product of
right handed Dehn twist along the vanishing cycles.

Figure 5. Critical values of the Lefschetz fibration $\pi$ in $D^{2}.$

In case if $\Sigma_{g}$ has non-empty boundary, the boundary of $X_{0}$ is a
closed $3$–manifold with an open book structure on it. More precisely,

$\displaystyle\partial
X_{0}=\partial((D^{2}\times\Sigma_{g})\cup(\cup_{i=1}^{n}H_{p_{i}}))\cup
D^{2}\times\partial\Sigma_{g}.$

Recall that an _open book decomposition_ of a manifold $M$ is a pair
$(B,\pi),$ where $B$ is a co-dimension $2$ submanifold with a trivial normal
bundle in $M$ and $\pi:M\setminus B\to S^{1}$ is a locally trivial fibration
such that the fibration in a neighborhood of $B$ looks like the trivial
fibration of $(B\times D^{2})\setminus B\times\\{0\\}\to S^{1}$ sending
$(x,r,\theta)$ to $\theta$, where $x\in B$ and $(r,\theta)$ are polar
coordinates on $D^{2}$. For each $\theta\in S^{1}$, the closure of
$\pi^{-1}(\theta)$ is called a page and the monodromy of the fibration is
called the monodromy of the open book. Thus, given a Lefschetz fibration
$\pi:X^{4}\to D^{2}$ and a radial projection
$\theta:D^{2}\setminus\\{0\\}\to\partial D^{2},$ we get an open book
decomposition on $\partial X$ by the map $\theta\circ\pi:(\partial
X\setminus\pi^{-1}(0))\to S^{1}$ with binding $\pi^{-1}(0)\cap\partial X,$
where $0$ is a base point with regular fiber.

Note that any locally trivial fiber bundle over $S^{1}$ with fibers $\Sigma$,
a closed manifold is canonically isomorphic to the fiber bundle

$\frac{\Sigma\times[0,1]}{(x,0)\sim(\phi(x),1)},$

for some diffeomorphism $\phi$ of $\Sigma$. Let us recall the definition of
mapping torus:

######  Definition 4.

(Mapping torus) Let $\Sigma$ be a manifold with non-empty boundary
$\partial\Sigma$ and $\varphi$ be a diffeomorphism of $\Sigma$ which is the
identity in a collar neighborhood of $\partial\Sigma$. Then, we define the
mapping torus $\mathcal{MT}(\Sigma,\partial\Sigma,\varphi)$ of $\Sigma$ as

$\Sigma\times[0,1]/\sim,$

where $\sim$ is the equivalence relation identifying $(x,0)$ with
$(\varphi(x),1)$.

##  3\. Co-dimension $2$ Lefschetz fibration embeddings

In this section, we will discuss Lefschetz fibration embeddings of
$4$–dimensional Lefschetz fibrations into a $6$–dimensional Lefschetz
fibration. First, recall the definition of the Lefschetz fibration embedding.

######  Definition 5.

Let $\pi_{1}:X\to D^{2}$ and $\pi_{2}:Y\to D^{2}$ be two Lefschetz fibration
on $X$ and $Y$ respectively. A proper embedding $f:X\to Y$ is said to be a
Lefschetz fibration embedding if $\pi_{2}\circ f=\pi_{1}.$

######  Theorem 6.

Let $\pi:X\to D^{2}$ be a Lefschetz fibration, where $X$ is a compact,
connected, orientable $4$–manifold with non-empty boundary such that for each
regular value, the fiber is a closed, connected, orientable surface of genus
$g\geq 0$. Then, there exists a Lefschetz fibration embedding of $X$ into a
trivial Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times D^{2}\to
D^{2}.$

The key idea in establishing this result is flexible embeddings of surfaces
$\Sigma_{g}$ in $\mathbb{C}P^{2}.$ The notion of flexibility of surfaces in
$4$-dimensional manifolds is studied in [7]. The idea of flexible embeddings
is used to produce open book embeddings of closed $3$–manifolds in
$5$–manifolds for orientable case in [16] and for non-orientable case in [3].
In [17], the flexibility of surfaces is used to study spinal open book
embeddings of closed orientable $3$-manifolds in $5$–manifolds. In [5],
flexibility is used to obtain Lefschetz fibration embedding of closed oriented
$4$–dimensional Lefschetz fibration over $\mathbb{C}P^{1}$ into a trivial
Lefschetz fibration $\mathbb{C}P^{2}\times\mathbb{C}P^{1}$ over
$\mathbb{C}P^{1}$, where fibration is given by the projection on
$\mathbb{C}P^{1}$. They also establish, the embeddings of closed orientable
$4$–manifolds in $\mathbb{C}P^{3}$ as well as in $\mathbb{R}^{7}.$

###  3.1. Flexibility of surfaces in $\mathbb{C}P^{2}$

######  Definition 7.

(Flexible mapping class) We call a diffeomorphism $\phi$ of a surface $F$ a
flexible diffeomorphism in a $4$–manifold $X$ if there exists an embedding $f$
of $F$ in $X$ and a diffeomorphism $\Phi$ of $X$ isotopic to the identity map
of $X$ such that $\phi=f^{-1}\circ\Phi\circ f$.

######  Definition 8.

An embedding of a surface $\Sigma$ in $X$ is said to be flexible if every
diffeomorphism of $\Sigma$ is flexible in $X$ with respect to $f$.

We recall the following result from [7] (also in [16]). For the sake of
completeness, we will give this construction here again as we use it
concretely.

Figure 6. The right and left twisted bands.

######  Lemma 9.

Let $\Sigma_{g}$ be a closed, connected, orientable genus of $g$ surface.
Then, there exists a flexible embedding of $\Sigma_{g}$ into
$\mathbb{C}P^{2}.$

###### Proof.

Recall that every diffeomorphism of the surface $\Sigma_{g}$ is isotopic to a
product of the Dehn twists along the Lickorish generating curves as described
in Figure 2. So, in order to establish the lemma, it is enough to show that
there exists an embedding of $\Sigma_{g}$ in $\mathbb{C}P^{2}$ such that each
Dehn twist along the Lickorish generating curve is flexible in
$\mathbb{C}P^{2}$.

Each Dehn twist $d_{c}$ along a simple closed curve $c$ in $\Sigma_{g}$ is
supported in an annular neighbourhood of $c$ in $\Sigma_{g}$. In [7] (Also in
[3] and [16] ), it is shown that there exists a flexible embedding of an
annulus $A=S^{1}\times[0,1]$ in $D^{4}$ and hence in $\mathbb{C}P^{2}$ as
$\mathbb{C}P^{2}$ has handle decomposition with a $0-$handle, a $2-$handle and
a $4-$handle. The proof of this is based on the fact that the $3$–sphere
$S^{3}$ admits an open book decomposition with pages a Hopf annulus and
monodromy a Dehn twist along the central curve of the Hopf annulus. By a Hopf
annulus, we mean an embedding of an annulus into $S^{3}$ such that the
boundary of the embedded annulus is a Hopf link, i.e., the linking number of
the boundary components of the embedded annulus is $\pm 1$. This implies that
there exists a flow $\phi_{t}$ on $S^{3}$ whose time $1$ map $\phi_{1}$
induces the Dehn twist along the central curve on each page–which is a Hopf
annulus– of the open book of $S^{3}$. We can choose any Hopf annulus page
$\mathcal{A}$ as an embedding of the annulus $A$ in $S^{3}$.

Using the flow $\phi_{s}$ on $S^{3}$, we can define a level preserving
diffeomorphism $\Psi:S^{3}\times[0,2]\to S^{3}\times[0,2]$ such that $\Psi$ is
the identity near both the boundary components of $S^{3}\times[0,2]$ and
$\Psi$ induces $\phi_{1}$ on $S^{3}\times 1$, i.e. it will induce the Dehn
twist along the central curve in the embedded annulus $A$ which is a positive
Hopf annulus page $\mathcal{A}$ of the open book decomposition of $S^{3}\times
1$. By considering $S^{3}\times[0,2]$ as a collar of the boundary $\partial
D^{4}=S^{3}\times 0$ of $D^{4}$, we can extend the diffeomorphism $\Psi$ to a
diffeomorphism of $D^{4}$ which is isotopic to the identity map of $D^{4}$ and
the identity near the boundary. Hence, it can be extended to a diffeomorphism
of $\mathbb{C}P^{2}$ which is isotopic to the identity map of
$\mathbb{C}P^{2}$. In fact, it can be chosen to be the identity on the
$2$-handle of $\mathbb{C}P^{2}$ attached to $D^{4}$ along an unknot with
framing $+1$ in the boundary of $D^{4}$ as well as the $4-$handle of
$\mathbb{C}P^{2}$. Thus, the Dehn twist along the central curve of the annulus
$A$ is flexible in $\mathbb{C}P^{2}$. This implies that this embedding of the
annulus $A$ in $S^{3}\times 1$ is a flexible embedding of $A$ in
$\mathbb{C}P^{2}$ as every diffeomorphism of $A$ is isotopic to a power of the
Dehn twist along the central curve of $A$.

In the light of above discussion, we construct an embedding of $\Sigma_{g}$ in
$\mathbb{C}P^{2}$ such that each Lickorish generating curve on $\Sigma_{g}$
admits a Hopf annulus regular neighbourhood in the embedded $\Sigma_{g}$ so
that each Dehn twist along the Lickorish generating curve is flexible in
$\mathbb{C}P^{2}$ and hence the embedding of $\Sigma_{g}$ is flexible in
$\mathbb{C}P^{2}$.

Figure 7. A Flexible embedding of $\Sigma_{g}$ in $\mathbb{C}P^{2}$ with an
isotopy of the curve $\gamma_{2}$ which has a positive Hopf annulus regular
neighbourhood

To construct such an embedding, we shall proceed as follows: Consider a collar
neighbourhood $S^{3}\times[0,2]$ of the boundary $\partial D^{4}=S^{3}\times
0$ in a $4-$ball $D^{4}$. Consider a standardly embedded closed orientable
surface $S$ of genus $g$ which bounds an unknotted untwisted handlebody of
genus $g$ – see Figure 2 – in $S^{3}\times 1$. Remove the interior of a disc
$D$ from $S$ and attach a positive full-twisted band to the $\partial D$ in
$S^{3}\times 1$ refer Figure 7. Thus, we get a new compact connected
orientable surface $S_{1}$ of genus $g$ with boundaries $l_{1}$ and $l_{2}$
such that $l_{1}\cup l_{2}$ is a positive Hopf link in $S^{3}\times 1$ refer
Figure 7. We attach tubes $l_{1}\times[0,1]$ and $l_{2}\times[0,1]$ to get a
proper embedding of $S_{1}$ in $D^{4}$. Now, we attach a $2$–handle $H$ to
$D^{4}$ along the boundary component $l_{2}$ of $S_{1}$ with framing $+1$ and
then attach a $4$–handle on the resulting boundary to obtain
$\mathbb{C}P^{2}$. The component $l_{2}$ of the surface $S_{1}$ bounds a
$2$–disc $D^{\prime}$ in the $2$–handle $H$, namely the core of the $2$–handle
and the boundary component $l_{1}$ of $S_{1}$ bounds a disc $D^{\prime\prime}$
in the $4-$handle of $\mathbb{C}P^{2}$. Thus, we get an embedding of the
surface $\Sigma_{g}=S_{1}\cup D^{\prime}\cup D^{\prime\prime}$ in
$\mathbb{C}P^{2}$ such that $\Sigma_{g}$ contains an embedded positive Hopf
annulus $A$ with boundary components $l_{1}$ and $l_{2}$, refer Figure 7.

Now, we can use this disc $D^{\prime}$ (or $D^{\prime\prime}$) to isotope each
Lickorish generating curve $c$ to another curve $c^{\prime}$ in $\Sigma_{g}$
so that $c^{\prime}$ travels over the Hopf annulus $A$ exactly once and it
admits a Hopf annulus regular neighbourhood of $\Sigma_{g}$ in $S^{3}\times 1$
of the collar of the boundary of the $0$–handle $D^{4}$ of $\mathbb{C}P^{2}$.
In order to get the isotopy of the flexibility of $c^{\prime}$, we push the
neighborhood of $c^{\prime}$ slightly towards $S^{3}\times 2$ in such a way
that at a fixed level between $1$ and $s_{0}$ the intersection of this pushed
regular neighborhood $\nu(c^{\prime})$ is a positive Hopf link and regular
neighborhood $\nu(c^{\prime})\cap S^{3}\times s_{0}$ is a positive Hopf
annulus in $S^{3}\times s_{0},\ s_{0}\in(1,2)$ containing the curve
$c^{\prime}$ as its central curve. We now perform an isotopy to induce a Dehn
twist along the pushed $c^{\prime}$ in such a way that this isotopy is
supported in a small neighborhood of $S^{3}\times s_{0}$ not intersecting
$S^{3}\times 1.$ After performing this isotopy, we further isotope the pushed
neighborhood $\nu(c^{\prime})$ back to its original place in $S^{3}\times 1$.

From this, one can then easily observe that each Dehn twist $d_{c^{\prime}}$
is flexible in $\mathbb{C}P^{2}.$ Moreover, the diffeomorphism of
$\mathbb{C}P^{2}$ which induces the Dehn twist $d_{c^{\prime}}$ on
$\Sigma_{g}$ can be chosen so that it is isotopic to the identity map of
$\mathbb{C}P^{2}$ and the identity on the $2-$ and $4-$handle of
$\mathbb{C}P^{2}$. Thus, we obtain a flexible embedding of $\Sigma_{g}$ in
$\mathbb{C}P^{2}$ as required. For more details, refer [16] and [17].

∎

Observe that we have constructed an embedding $f$ of $\Sigma_{g}$ in
$\mathbb{C}P^{2}$ such that for every Lickorish generating curve $\gamma$, the
curve $f(\gamma)$ can be isotoped in $f(\Sigma_{g})$ to a curve
$\gamma^{\prime}$ such that

  1. (1)

the curve $\gamma^{\prime}$ admits a regular neighbourhood $\nu(\gamma)$ in
$f(\Sigma_{g})$ which is positive Hopf annulus in $S^{3}\times 1$,

  2. (2)

$\gamma^{\prime}$ bounds the disc $D$ in $\mathbb{C}P^{2}$ which intersects
$f(\Sigma_{g})$ only in $\gamma^{\prime}$,

  3. (3)

a tubular neighbourhood $\mathcal{N}(D)=D^{2}\times D^{2}$ intersects
$f(\Sigma_{g})$ in the regular neighbourhood $\nu(\gamma^{\prime})$ of
$\gamma^{\prime}$ in $f(\Sigma_{g})$,

  4. (4)

we can choose coordinates $(z_{1},z_{2})$ on $\mathcal{N}(D)$ such that
$\nu(\gamma^{\prime})=g^{-1}(1)\cap f(\Sigma_{g})$, where
$g:\mathbb{C}^{2}\to\mathbb{C}$ is the map defined by
$g(z_{1},z_{2})=z_{1}^{2}+z_{2}^{2}$ and the plane
$\mathbb{R}^{2}=\\{(z_{1},z_{2})\in\mathbb{C}^{2}:z_{1}=x_{1}+i0,z_{2}=x_{2}+i0\\}$
in $\mathbb{C}^{2}$ intersects $\mathcal{N}(D)$ in the disc $D.$ Note that
$\gamma^{\prime}$ is the circle $x_{1}^{2}+x_{2}^{2}=1$ and the disc $D$ is
$x_{1}^{2}+x_{2}^{2}\leq 1$ in $\mathbb{R}^{2}.$

Any simple closed curve $\gamma$ in $\Sigma_{g}$ is said to be in _the
standard position_ with respect to the embedding $f_{pr}$ if $f_{pr}(\gamma)$
can be isotoped to a curve $\gamma^{\prime}$ in $f_{pr}(\Sigma_{g})\cap
S^{3}\times 1$ such that $\gamma^{\prime}$ admits a regular neighbourhood
$\nu(\gamma^{\prime})$ in $f_{pr}(\Sigma_{g})$ which is a positive Hopf
annulus in $S^{3}\times 1$. Any such curve will satisfy the conditions
$(1),(2),(3)$ and $(4)$ satisfied by the Lickorish generating curves as
described above.

We call the flexible embedding of $\Sigma_{g}$ in $\mathbb{C}P^{2}$
constructed in the proof of Lemma 9, _the preferred embedding_ of $\Sigma_{g}$
in $\mathbb{C}P^{2}$. We denote this embedding by
$f_{pr}:\Sigma_{g}\to\mathbb{C}P^{2}$ and the embedded surface $\Sigma_{g}$ by
$\Sigma_{pr}$. We call the pair $(\mathbb{C}P^{2},\Sigma_{pr})$ _the preferred
$\mathbb{C}P^{2}$_ and we denote it by $\mathbb{C}P_{pr}^{2}.$

######  Remark 10.

Note that the simple closed curves shown by red color in Figure 8, are
separating curves and they are in the standard position with respect to
$f_{pr}$.

######  Remark 11.

  1. (1)

The monodromy of the singular fibration
$g:\mathcal{N}(D)=\mathbb{C}^{2}\to\mathbb{C}=D^{2}$ with the singular point
$(0,0)$ is the positive Dehn twist $d_{\gamma^{\prime}}$ along the curve
$\gamma^{\prime}$ which lies in the regular fiber
$g^{-1}(1)\cap\Sigma_{pr}=\nu(\gamma^{\prime}).$

  2. (2)

The isotopy of $\mathcal{N}(D)$ which produces the positive Dehn twist
$d_{\gamma^{\prime}}$ on $g^{-1}(1)\cap\Sigma_{pr}$ can be extended to an
isotopy $\Psi_{s}$, $s\in[0,1]$, of $\mathbb{C}P^{2}$ such that the isotopy is
supported in the tubular neighbourhood $\mathcal{N}(D)$. Then, we have
$f_{pr}^{-1}\circ\Psi_{1}\circ f_{pr}=d_{\gamma}$ up to isotopy. Thus, we have
isotopy of $\mathbb{C}P^{2}$ supported in the tubular neighbourhood
$\mathcal{N}(D)$ of $D$ which produces Dehn twist $d_{\gamma^{\prime}}$ on
$\Sigma_{pr}.$

###  3.2. Proof of Theorem 6 (Lefschetz fibration embedding)

Consider a Lefschetz fibration $\pi:X^{4}\to D^{2}$ on a compact, connected,
oriented $4$–manifold $X$ with non-empty boundary. Suppose that
$q_{1},q_{2},...,q_{n}$ and $p_{1},p_{2},...,p_{n}$ are the critical points
and the critical values of $\pi$ respectively, such that $\pi(q_{i})=p_{i}$.
Fix a regular value $p$ in the interior of $D^{2}$ and an identification of a
fiber $\pi^{-1}(p)$ with a closed surface $\Sigma_{g}$. For each $i=1,2..,n$,
Choose a small open disc $D_{p_{i}}$ centered at the critical value $p_{i}$
and a disc $D_{p}$ centered at $p$ in $D^{2}$ such that these discs are
disjoint from each other. Consider an arc $\alpha_{i}$ in the interior of
$D^{2}$ connecting $p$ to a point $p_{i}^{\prime}$ on $\partial D_{p_{i}}$ for
each $i$ such that $\alpha_{i}$’s meet only at the point $p$. The labels
$p_{i}$’s for the singular values and the arcs $\alpha_{i}$’s can be chosen in
a cyclic order with respect to a small circle around $p$ with the
anticlockwise orientation on the circle. Note that

$D_{p}\cup(\cup_{i=1}^{n}\nu(\alpha_{i}))\cup(\cup_{i=1}^{n}D_{p_{i}})$

is a $2$-disc. Let $\gamma_{i}$ be the vanishing cycle corresponding to the
critical value $p_{i}$ on $\Sigma_{g}$. Note that there is a unique lift of
$\gamma_{i}$ in the fiber on $p_{i}^{\prime}$.

Firstly, we will construct Lefschetz fibration embedding of the Lefschetz
fibration of $X$ restricted to the Lefschetz fibration over $D_{p_{i}}$ into
the trivial Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times
D_{p_{i}}\to D_{p_{i}}$ for each singular value $p_{i}$. Then, we will
construct Lefschetz fibration embedding of the Lefschetz fibration of $X$
restricted to the Lefschetz fibration over
$\tilde{D}=D_{p}\cup(\cup_{i=1}^{n}\nu(\alpha_{i}))$ into the trivial
Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times\tilde{D}\to\tilde{D}$.
Finally, we will glue these Lefschetz fibration embeddings to get the required
Lefschetz fibration embedding of $\pi:X\to D^{2}$ into the trivial Lefschetz
fibration $\tilde{\pi}:\mathbb{C}P^{2}\times D^{2}\to D^{2}.$

Case -$(I)$: When the vanishing cycle $\gamma_{i}$ is in the standard position
with respect to the preferred embedding of $\Sigma_{g}$ :

As the vanishing cycle $\gamma_{i}$ on $\Sigma_{g}$ is in the standard
position with respect to $f_{pr}$, $f_{pr}(\gamma_{i})$ can be isotoped to
$\gamma_{i}^{\prime}$ in $\Sigma_{pr}$ such that the regular neighbourhood of
$\gamma_{i}^{\prime}$ in $\Sigma_{pr}\cap S^{3}\times 1$ is a positive Hopf
annulus $S^{3}\times 1$. The curve $\gamma_{i}^{\prime}$ bounds a disc $D^{i}$
in $\mathbb{C}P^{2}$ such that $\partial D^{i}=\gamma_{i}^{\prime}$ and
$D^{i}$ intersects $\Sigma_{pr}$ only in $\gamma_{i}^{\prime}$. Also, we have
a tubular neighbourhood $\mathcal{N}(D^{i})=D^{2}\times D^{2}$ intersects
$\Sigma_{pr}$ in a regular neighbourhood $\nu(\gamma_{i}^{\prime})$ of
$\gamma_{i}^{\prime}$ in $\Sigma_{pr}$. The tubular neighbourhood
$\mathcal{N}(D^{i})$ admits the coordinates $(z_{1},z_{2})$ on
$\mathcal{N}(D)$ such that $\nu(\gamma^{\prime})=g^{-1}(1)\cap\Sigma_{pr}$,
where $g:\mathbb{C}^{2}\to\mathbb{C}$ is a map defined by
$g(z_{1},z_{2})=z_{1}^{2}+z_{2}^{2}$ and the plane
$\mathbb{R}^{2}=\\{(z_{1},z_{2})\in\mathbb{C}^{2}:z_{1}=x_{1}+i0,z_{2}=x_{2}+i0\\}$
in $\mathbb{C}^{2}$ intersect $\Sigma_{pr}$ in the disc
$D^{i}=\\{(x_{1},0,x_{2},0):x_{1}^{2}+x_{2}^{2}\leq
1\\}\subset\mathbb{R}^{2}$.

Recall that the Dehn twist $d_{\gamma_{i}^{\prime}}$ is flexible in
$\mathbb{C}P^{2}$. Therefore, as described in Remark 11, there exists an
isotopy $\Psi_{t}^{i}$: $\mathbb{C}P^{2}\to\mathbb{C}P^{2}$ such that
$f_{pr}^{-1}\circ\Psi_{1}^{i}\circ f_{pr}=d_{\gamma_{i}}$ (up to isotopy) and
this isotopy is supported in the tubular neighbourhood of $D^{i}$. In other
words, $\Psi_{1}^{i}$ induces the Dehn twist $d_{\gamma_{i}^{\prime}}$ on
$\Sigma_{pr}$.

The map $\pi_{|_{\pi^{-1}(D_{p_{i}})}}:\pi^{-1}(D_{p_{i}})\to D_{p_{i}}$ is a
Lefschetz fibration with $p_{i}$ is the only critical value. We shall
construct a Lefschetz fibration embedding of the Lefschetz fibration
$\pi_{|_{\pi^{-1}(D_{p_{i}})}}:\pi^{-1}(D_{p_{i}})\to D_{p_{i}}$ into a
trivial Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times D_{p_{i}}\to
D_{p_{i}}$.

We identify the disc $D_{p_{i}}$ as well as the disc $D^{i}$ as $\\{te^{2\pi
is}|s,t\in[0,1]\\}$. The point $p_{i}^{\prime}$ is $(0,1)=1.e^{2\pi
i0}\in\partial D^{i}$. For each $(s,t)\in[0,1]\times(0,1]$, we shall define a
flexible embedding $f^{i}_{s,t}$ of $\Sigma_{g}$ in $\mathbb{C}P^{2}$ as
follows: We shall denote the image of $\Sigma_{g}$ under the embedding
$f^{i}_{s,t}$ by $\Sigma^{i}_{s,t}$.

  * •

We consider $f^{i}_{0,1}$ as the preferred embedding $f_{pr}$. Note that
$\Sigma_{0,1}^{i}=\Sigma_{pr}.$

  * •

We consider a smooth family $f^{i}_{0,t}$, $t\in(0,1]$ of the flexible
embeddings of $\Sigma_{g}$ in $\mathbb{C}P^{2}$ such that

    1. (1)

there exists an isotopy $\\{\Psi^{i}_{0,t}\\}_{t\in(0,1]}$ of
$\mathbb{C}P^{2}$ supported in $\mathcal{N}(D^{i})=\mathcal{N}(D^{i}_{0,1})$
such that the embedding $f^{i}_{0,t}=\Psi^{i}_{0,t}\circ f_{pr}$,

    2. (2)

the simple closed curve $\gamma^{i}_{0,t}=f^{i}_{0,t}(\gamma_{i})$ bounds the
disc $D^{i}_{0,t}$ which is the image of the disc $D^{i}_{0,1}$ under the
isotopy . The disc $D^{i}_{0,t}$ intersects $\Sigma_{0,t}^{i}$ only in
$\gamma^{i}_{0,t}$. Here,
$\gamma^{i}_{0,t}=g^{-1}(t)\cap\mathbb{R}^{2}=\\{(x_{1},0,x_{2},0):x_{1}^{2}+x_{2}^{2}=t\\}$
and $D^{i}_{0,t}=\\{(x_{1},0,x_{2},0):x_{1}^{2}+x_{2}^{2}\leq
t\\}\subset\mathbb{R}^{2}$,

    3. (3)

we have a tubular neighbourhood
$\mathcal{N}(D^{i}_{0,t})\subset\mathcal{N}(D^{i}_{0,1})$ intersects
$\Sigma_{0,t}^{i}$ in the regular neighbourhood
$\nu(\gamma_{0,t}^{i})=g^{-1}(t)\cap\Sigma_{0,t}^{i}$ of $\gamma_{0,t}^{i}$ in
$\Sigma_{0,t}^{i}$.

  * •

Note that each embedding $f^{i}_{0,t},\ t\in(0,1]$ is flexible. For, if
$h:\Sigma\to\Sigma$ is a diffeomorphism then, there exists
$\Phi_{pr}:\mathbb{C}P^{2}\to\mathbb{C}P^{2}$ such that
$f_{pr}^{-1}\circ\Phi_{pr}\circ f_{pr}=h$ as the embedding $f_{pr}$ is a
flexible embedding in $\mathbb{C}P^{2}$. Let
$\Phi=\Psi_{0,t}^{i}\circ\Phi_{pr}\circ\Psi_{0,t}^{i^{-1}}$. Then,
${f^{i}_{0,t}}^{-1}\circ\Phi\circ f_{0,t}^{i}=h.$

  * •

We define the embedding $f^{i}_{s,t}$ as the map $\Psi^{i}_{s,t}\circ
f^{i}_{0,t}$, where $\Psi^{i}_{s,t}=\Psi^{i}_{s}\circ\Psi^{i}_{0,t}$ with the
isotopy $\Psi^{i}_{s}$ of the flexibility of $d_{\gamma_{i}}$ described above
as well as in Remark 11. Note that
$\gamma^{i}_{s,t}=\Psi^{i}_{s,t}(\gamma_{i}^{\prime})$ bounds the disc
$D^{i}_{s,t}=\Psi^{i}_{s,t}(D^{i})$ in $\mathbb{C}P^{2}$. The disc
$D^{i}_{s,t}$ intersects $\Sigma^{i}_{s,t}$ only in $\gamma^{i}_{s,t}$ and the
tubular neighbourhood
$\mathcal{N}(D^{i}_{s,t})=\Psi^{i}_{s,t}(\mathcal{N}(D^{i}))$ intersects
$\Sigma^{i}_{s,t}$ in a regular neighbourhood
$\nu(\gamma_{s,t}^{i})=g^{-1}(t)\cap\Sigma^{i}_{s,t}$ of $\gamma^{i}_{s,t}$ in
$\Sigma^{i}_{s,t}$. Note the the restriction of $\Psi^{i}_{1,t}$ to
$\Sigma^{i}_{0,t}$ induces the Dehn twist $d_{\gamma^{i}_{0,t}}$ on
$\Sigma^{i}_{0,t}$.

  * •

Note that for each $t$, $\Sigma^{i}_{0,t}=\Sigma^{i}_{1,t}$. From this, we can
observe that the mapping torus
$\mathcal{MT}(\Sigma^{i}_{0,t},d_{\gamma^{i}_{0,t}})$ is embedded fiberwise in
the mapping torus $\mathcal{MT}(\mathbb{C}P^{2},\Psi^{i}_{1,t})$. The mapping
torus $\mathcal{MT}(\mathbb{C}P^{2},\Psi^{i}_{1,t})$ is a trivial fiber bundle
over the circle $\\{te^{2\pi is},\ s\in[0,1]\\}$ with fiber $\mathbb{C}P^{2}$
containing the fiber $\Sigma^{i}_{0,t}$ of the locally trivial fiber bundle
$\mathcal{MT}(\Sigma^{i}_{0,t},d_{\gamma^{i}_{0,t}})$ over the same circle.

  * •

For each $s\in[0,1]$, if we travel through the family $\Psi^{i}_{s,1-t},\
t\in[0,1)$ of diffeomorphisms of $\mathbb{C}P^{2}$, the disc $D^{i}_{s,1}$
shrinks through the discs $D^{i}_{s,1-t}$ to the origin of $D^{i}_{0,1}$.
Thus, the embedded surface $\Sigma^{i}_{s,1}$ can be collapsed through
$\Sigma^{i}_{s,t}$ to get a singular surface $\Sigma^{i}_{0}$ which is
obtained by pinching the vanishing cycle $\gamma^{i}_{s,1}$ to the origin of
the disc $D^{i}_{0,1}$. The vanishing cycle $\gamma^{i}_{s,1}$ in
$\Sigma^{i}_{s,1}$ corresponds to the vanishing cycle
$\gamma_{i}\subset\Sigma_{g}$ of the critical value $p_{i}$ of the Lefschetz
fibration $\pi:X^{4}\to D^{2}$. The singular surface $\Sigma^{i}_{0}$ gives an
embedding of the singular fiber $\pi^{-1}(p_{i})$ in $\mathbb{C}P^{2}$.

From this, one can see that we get a Lefschetz fibration embedding of the
Lefschetz fibration $\pi_{|_{\pi^{-1}(D_{p_{i}})}}:\pi^{-1}(D_{p_{i}})\to
D_{p_{i}}$ into the trivial Lefschetz fibration
$\tilde{\pi}:\mathbb{C}P^{2}\times D_{p_{i}}\to D_{p_{i}}$.

Case -$(II)$: When the vanishing cycle $\gamma_{j}$ is not in the standard
position with respect to the preferred embedding of $\Sigma_{g}$:

Now, we consider the case where the vanishing cycle $\gamma_{j}$ in
$\Sigma_{g}$ corresponding to the critical value $p_{j}$ of the Lefschetz
fibration $\pi:X^{4}\to D^{2}$ has the property that $f_{pr}(\gamma_{j})$ can
not be isotoped in $\Sigma_{pr}$ to a closed curve which admits a positive
Hopf annulus neighbourhood in $\Sigma_{pr}$.

Recall the following fact that given any two simple closed non-separating
curves in $\Sigma_{g}$, there exists a diffeomorphism of $\Sigma_{g}$ taking
one to another. Also, given any two separating simple closed curves in
$\Sigma_{g}$, there exists a diffeomorphism of $\Sigma_{g}$ taking one to
another, provided they separate the surface $\Sigma_{g}$ into the surfaces of
the same genera.

Figure 8. The red curves drawn here are some separating curves of a closed
surface of genus $g>1$.

Hence, if $\gamma_{j}$ is non-separating, then there exists a diffeomorphism
$\phi$ of $\Sigma_{g}$ which maps $\gamma_{j}$ to one of the Lickorish
generators and if it is separating, then there exists a diffeomorphism $\phi$
of surface $\Sigma_{g}$ which maps $\gamma_{j}$ to one of the red colored
separating closed curve shown in Figure 8, provided they separate the surface
$\Sigma_{g}$ into the surfaces of the same genera, refer Remark 10. Since
$f_{pr}$ is a flexible embedding, there exists an isotopy
$\\{\Phi_{r}\\}_{r\in[0,1]}$ of diffeomorphisms of $\mathbb{C}P^{2}$ such that
$f_{pr}^{-1}\circ\Phi_{1}\circ f_{pr}=\phi.$

We consider a family of embeddings $\\{f_{r}\\}_{r\in[0,1]}$, where
$f_{r}=\Phi_{r}\circ f_{pr}$ and we denote the image of embeddings $f_{r}$ by
$\Sigma^{r}$, where $\Sigma^{0}=\Sigma_{pr}.$ Note that $f_{pr}$ and $f_{1}$
have the same image $\Sigma_{pr}$ but $f_{pr}\neq f_{1}.$ Now, the curve
$f_{1}(\gamma_{j})$ can be isotoped to a simple closed curve
$\gamma_{j}^{\prime}$ in $\Sigma^{1}$ so that $\gamma_{j}$ is in the standard
position with respect to $f_{1}.$

Now, we can construct a Lefschetz fibration embedding of the Lefschetz
fibration $\pi_{|_{\pi^{-1}(D_{p_{j}})}}:\pi^{-1}(D_{p_{j}})\to D_{p_{j}}$
into a trivial Lefschetz fibration of $\mathbb{C}P^{2}\times D_{p_{j}}$ over
$D_{p_{j}}$ by constructing smooth family of diffeomorphism $\Psi^{j}_{s,t}$
of $\mathbb{C}P^{2}$ and embeddings $f^{j}_{s,t}$ of $\Sigma_{g}$ as described
above with the properties that $\Psi^{j}_{0,1}=Id_{\mathbb{C}P^{2}}$ and
$f^{j}_{0,1}=f_{1}$ from the family $\\{f_{r}\\}_{r\in[0,1]}$ described above.

Thus, we have a Lefschetz fibration embedding of the local Lefschetz fibration
$\pi_{|_{\pi^{-1}(D_{p_{k}})}}:\pi^{-1}(D_{p_{k}})\to D_{p_{k}}$ into a
trivial Lefschetz fibration of $\mathbb{C}P^{2}\times D_{p_{k}}$ over
$D_{p_{k}}$ for each critical value $p_{k}.$

Figure 9. The region $D_{0}$ which is constructed by discs $D_{p_{i}}$, the
region $B$ and strips $S_{i},$ $R_{p_{i}}$ and $R_{p_{j}}^{\prime}$.

Hence, we have a Lefschetz fibration embedding of the Lefschetz fibration

$\pi:\pi^{-1}(\cup_{i=1}^{n}D_{p_{i}})\to\cup_{i=1}^{n}D_{p_{i}}$

into a trivial Lefschetz fibration

$\tilde{\pi}:\mathbb{C}P^{2}\times\cup_{i=1}^{n}D_{p_{i}}\to\cup_{i=1}^{n}D_{p_{i}}.$

Let $\tilde{D}=\cup_{i=1}^{n}\nu(\alpha_{i})\cup D_{p}.$ We construct a
Lefschetz fibration embedding of the the Lefschetz fibration

$\pi:\pi^{-1}(\tilde{D})\to\tilde{D}$

into a trivial Lefschetz fibration

$\tilde{\pi}:\mathbb{C}P^{2}\times\tilde{D}\to\tilde{D}.$

Then, we glue all Lefschetz fibration embeddings constructed above to get the
Lefschetz fibration embedding of the Lefschetz fibration
$\pi:\pi^{-1}(D_{0})\to D_{0}$ into the Lefschetz fibration
$\tilde{\pi}:\mathbb{C}P^{2}\times D_{0}\to D_{0}$, where
$D_{0}=\tilde{D}\cup(\cup_{i=1}^{n}D_{p_{i}}).$

Lefschetz fibration embedding over $D_{0}$:

Consider the region $B$ as shown in Figure 9, which is the union of the disc
$D_{p}$ with the strips $S_{1},S_{2},...,S_{n}$, where
$S_{i}=[0,\epsilon]\times[0,1]$ identified with $D_{p}$ along
$[0,\epsilon]\times\\{1\\}$. We consider the trivial Lefschetz fibration
$\tilde{\pi}:\mathbb{C}P^{2}\times B\to B$, where the fibers are
$\mathbb{C}P^{2}_{pr}$, i.e., $\mathbb{C}P^{2}$ together with the preferred
embedding $f_{pr}$ of $\Sigma_{g}.$

Now, for the critical point $p_{i}$ whose vanishing cycle $\gamma_{i}$ in
$\Sigma_{g}$ is in the standard position with respect to the embedding
$f_{pr}$, we consider the strip $R_{\gamma_{i}}=[0,\epsilon]\times[0,1]$ as
shown in Figure 9. The edge $[0,\epsilon]\times\\{1\\}\subset R_{\gamma_{i}}$
is identified with the arc $[0,\epsilon]\subset\partial D_{p_{i}}$ of the disc
$D_{p_{i}}$ and the edge $[0,\epsilon]\times\\{0\\}\subset R_{\gamma_{i}}$ is
identified with the edge $[0,\epsilon]\times\\{0\\}\subset S_{i}$. Note that
one of the boundary component of $S_{i}\cup R_{\gamma_{i}}$ is a part arc
$\alpha_{i}$.

Now, we consider the trivial Lefschetz fibration
$\tilde{\pi}:\mathbb{C}P^{2}\times R_{\gamma_{i}}\to R_{\gamma_{i}}$ with the
fibers $\Psi^{i}_{ts,1}(\mathbb{C}P^{2})$ for $(s,t)\in
R_{\gamma_{i}}=[0,\epsilon]\times[0,1].$ One can observe that when $t=0$, the
fibers $\Psi^{i}_{0,1}(\mathbb{C}P^{2})$ is $\mathbb{C}P^{2}_{pr}$ which
coincide with the fibers on the edge $[0,\epsilon]\times\\{0\\}\subset S_{i}$
and when $t=1$, the fibers $\Psi^{i}_{s,1}(\mathbb{C}P^{2})$ coincide with the
fibers on the arc $[0,\epsilon]\subset\partial D_{p_{i}}$. This gives a
Lefschetz fibration embedding of

$\pi_{|_{\pi^{-1}(R_{\gamma_{i}})}}:\pi^{-1}(R_{\gamma_{i}})\to
R_{\gamma_{i}}$

into the trivial Lefschetz fibration

$\tilde{\pi}:\mathbb{C}P^{2}\times R_{\gamma_{i}}\to R_{\gamma_{i}}$

with fibers $\Psi^{i}_{ts,1}(\mathbb{C}P^{2})$.

For the critical point $p_{j}$ whose vanishing cycle $\gamma_{j}$ in
$\Sigma_{g}$ is not in the standard position with respect to the embedding
$f_{pr}$, we consider the strips $R_{\gamma_{j}}=[0,\epsilon]\times[0,1]$ and
$R_{\gamma_{j}}^{\prime}=[0,\epsilon]\times[0,1]$ as shown in Figure 9. The
edge $[0,\epsilon]\times\\{1\\}\subset R_{\gamma_{j}}$ is identified with the
arc $[0,\epsilon]\subset\partial D_{\gamma_{j}}$ of the disc $D_{\gamma_{j}}.$
The edge $[0,\epsilon]\times\\{0\\}\subset R_{\gamma_{j}}$ is identified with
the edge $[0,\epsilon]\times\\{1\\}\subset R_{\gamma_{j}}^{\prime}$ and the
edge $[0,\epsilon]\times\\{0\\}\subset R_{\gamma_{j}}^{\prime}$ is identified
with the edge $[0,\epsilon]\times\\{0\\}\subset S_{j}$. Note that one of the
boundary components of $S_{j}\cup R_{\gamma_{j}}\cup R_{\gamma_{j}}^{\prime}$
is a part of the arc $\alpha_{j}$.

Now, we consider the trivial Lefschetz fibration
$\tilde{\pi}:\mathbb{C}P^{2}\times R_{\gamma_{j}}^{\prime}\to
R_{\gamma_{j}}^{\prime}$ with fibers $\Phi_{t}(\mathbb{C}P^{2}_{pr})$, for
$(s,t)\in R^{\prime}_{\gamma_{j}}=[0,\epsilon]\times[0,1]$ and the trivial
Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times R_{\gamma_{j}}\to
R_{\gamma_{j}}$ with the fibers
$\Psi^{j}_{ts,1}(\Phi_{1}(\mathbb{C}P^{2}_{pr})),$ for $(s,t)\in
R_{\gamma_{j}}=[0,\epsilon]\times[0,1]$.

One can observe that when $t=0$, the fiber $\Phi_{t}(\mathbb{C}P^{2}_{pr})$ of
$[0,\epsilon]\times\\{0\\}\subset R_{\gamma_{j}}^{\prime}$ are
$\mathbb{C}P^{2}_{pr}$, which coincide with the fibers on the edge
$[0,\epsilon]\times\\{0\\}\subset S_{j}$ for each $s\in[0,\epsilon]$. The
fiber $\Psi^{j}_{s,1}(\Phi_{1}(\mathbb{C}P^{2}))$ of the edge
$[0,\epsilon]\times\\{0\\}\subset R_{\gamma_{j}}$ is
$\Phi_{1}(\mathbb{C}P^{2})$ coincides with the fiber on the edge
$[0,\epsilon]\times\\{1\\}\subset R_{\gamma_{j}}^{\prime}$ for each $s$.
Finally, when $t=1$, the fiber
$\Psi^{j}_{s,1}(\Phi_{1}(\mathbb{C}P^{2}_{pr}))$ of
$[0,\epsilon]\times\\{1\\}\subset R_{\gamma_{j}}$ coincide with the fiber on
the arc $[0,\epsilon]\subset\partial D_{p_{i}}$ for each $s$. This gives a
Lefschetz fibration embedding of

$\pi_{|_{\pi^{-1}(R_{\gamma_{j}}\cup
R_{\gamma_{j}}^{\prime})}}:\pi^{-1}(R_{\gamma_{j}}\cup
R_{\gamma_{j}}^{\prime})\to R_{\gamma_{j}}\cup R_{\gamma_{j}}^{\prime}$

into the trivial Lefschetz fibration

$\tilde{\pi}:\mathbb{C}P^{2}\times(R_{\gamma_{j}}\cup
R_{\gamma_{j}}^{\prime})\to R_{\gamma_{j}}\cup R_{\gamma_{j}}^{\prime}$

with fibers described on $R_{\gamma_{j}}$ and $R_{\gamma_{j}}^{\prime}$ as
above.

Now, we can glue all the Lefschetz fibration embeddings constructed above to
get a required Lefschetz fibration embedding of

$\pi_{|_{\pi^{-1}(D_{0})}}:\pi^{-1}(D_{0})\to D_{0}$

into the trivial Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times
D_{0}\to D_{0}$. where

$D_{0}=D_{p}\cup(\cup_{i=1}^{n}\nu(\alpha_{i}))\cup(\cup_{i=1}^{n}D_{p_{i}}).$

Note that $D^{2}\setminus D_{0}$ is just the collar $S^{1}\times[0,1]$ of the
boundary of $D^{2}$ with $S^{1}\times\\{0\\}=\partial D^{2}$, one can easily
extend both the Lefschetz fibrations
$\pi_{|_{\pi^{-1}(D_{0})}}:\pi^{-1}(D_{0})\to D_{0}$ and
$\tilde{\pi}:\mathbb{C}P^{2}\times D_{0}\to D_{0}$ to the Lefschetz fibration
$\pi:X\to D^{2}$ and the trivial Lefschetz fibration
$\tilde{\pi}:\mathbb{C}P^{2}\times D^{2}\to D^{2}$ in such a way that we will
get the Lefschetz fibration embedding of the Lefschetz fibration $\pi:X\to
D^{2}$ to the trivial Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times
D^{2}\to D^{2}.$ This completes the arguments of Theorem 6.

##  4\.  Achiral Lefschetz fibration

In this section, we shall discuss Lefschetz fibration embeddings of
$4$–dimensional achiral Lefschetz fibration into the trivial Lefschetz
fibration $\tilde{\pi}:\mathbb{C}P^{2}\times D^{2}\to D^{2}$. Recall the
definition of an achiral Lefschetz fibration on $4$–manifolds.

######  Definition 12.

(Achiral Lefschetz fibration on $4$–manifold) Let $X$ be a compact, connected,
orientable smooth $4$–manifold with non-empty boundary. A proper smooth map
$\pi:X\to D^{2}$ is said to be an achiral Lefschetz fibration if

  1. (1)

the map $\pi$ has finitely many critical (or singular) points $q_{i},$ all lie
in the interior of $X$ with critical (or singular) values $f(q_{i})=p_{i}$,
$1\leq i\leq n$ and $p_{i}\neq p_{j}$, for $i\neq j$,

  2. (2)

for each critical point $q_{i}$ and its corresponding critical value $p_{i}$,
there exist coordinate charts $U_{q_{i}}=\mathbb{C}^{n}$ around
$q_{i}=(0,...,0)$ and $V_{p_{i}}=\mathbb{C}$ around $p_{i}=0$ in $X$ and
$D^{2}$ respectively, such that the restriction of $\pi$ on $U_{q_{i}}$ is the
map $\pi:U_{q_{i}}\to V_{p_{i}}$ defined as either
$\pi(z_{1},z_{2})=z_{1}^{2}+z_{2}^{2}$ or
$\pi(z_{1},z_{2})=z_{1}^{2}+\bar{z}_{2}^{2}$,

  3. (3)

for a regular value $p$, the fiber is a closed, connected, oriented surface of
$X.$

Note that an achiral Lefschetz fibration is a Lefschetz fibration which
satisfies the conditions (1) and (2) of Definition 3 without requiring the
coincidence of the canonical orientation determined by $(z_{1},z_{2})$ and the
orientation of $X$. For a critical value $p_{i}$ which has the coordinate map
$\pi(z_{1},z_{2})=z_{1}^{2}+\bar{z}_{2}^{2}$ defined in the above definition
corresponds to attaching a $2$–handle to a near by regular fiber along the
vanishing cycle $\gamma$ with framing $+1$. From the handle decomposition of a
$4-$manifold associated to a Lefschetz fibration given in Subsection 2.2, one
can see that the critical value $p_{i}$ contributes a negative Dehn twist
along the vanishing cycle $\gamma$ to the boundary monodromy of the Lefschetz
fibration. Such a singularity, we call an _achiral singularity_. For more
details, see [6] and [15].

Any simple closed curve $\gamma$ in $\Sigma_{g}$ is said to be in _the achiral
standard position_ with respect to the embedding $f_{pr}$ if $f_{pr}(\gamma)$
can be isotoped to a curve $\gamma^{\prime}$ in $f_{pr}(\Sigma_{g})\cap
S^{3}\times 1$ such that $\gamma^{\prime}$ admits a regular neighbourhood
$\nu(\gamma^{\prime})$ in $f_{pr}(\Sigma_{g})$ which is a negative Hopf
annulus in $S^{3}\times 1$. Any such curve will satisfy the following
conditions.

  * •

the curve $\gamma^{\prime}$ admits a regular neighbourhood $\nu(\gamma)$ in
$f_{pr}(\Sigma_{g})$ which is a negative Hopf annulus in $S^{3}\times 1$

  * •

$\gamma^{\prime}$ bounds a disc $D$ in $\mathbb{C}P^{2}$ which intersects
$f_{pr}(\Sigma_{g})$ only in $\gamma^{\prime}$

  * •

a tubular neighbourhood $\mathcal{N}(D)=D^{2}\times D^{2}$ intersects
$f_{pr}(\Sigma_{g})$ in a regular neighbourhood $\nu(\gamma^{\prime})$ of
$\gamma^{\prime}$ in $f_{pr}(\Sigma_{g})$,

  * •

we can choose coordinates $(z_{1},z_{2})$ on $\mathcal{N}(D)$ such that
$\nu(\gamma^{\prime})=g^{-1}(1)\cap f_{pr}(\Sigma_{g})$, where
$g:\mathbb{C}^{2}\to\mathbb{C}$ is a map defined by
$g(z_{1},z_{2})=z_{1}^{2}+\bar{z}_{2}^{2}$ and the plane
$\mathbb{R}^{2}=\\{(z_{1},z_{2})\in\mathbb{C}^{2}:z_{1}=x_{1}+i0,z_{2}=x_{2}+i0\\}$
in $\mathbb{C}^{2}$ intersects $\mathcal{N}(D)$ in the disc $D.$ Note that
$\gamma^{\prime}$ is the circle $x_{1}^{2}+x_{2}^{2}=1$ and the disc $D$ is
$x_{1}^{2}+x_{2}^{2}\leq 1$ in $\mathbb{R}^{2}.$

The monodromy of the achiral singular fibration
$g:\mathcal{N}(D)=\mathbb{C}^{2}\to\mathbb{C}=D^{2}$ with the achiral singular
point $(0,0)$ is the negative Dehn twist ${d_{\gamma^{\prime}}}^{-1}$ along
the curve $\gamma^{\prime}$ which lies in the regular fiber
$g^{-1}(1)\cap\Sigma_{pr}=\nu(\gamma^{\prime}).$ The isotopy of
$\mathcal{N}(D)$ which produces the negative Dehn twist
${d_{\gamma^{\prime}}}^{-1}$ on $g^{-1}(1)\cap\Sigma_{pr}$ can be extended to
an isotopy $\Psi_{s}$, $s\in[0,1]$, of $\mathbb{C}P^{2}$ such that the isotopy
is supported in the tubular neighbourhood $\mathcal{N}(D)$. Then, we have
$f_{pr}^{-1}\circ\Psi_{1}\circ f_{pr}={d_{\gamma}}^{-1}$ up to isotopy. Thus,
we have isotopy of $\mathbb{C}P^{2}$ supported in the tubular neighbourhood
$\mathcal{N}(D)$ of $D$ which produces Dehn twist ${d_{\gamma^{\prime}}}^{-1}$
on $\Sigma_{pr}.$

Now, we have the following theorem.

######  Theorem 13.

Let $\pi:X\to D^{2}$ be an achiral Lefschetz fibration, where $X$ is a
compact, connected, orientable $4$–manifold with non-empty boundary such that
for each regular value, the fiber is a closed, connected, orientable surface
$\Sigma_{g}$ of genus $g\geq 1$. If the vanishing cycle corresponding to each
achiral singularity is non-separating in $\Sigma_{g}$. Then, there exists a
Lefschetz fibration embedding of $X$ into a trivial Lefschetz fibration
$\tilde{\pi}:\mathbb{C}P^{2}\times D^{2}\to D^{2}.$

###### Proof.

We use the preferred embedding $f_{pr}$ of $\Sigma_{g}$ into $\mathbb{C}P^{2}$
the same as the one used in Theorem 6.

For a non-achiral Lefschetz singularity $p_{i}$ with a neighbourhood
$D_{p_{i}}$ not containing any other singularity, we can produce local
Lefschetz fibration embedding of the Lefschetz fibration
$\pi_{|_{\pi^{-1}(D_{p_{i}})}}:\pi^{-1}(D_{p_{i}})\to D_{p_{i}}$ into the
trivial Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times D_{p_{i}}\to
D_{p_{i}}$ as described in Subsection 3.2.

Now, we consider the case of an achiral singular point $p_{j}$ with
corresponding vanishing cycle $\gamma_{j}$ which is non-separating. Suppose
$f_{pr}(\gamma_{j})$ is in the achiral standard position. Then, using the
arguments given in the proof of Theorem 6 and replacing positive Hopf annulus
regular neighbourhood by negative Hopf annulus regular neighbourhood in the
arguments, we get the local Lefschetz fibration embedding of the Lefschetz
fibration $\pi_{|_{\pi^{-1}(D_{p_{j}})}}:\pi^{-1}(D_{p_{j}})\to D_{p_{j}}$
into the trivial Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times
D_{p_{j}}\to D_{p_{j}}$ as described in Subsection 3.2.

Figure 10. The curve $\beta$ in the surface $\Sigma_{g}$ of genus $g>0$ admits
a negative Hopf annulus regular neighbourhood in $\Sigma_{g}.$

Suppose $\gamma_{j}$ is not in the achiral standard position with respect to
$f_{pr}$. We consider the simple closed curve $\beta$ shown in Figure 10. The
curve $\beta$ is non-separating in $\Sigma_{g}$. Note that $\beta$ is in the
achiral standard position with respect to the embedding $f_{pr}$. As we know
that given any two non-separating simple closed curve there exists a
diffeomorphism mapping one to another, we choose a diffeomorphism $\phi$ which
maps $\gamma_{j}$ to $\beta$ in $\Sigma_{g}.$ As the embedding
$f_{pr}:\Sigma_{g}\to\mathbb{C}P^{2}$ is flexible, there exists an isotopy
$\\{\Gamma_{r}\\}_{r\in[0,1]}$ such that $f_{pr}^{-1}\circ\Gamma_{1}\circ
f_{pr}=\phi$.

We consider a family of embeddings $\\{f_{r}\\}_{r\in[0,1]}$, where
$f_{r}=\Gamma_{r}\circ f_{pr}$ and we denote the image of embeddings $f_{r}$
by $\Sigma^{r}$, where $\Sigma^{0}=\Sigma_{pr}.$ Note that $f_{pr}$ and
$f_{1}$ have the same image $\Sigma_{pr}$. Now, the curve $\gamma_{j}$ is in
the achiral standard position with respect to the embedding $f_{1}$.

In this way, given any vanishing cycle $\gamma_{j}$, we can find a flexible
embedding of $\Sigma_{g}$ ambient isotopic to $f_{pr}$ in $\mathbb{C}P^{2}$
such that $\gamma_{j}$ either in the standard position or in the achiral
standard position with respect to that embedding.

Now, using the arguments given in Theorem 6, we obtain the Lefschetz fibration
embedding of the achiral Lefschetz fibration $\pi:X\to D^{2}$ to the trivial
Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times D^{2}\to D^{2}.$

∎

We call a Lefschetz fibration _allowable_ if none of its vanishing cycle is
null-homologous in the regular fiber.

######  Remark 14.

Thus, given an allowable achiral Lefschetz fibration over $D^{2}$ it admits a
Lefschetz fibration embedding into a trivial Lefschetz fibration
$\tilde{\pi}:\mathbb{C}P^{2}\times D^{2}\to D^{2}.$

######  Theorem 15.

Let $\pi:X\to D^{2}$ be an achiral Lefschetz fibration, where $X$ is a
compact, connected, orientable $4$–manifold with non-empty boundary such that
for each regular value, the fiber is a closed, connected, orientable surface
$\Sigma_{g}$ of genus $g\geq 0$. Then, there exists a Lefschetz fibration
embedding of $X$ into a trivial Lefschetz fibration
$\tilde{\pi}:(\mathbb{C}P^{2}\\#\mathbb{C}P^{2})\times D^{2}\to D^{2}$ as well
into a trivial Lefschetz fibration
$\tilde{\pi}:(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}})\times D^{2}\to
D^{2}$

###### Proof.

In this case, we construct a preferred embedding of $\Sigma_{g}$ in
$\mathbb{C}P^{2}$ as described in Lemma 9 with following modification:

Figure 11. A flexible embedding of $\Sigma_{g}$ in
$\mathbb{C}P^{2}\\#\mathbb{C}P^{2}$ with an isotopies of the curve
$\gamma_{2}$ which has a positive Hopf annulus regular neighbourhood and a
negative Hopf annulus regular neighbourhood in $S^{3}\times 1$.

Consider a collar neighbourhood $S^{3}\times[0,2]$ of the boundary $\partial
D^{4}=S^{3}\times 0$ in a $4-$ball $D^{4}$. Consider a standardly embedded
closed orientable surface $S$ of genus $g$ which bounds an unknotted untwisted
handlebody of genus $g$ – see Figure 2 – in $S^{3}\times 1$. Remove the
interiors of two disjoint discs $D_{1}$ and $D_{2}$ from $S$ and attach one
positive full-twisted band and one negative full-twisted band to $\partial
D_{1}$ and $\partial D_{2}$ respectively, in $S^{3}\times 1,$ refer Figure 11.
Thus, we get a new compact, connected, orientable surface $S_{1}$ of genus $g$
with boundaries $l_{1},l_{2},l_{3}$ and $l_{4}$ such that $l_{1}\cup l_{2}$ is
a positive Hopf link and $l_{3}\cup l_{4}$ is a negative Hopf link in
$S^{3}\times 1,$ refer Figure 11. We attach tubes $l_{i}\times[0,1],\
i=1,2,3,4$ to get a proper embedding of $S_{1}$ in $D^{4}$. Now, we attach
$2$–handles $H_{2}$ and $H_{4}$ to $D^{4}$ along the boundary components
$l_{2}$ and $l_{4}$ of $S_{1}$ with the framing $+1$ and $+1$ (respectively
$-1$) respectively, and then attach a $4$–handle on the resulting boundary to
obtain $\mathbb{C}P^{2}\\#\mathbb{C}P^{2}$ (respectively,
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$). The components $l_{i}$ of the
surface $S_{1}$ bounds a $2$–disc $D_{i}^{\prime}$ in the $2$–handle $H_{i}$
for $i=2,4$, namely the core of the $2$–handles and the boundary component
$l_{i},\ i=1,3$ of $S_{1}$ bounds a disc $D_{i}^{\prime\prime}$ in the
$4-$handle of $\mathbb{C}P^{2}\\#\mathbb{C}P^{2}$ (respectively,
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$). Thus, we get an embedding of
the surface $\Sigma_{g}=S_{1}\cup D_{1}^{\prime}\cup D_{2}^{\prime\prime}\cup
D_{3}^{\prime}\cup D_{4}^{\prime\prime}$ in
$\mathbb{C}P^{2}\\#\mathbb{C}P^{2}$ (respectively,
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$) such that $\Sigma_{g}$
contains an embedded positive Hopf annulus $A$ and negative Hopf annulus
$A^{\prime}$ with boundary components $l_{1}\cup l_{2}$ and $l_{3}\cup l_{4}$,
refer Figure 11. We consider this embedding as our preferred embedding
$f_{pr}$ of $\Sigma_{g}.$

Then, using the arguments given in Theorem 6, we obtain the Lefschetz
fibration embedding of the achiral Lefschetz fibration $\pi:X\to D^{2}$ to the
trivial Lefschetz fibration
$\tilde{\pi}:(\mathbb{C}P^{2}\\#\mathbb{C}P^{2})\times D^{2}\to D^{2}$
(respectively, $\tilde{\pi}:\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}\times
D^{2}\to D^{2}$).

∎

###  4.1.  Achiral Lefschetz fibration over $\mathbb{C}P^{1}$

Let $X^{4}$ be a closed connected orientable smooth $4$–manifold. A smooth map
$\pi:X^{4}\to S^{2}=\mathbb{C}P^{1}$ is said to be a Lefschetz fibration if
the conditions (1), (2) and (3) of Definition 3 are satisfied. Similarly, we
have the notion of an achiral Lefschetz fibration $\pi:X^{4}\to
S^{2}=\mathbb{C}P^{1}$.

If we consider a disc $D^{2}$ in $\mathbb{C}P^{1}$ containing all the critical
values of $\pi$, then we can construct a Lefschetz fibration embedding of the
Lefschetz fibration $\pi_{|_{\pi^{-1}(D^{2})}}:\pi^{-1}(D^{2})\to D^{2}$ in a
trivial Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times D^{2}\to
D^{2}.$

Now, $\pi$ is a Lefschetz fibration over $\mathbb{C}P^{1}$, the product of the
Dehn twist along the vanishing cycles taken in the same way the corresponding
critical values lie on a small counter clockwise oriented circle around the
preferred regular value in $D^{2}$ as described in Subsection 3.2, is isotopic
to the identity map of $\Sigma_{g}$, i.e., the monodromy of the $\Sigma_{g}$
fiber bundle over $\partial D^{2}$ is isotopic to the identity of
$\Sigma_{g}$. This implies that we get a trivial $\Sigma_{g}$ fiber bundle
over $\partial D^{2}$. As the embedding $f_{pr}:\Sigma_{g}\to\mathbb{C}P^{2}$
is flexible in $\mathbb{C}P^{2}$ and the monodromy of the $\Sigma_{g}$ bundle
over $\partial D^{2}$ is flexible in $\mathbb{C}P^{2}$, we get a trivial
$\mathbb{C}P^{2}$ bundle $\partial D^{2}$. Using the isotopy of the monodromy
of $\Sigma_{g}$ with the identity map of $\Sigma_{g}$ and the flexibility of
the preferred embedding, we can extend the Lefschetz fibration embedding of
$\pi_{|_{D^{2}\cup A}}:\pi^{-1}(D^{2}\cup A)\to D^{2}\cup A$ into
$\tilde{\pi}_{|_{\mathbb{C}P^{2}\times(D^{2}\cup
A)}}:\mathbb{C}P^{2}\times(D^{2}\cup A)\to D^{2}\cup A$, where $A$ is the
collar of $\partial(\mathbb{C}P^{1}\setminus D^{2})$. Finally, we can extend
this bundle trivially to the removed disc of $\mathbb{C}P^{2}$ over
$\mathbb{C}P^{1}$. Therefore, both $\Sigma_{g}$ bundle over $\partial D^{2}$
and $\mathbb{C}P^{2}$ bundle over $\partial D^{2}$ can be extended to a
trivial Lefschetz fibrations over $\mathbb{C}P^{1}\setminus D^{2}$ so that we
get a Lefschetz fibration embedding of Lefschetz fibration $\pi:X^{4}\to
S^{2}=\mathbb{C}P^{1}$ into a trivial
$\pi:\mathbb{C}P^{2}\times\mathbb{C}P^{1}\to S^{2}=\mathbb{C}P^{1}$.

Thus, we have the following theorems:

######  Theorem 16.

Let $\pi:X^{4}\to\mathbb{C}P^{1}$ be an achiral Lefschetz fibration, where $X$
is a closed, connected, orientable $4$–manifold such that for each regular
value, the fiber is a closed, connected, orientable surface $\Sigma_{g}$ of
genus $g\geq 2$. If the vanishing cycle corresponding to each achiral
singularity is non-separating on $\Sigma_{g}$, then there exists a Lefschetz
fibration embedding of $\pi:X^{4}\to\mathbb{C}P^{1}$ into a trivial Lefschetz
fibration
$\tilde{\pi}:\mathbb{C}P^{2}\times\mathbb{C}P^{1}\to\mathbb{C}P^{1}.$

######  Theorem 17.

Let $\pi:X^{4}\to CP^{1}$ be an achiral Lefschetz fibration, where $X$ is a
closed, connected, orientable $4$–manifold such that for each regular value,
the fiber is a closed, connected, orientable surface $\Sigma_{g}$ of genus
$g\geq 2$. Then, there exists a Lefschetz fibration embedding of $X$ into a
trivial Lefschetz fibration
$\tilde{\pi}:(\mathbb{C}P^{2}\\#\mathbb{C}P^{2})\times\mathbb{C}P^{1}\to\mathbb{C}P^{1}$
as well as into a trivial Lefschetz fibration
$\tilde{\pi}:(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}})\times\mathbb{C}P^{1}\to\mathbb{C}P^{1}.$

##  5\. Broken Lefschetz fibration

In this section, we shall discuss Lefschetz fibration embeddings of broken
Lefschetz fibrations.

######  Definition 18.

Let $X$ be a compact, connected, oriented $4$–manifold with non-empty
boundary. A proper smooth surjective map $\pi:X\to D^{2}$ is said have a round
singularity along an embedded $1-$manifold $Z\subset int(X)\setminus C$, where
$C$ is the set of Lefschetz singularities of $\pi$, if around every $q\in Z$,
there are coordinates $(t,x_{1},x_{2},x_{3})$ with $t$ a local coordinate on
$Z$ such that $\pi$ is given by the map
$(t,x_{1},x_{2},x_{3})\mapsto(t,x_{1}^{2}+x_{2}^{2}-x_{3}^{2})$. We call the
image $\pi(Z)\subset int(D^{2})$ the round image.

######  Definition 19.

A proper smooth surjective map $\pi:X\to D^{2}$ defined on a compact,
connected, oriented $4$–manifold $X$ with non-empty boundary is said to be a
broken Lefschetz fibration (BLF) if $\pi$ has both Lefschetz and round
singularities.

A broken Lefschetz fibration is said to be a _simplified broken Lefcshetz
fibration (SBLF)_ if

  * •

the round singular set $Z$ is connected and its image on $D^{2}$ is a boundary
parallel circle,

  * •

the regular fibers of $\pi$ are connected,

  * •

the disc containing all the Lefschetz singular value is contained in the
interior of the disc bounded by $\pi(Z)$.

Thus, we get the decomposition of the SBLF into $3$ pieces which are

  1. (1)

a genus $g-1$ Lefschetz fibration over an annulus, we call this piece a lower
genus piece,

  2. (2)

a genus $g$ Lefschetz fibration over a $2$–disc, we call this piece the higher
genus piece and

  3. (3)

a round cobordism between the boundaries of the the higher genus piece and the
lower genus piece containing the round singular set in the interior.

A handle description of a SBLF is the higher genus Lefschetz fibration
together with a round $2$–handle attached to it. A round $2$–handle is a
fiberwise $2$–handle attachment parameterized along $Z$. Thus, a (untwisted)
round $2$–handle is $S^{1}\times D^{2}\times D^{1}=S^{1}\times D^{3}$. Note
that the (untwisted) round $2$–handle $S^{1}\times D^{3}$ can be regarded as
the union of a $0$–handle and a $1$–handle. So, we can express a (untwisted)
round $2$–handle in the $4$–manifold as attaching a $4$–dimensional $2$–handle
and a $3$–handle. The attaching circle of the $2$–handle of a round $2$–handle
is a simple closed curve $\gamma$ on a regular fiber, which is preserved under
the monodromy of this fibration up to isotopy. Since this attachment comes
from the fiberwise handle attachments, the $2$–handle should have the fiber
framing zero. Hence, the framing of the handle attachments are trivial. Here,
we will consider attaching untwisted $2$–handle, hence the monodromy $\phi$
maps $\gamma$ to $\gamma$, i.e., we assume $\phi(\gamma)=\gamma$. For more
details, refer [1] and [2].

Now, we prove the following theorem.

######  Theorem 20.

Let $\pi:X\to D^{2}$ be a simplified broken Lefschetz fibration, where $X$ is
a compact, connected, orientable $4$–manifold with non-empty boundary such
that each regular fiber in the higher genus piece is a closed, connected,
orientable surface of genus $g>2$, then there exists a Lefschetz fibration
embedding of $X$ into a trivial Lefschetz fibration
$\tilde{\pi}:\mathbb{C}P^{2}\times D^{2}\to D^{2}.$

###### Proof.

Consider a SBLF $\pi:X\to D^{2}$. Let $q_{1},...,q_{n}$ denote the Lefschetz
singularities in the interior of $X$ and $p_{1},...,p_{n}$ be the
corresponding singular values of $q_{i}$’s. Let $Z$ be the round singular
circle in the interior of $X$ and $Z^{\prime}=\pi(Z)$ be the image of $Z$ in
$D^{2}.$ The circle $Z^{\prime}$ encloses all the singular values and is
boundary parallel. Let $D$ be the disc containing all the singular values and
contained in the disc bounded by $Z^{\prime}$, see Figure 12.

Figure 12. In the figure the circle $Z^{\prime}$ is a round singular image of
broken Lefschetz fibration.

Now, as shown in Theorem 6, we have a Lefschetz fibration embedding of the
Lefschetz fibration $\pi_{|_{\pi^{-1}(D)}}:\pi^{-1}(D)\to D$ into a trivial
Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times D\to D.$ Let $\phi$ be
the boundary monodromy of the Lefschetz fibration
$\pi_{|_{\pi^{-1}(D)}}:\pi^{-1}(D)\to D$ and let $\Psi$ be the boundary
monodromy of the Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times D\to
D.$ Then, $f_{pr}^{-1}\circ\Psi\circ f_{pr}=\phi$.

Consider the annulus $A=D^{2}\setminus
D=S^{1}\times[-1,1]=\\{((e^{i\theta},t):\theta\in[0,2\pi],t\in[-1,1]\\}$ with
$S^{1}\times-1=\partial D^{2}$, $S^{1}\times 1=\partial D$ and $S^{1}\times
0=Z^{\prime}.$ Suppose $\pi^{-1}((0,1))$ is $\mathbb{C}P^{2}_{pr}$, i.e.,
$\mathbb{C}P^{2}$ with preferred embedding $f_{pr}$ of $\Sigma_{g}.$

Let $\gamma$ be a simple closed curve on $\Sigma_{g}$ preserved by the
monodromy $\phi$ of $\pi_{|_{\pi^{-1}(D)}}$. Suppose that
$\gamma^{\prime}=f_{pr}(\gamma)$ up to isotopy in $\Sigma_{pr}$ has following
properties:

  * •

The curve $f_{pr}(\gamma)$ bounds a disc $D^{\prime}$ in $\mathbb{C}P^{2}$
which intersects the surface $\Sigma_{pr}$ only in $\gamma^{\prime}$.

  * •

A tubular neighbourhood $\mathcal{N}(D^{\prime})=D^{2}\times D^{2}$ of
$D^{\prime}$ intersects $\Sigma_{pr}$ in the annulus
$\nu(\gamma^{\prime})=\gamma^{\prime}\times[-1,1]$ and $\gamma^{\prime}\times
0$ and $\gamma^{\prime}\times 1$ are unknots and unlinked in $S^{3}\times-1$
as described in the construction in the Lemma 9.

  * •

Moreover, $D^{\prime}\times[-1,1]\subset\mathcal{N}(D^{\prime})$ with
$\partial D^{\prime}\times[-1,1]=\gamma^{\prime}\times[-1,1]$.

  * •

We can choose coordinates $(\theta,x_{1},x_{2},x_{3})$ on
$\mathcal{N}(D^{\prime})$ such that $\Sigma_{pr}\cap
g^{-1}(0,1)=\nu(\gamma^{\prime})$ with
$D^{\prime}\times[-1,1]=\\{(0,x_{1},x_{2},x_{3}):x_{1}^{2}+x_{2}^{2}\leq 1,\
x_{3}\in[-1,1]\\}$, where $g:\mathbb{R}^{4}\to\mathbb{R}^{2}$ is a map defined
by $(\theta,x_{1},x_{2},x_{3})\mapsto(\theta,x_{1}^{2}+x_{2}^{2}-x_{3}^{2})$
and the plane
$\mathbb{R}^{2}=\\{(0,x_{1},x_{2},0)\in\mathbb{R}^{4}:x_{1},x_{2}\in\mathbb{R}\\}$
in $\mathbb{R}^{4}$ intersect $\mathcal{N}(D^{\prime})$ in the disc
$D^{\prime}$.

We call such a simple closed curve $\gamma$ is in the SBLF standard position
with respect to the embedding $f_{pr}.$

We can consider $[0,2\pi]\times
D^{2}\times[-1,1]\subset\mathcal{N}(D^{\prime})$. Note that all the Lickorish
generating curves admit this property with respect to the preferred embedding.
First, we consider the case where $\gamma$ is in the SBLF standard position
with respect to the embedding $f_{pr}.$

Figure 13.  Flexible embedding of $\Sigma_{g-1}$ from $\Sigma_{g}$.

Now, we consider the flexible embeddings $f_{0,t},\ t\in(0,1]$ with
$f_{0,1}=f_{pr}$ as follows: We will consider isotopy $\Psi_{t}$, $t\in(0,1]$
of diffeomorphism of $\mathbb{C}P^{2}$ supported in $\mathcal{N}(D^{\prime})$
with $\Psi_{1}=id$ such that

  * •

the embeddings $f_{0,t}=\Psi_{t}\circ f_{pr}$,

  * •

the curve $f_{0,t}(\gamma)$ bounds a disc
$\Psi_{t}(D^{\prime})=D^{\prime}_{t}\subset D^{\prime}$ in $\mathbb{C}P^{2}$
which intersect the surface $f_{0,t}(\Sigma_{pr})$ only in
$f_{0,t}(\gamma^{\prime})$,

  * •

the tubular neighbourhood $\Psi_{t}(\mathcal{N}(D_{t}^{\prime}))$ of
$D^{\prime}_{t}$ intersect $f_{0,t}(\Sigma_{pr})$ in the annulus
$f_{0,t}(\nu(\gamma^{\prime}))=\gamma_{t}^{\prime}\times[-1,1]$,

  * •

moreover, $\Psi_{t}(D^{\prime}\times[-1,1])\subset\mathcal{N}(D_{t}^{\prime})$
with $\partial D_{t}^{\prime}\times[-1,1]=\gamma_{t}^{\prime}\times[-1,1]$,

  * •

we have $f_{0,t}(\Sigma_{pr})\cap g^{-1}(0,t)=\nu(\gamma_{t}^{\prime})$ with
$D_{t}^{\prime}\times[-1,1]=\\{(0,x_{1},x_{2},x_{3}):x_{1}^{2}+x_{2}^{2}\leq
t,\ x_{3}\in[-1,1]\\}$, see Figure 13.

For each $t\in(0,1]$, if we travel through the family $\Psi_{t}$ of
diffeomorphisms of $\mathbb{C}P^{2}$, the disc $D^{\prime}=D_{1}^{\prime}$
shrinks through the discs $D^{\prime}_{1}$ to the origin of $D^{\prime}_{1}$.
Thus, the embedded surface $f_{0,1}(\Sigma_{pr})$ can be collapsed through
$f_{0,t}(\Sigma_{pr})$ to get a singular surface $\Sigma^{\prime}$ which is
obtained by pinching the circle $\gamma^{\prime}$ to the origin of the disc
$D^{\prime}$. The singular surface $\Sigma_{0}$ gives a flexible embedding of
the singular fiber $\pi^{-1}(0,0)$ in $\mathbb{C}P^{2},$ where $(0,0)\in
Z^{\prime}\subset A$ as the monodromy preserves $\gamma^{\prime}$ up to
isotopy. Hence, the isotopy $\\{\Psi_{t}\\}_{t\in(0,1]}$ clearly extends to
the isotopy $\\{\Psi_{t}\\}_{t\in[0,1]}$.

Now, we construct flexible embeddings of the lower genus surface
$\Sigma_{g-1}$ from the preferred embedding
$f_{pr}:\Sigma_{g}\to\mathbb{C}P^{2}.$ Consider the surface
$\Sigma_{pr}\setminus(\gamma^{\prime}\times(-1,1))$ and cap off by the discs
$D^{\prime}\times-1$ and $D^{\prime}\times 1$ to get the preferred embedding
$f_{pr}^{g-1}:\Sigma_{g-1}\to\mathbb{C}P^{2}$ of the surface $\Sigma_{g-1}$
and we denote the image $f_{pr}^{g-1}(\Sigma_{g-1})$ by $\Sigma^{g-1}_{pr}$.
Now, consider $\Sigma_{pr}^{g-1}\cup D^{\prime}\times[-1,1]$ and consider the
isotopy $\Psi_{t},\ t\in[-1,0]$, where $\Psi_{t}=\Psi_{-t}.$ Consider the
embeddings $f^{g-1}_{0,t}=\Psi_{t}\circ f^{g-1}_{pr},\ t\in[-1,0]$ of
$\Sigma^{g-1}$. Note that by construction these embeddings are flexible.

Now, we can construct flexible embeddings of $f_{\eta,t}$, $t\in(0,1]$,
$f_{\eta,0}$ and $f^{g-1}_{\eta,t},$ $t\in[-1,0)$ as well as the family
$\Psi_{\eta,t},\ t\in[-1,1]$ as discussed in proof of Theorem 6. From this, we
can see that the mapping tori

  * •

$\mathcal{MT}(f_{0,t}(\Sigma_{pr}),\phi),\ t\in(0,1]$ is embedded fiberwise in
$\mathcal{MT}(\mathbb{C}P^{2},\Psi_{0,t}\circ\Psi\circ\Psi_{0,t}^{-1}),$
$t\in(0,1]$,

  * •

$\mathcal{MT}(\Sigma_{0},\phi)$ is embedded fiberwise in
$\mathcal{MT}(\mathbb{C}P^{2},\Psi_{0,0}\circ\Psi\circ\Psi_{0,0}^{-1})$ and

  * •

$\mathcal{MT}(f_{0,t}(\Sigma^{g-1}_{pr}),\phi),\ t\in[-1,0)$ is embedded
fiberwise in
$\mathcal{MT}(\mathbb{C}P^{2},\Psi_{0,t}\circ\Psi\circ\Psi_{0,t}^{-1}),$
$t\in[-1,0)$.

From this, we can see that the Lefschetz fibration embedding of the Lefschetz
fibration $\pi_{|_{\pi^{-1}(D)}}:\pi^{-1}(D)\to D$ to the trivial Lefschetz
fibration $\tilde{\pi}:\mathbb{C}P^{2}\times D\to D$ which we constructed
extends into the Lefschetz fibration embedding of the Lefschetz fibration
$\pi:X\to D^{2}$ into a trivial Lefschetz fibration
$\tilde{\pi}:\mathbb{C}P^{2}\times D^{2}\to D^{2}.$

If $\gamma$ is not in the SBLF standard position with respect to the embedding
$f_{pr},$ then we can choose a diffeomorphism $h$ of the surface $\Sigma_{g}$
such that $h$ maps $\gamma$ to one of the Lickorish generating curves. Since
$f_{pr}$ is a flexible embedding, there exists an isotopy
$\\{\Phi_{r}\\}_{r\in[0,1]}$ of diffeomorphisms of $\mathbb{C}P^{2}$ such that
$f_{pr}^{-1}\circ\Phi_{1}\circ f_{pr}=h.$ We consider a family of embeddings
$\\{f_{r}\\}_{r\in[0,1]}$, where $f_{r}=\Phi_{r}\circ f_{pr}$ and we denote
the image of embeddings $f_{r}$ by $\Sigma^{r}$, where
$\Sigma^{0}=\Sigma_{pr}.$ Now, $f_{1}(\gamma)$ has the require property with
respect to embedding $f_{1}$, i.e., $\gamma$ is is in the SBLF standard
position with respect to $f_{1}.$ As $\gamma$ preseved by $\phi,$ $h(\gamma)$
is preserved by $h\circ\phi\circ h^{-1}.$ Hence, we have fiberwise embedding
of $\mathcal{MT}(\Sigma^{r},h\circ\phi\circ h^{-1})$ into
$\mathcal{MT}(\mathbb{C}P^{2},\Phi_{r}\circ\Phi\circ\Phi_{r}^{-1})$, for every
$r$.

From this, we can see that the Lefschetz fibration embedding of the Lefschetz
fibration $\pi_{|_{\pi^{-1}(D)}}:\pi^{-1}(D)\to D$ into the trivial Lefschetz
fibration $\tilde{\pi}:\mathbb{C}P^{2}\times D\to D$ which we constructed
extends into the Lefschetz fibration embedding of the Lefschetz fibration
$\pi_{|_{\pi^{-1}(D\cup C_{0})}}:\pi^{-1}(D\cup C_{0})\to D\cup C_{0}$ into
the trivial Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times(D\cup
C_{0})\to D\cup C_{0}$, where $C_{0}$ is a small collar of the boundary
$S^{1}\times 1$ of the annulus $A$. Note that the boundary monodromy of the
Lefschetz fibration $\pi_{|_{\pi^{-1}(D\cup C_{0})}}:\pi^{-1}(D\cup C_{0})\to
D\cup C_{0}$ is $h\circ\phi\circ h^{-1}$ and the boundary monodromy of the
Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times(D\cup C_{0})\to D\cup
C_{0}$ is $\Phi_{1}\circ\Psi\circ\Phi_{1}^{-1}$ which is isotopic to the
identity of $\mathbb{C}P^{2}$.

Finally, using the similar argument given above, we can extend this embedding
into the Lefschetz fibration embedding of the Lefschetz fibration $\pi:X\to
D^{2}$ into a trivial Lefschetz fibration $\tilde{\pi}:\mathbb{C}P^{2}\times
D^{2}\to D^{2}.$ This completes the arguments of Theorem 20.

∎

Using the arguments in Theorem 13, 15 and 20, we have the following theorems.

######  Theorem 21.

Let $\pi:X\to D^{2}$ be a simplified broken achiral Lefschetz fibration, where
$X$ is a compact, connected, orientable $4$–manifold with non-empty boundary
such that each regular fiber in the higher genus piece is a closed, connected,
orientable surface of genus $g\geq 1$. If the vanishing cycles corresponding
to each achiral singularity is non-separating on $\Sigma_{g}$, then there
exists a Lefschetz fibration embedding of $X$ into a trivial Lefschetz
fibration $\tilde{\pi}:\mathbb{C}P^{2}\times D^{2}\to D^{2}.$

######  Theorem 22.

Let $\pi:X\to D^{2}$ be a simplified broken achiral Lefschetz fibration, where
$X$ is a compact, connected, orientable $4$–manifold with non-empty boundary
such that each regular fiber in the higher genus piece is a closed, connected,
orientable surface $\Sigma_{g}$ of genus $g\geq 1$. Then, there exists a
Lefschetz fibration embedding of $X$ into a trivial Lefschetz fibration
$\tilde{\pi}:(\mathbb{C}P^{2}\\#\mathbb{C}P^{2})\times D^{2}\to D^{2}$ as well
as into a trivial Lefschetz fibration
$\tilde{\pi}:(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}})\times D^{2}\to
D^{2}$

For a simplified broken achiral Lefschetz fibration
$\pi:X^{4}\to\mathbb{C}P^{1}$, we can easily obtain the following results
using arguments given in the proof of Theorems 16, 17.

######  Theorem 23.

Let $\pi:X^{4}\to\mathbb{C}P^{1}$ be a simplified broken achiral Lefschetz
fibration, where $X$ is a closed, connected, orientable $4$–manifold such that
each regular fiber in the higher genus piece is a closed, connected,
orientable surface $\Sigma_{1}$ genus $g>2$. If the vanishing cycle
corresponding to each achiral singularity is non-separating on $\Sigma_{g}$,
then there exists a Lefschetz fibration embedding of
$\pi:X^{4}\to\mathbb{C}P^{1}$ into a trivial Lefschetz fibration
$\tilde{\pi}:\mathbb{C}P^{2}\times\mathbb{C}P^{1}\to\mathbb{C}P^{1}.$

As every closed, connected, oriented $4$–manifold admits a simplified broken
Lefschetz fibration over $\mathbb{C}P^{1}$ with the higher genus regular
fiber, a surface $\Sigma_{g}$ of genus $g>2$, we have the following result.

######  Corollary 24.

Every closed, connected, orientable $4$–manifold admits a smooth embedding in
$\mathbb{C}P^{2}\times\mathbb{C}P^{1}.$

######  Theorem 25.

Let $\pi:X^{4}\to CP^{1}$ be a simplified broken achiral Lefschetz fibration,
where $X$ is a closed, connected, orientable $4$–manifold such that each
regular fiber in the higher genus piece is a closed, connected, orientable
surface $\Sigma_{g}$ of genus $g>2$. Then, there exists a Lefschetz fibration
embedding of $X$ into a trivial Lefschetz fibration
$\tilde{\pi}:(\mathbb{C}P^{2}\\#\mathbb{C}P^{2})\times\mathbb{C}P^{1}\to\mathbb{C}P^{1}$
as well into a trivial Lefschetz fibration
$\tilde{\pi}:(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}})\times\mathbb{C}P^{1}\to\mathbb{C}P^{1}.$

##  6\.  Bordered Lefschetz fibration

In this section, we discuss the notion of bordered Lefschetz fibration.

######  Definition 26.

(Bordered Lefschetz fibration on $4$–manifold) Let $X$ be a compact,
connected, orientable smooth $4$–manifold with non-empty boundary. A smooth
map $\pi:X\to D^{2}$ is said to be a bordered Lefschetz fibration if

  1. (1)

the map $\pi$ has finitely many critical (or singular) points $q_{i},$ all lie
in the interior of $X$ with critical (or singular) values $f(q_{i})=p_{i}$,
$1\leq i\leq n$ and $p_{i}\neq p_{j}$, for $i\neq j$,

  2. (2)

for each critical point $q_{i}$ and its corresponding critical value $p_{i}$,
there exist coordinate charts $U_{q_{i}}=\mathbb{C}^{2}$ around $q_{i}=(0,0)$
and $V_{p_{i}}=\mathbb{C}$ around $p_{i}=0$ in $X$ and $D^{2}$ respectively,
which agree with the orientations of $X$ and $D^{2}$, such that the
restriction of $\pi$ on $U_{q_{i}}$ is the map $\pi:U_{q_{i}}\to V_{p_{i}}$
defined as $\pi(z_{1},z_{2})=z_{1}^{2}+z_{2}^{2},$

  3. (3)

for a regular value $p$, the fiber is a compact, connected, oriented proper
surface of $X$ with non-empty boundary.

Note that the bordered (achiral) Lefschetz fibration gives an open book
structure on its boundary.

Note that using the construction given in Lemma 9, one can obtain a proper
flexible embeddings of a surface $\Sigma_{g,n}$ of genus $g$ with $n$ boundary
components in $\mathcal{D}E(m)$ as well as in
$\mathcal{D}E(m)\\#\mathcal{D}E(m)$, where $\mathcal{D}E(m)$ is the $D^{2}$
bundle over $S^{2}$ with the Euler number an integer $m,$ for more details
refer [16], [3] and [17].

Now, the following are the consequences of the arguments given in Theorem 6.

######  Theorem 27.

Let $\pi:X\to D^{2}$ be a bordered achiral Lefschetz fibration, where $X$ is a
compact, connected, orientable $4$–manifold with non-empty boundary such that
for each regular value, the fiber is a compact, connected, orientable surface
$\Sigma_{g,n}$ of genus $g\geq 1$ with $n$ boundary components. If the
vanishing cycle corresponding to each achiral singularity is non-separating on
$\Sigma_{g,n}$, then there exists a Lefschetz fibration embedding of $X$ into
a trivial Lefschetz fibration $\tilde{\pi}:\mathcal{D}E(m)\times D^{2}\to
D^{2}.$

######  Theorem 28.

Let $\pi:X\to D^{2}$ be a bordered achiral Lefschetz fibration, where $X$ is a
compact, connected, orientable $4$–manifold with non-empty boundary such that
for each regular value, the fiber is a compact, connected, orientable surface
$\Sigma_{g}$ of genus $g\geq 0$ with $n>1$ boundary components. Then, there
exists a Lefschetz fibration embedding of $X$ into a trivial Lefschetz
fibration $\tilde{\pi}:(\mathcal{D}E(m)\\#\mathcal{D}E(m))\times D^{2}\to
D^{2}.$

###  6.1. Lefschetz fibration embedding in $D^{4}\times D^{2}$

In this subsection, we give Lefschetz fibration embedding of a huge collection
of bordered Lefschetz fibrations $\pi:X^{4}\to D^{2}$ into the trivial
Lefschetz fibration $D^{4}\times D^{2}$ over $D^{2}$.

First, consider a collar neighbourhood $S^{3}\times[0,2]$ of the boundary
$\partial D^{4}=S^{3}\times 0$ in a $4-$ball $D^{4}$. Consider a standardly
embedded closed orientable surface $S$ of genus $g$ which bounds an unknotted
untwisted handlebody of genus $g$ – see Figure 2 – in $S^{3}\times 1$. Remove
the interior of $n$ disjointly embedded discs $D_{1},...D_{n}$ with $\partial
D_{i}=l_{i}$ from $S$ to get a surface $S_{g,n}$ of genus $g$ with $n$
boundary components and attach tubes $l_{1}\times[0,1],...,l_{n}\times[0,1]$
to get a proper embedding of the surface $\Sigma_{g,n}$ of genus $g$ with $n$
boundary components in $D^{4}$. We consider this embedding as the preferred
embedding $f_{pr}$ of $\Sigma_{g,n}$ in $D^{4}.$

Consider a collection $\mathcal{A}=\\{a_{i}\\}$ and $\mathcal{B}=\\{b_{i}\\}$
of simple closed curves in $\Sigma_{g,n}$ such that each $a_{i}$ admit a
regular neighbourhood in $\Sigma_{g,n}$ which is a positive Hopf annulus in
$\S^{3}\times 1$ and each $b_{i}$ admit a regular neighbourhood in
$\Sigma_{g,n}$ which is a negative Hopf annulus in $S^{3}\times 1$. For
example, see Figure 14. Note that all the Dehn twists
$d_{a_{i}},d_{a_{i}}^{-1},d_{b_{i}}$ and $d_{b_{i}}^{-1}$ are flexible in
$D^{4}.$ The curves $a_{i}$’s are in the standard position and the curves
$b_{i}$’s are in the achiral standard position with respect to the preferred
embedding $f_{pr}$ in $D^{4}$.

Figure 14. Some of the curves which admits positive or negative Hopf annulus
in $\Sigma_{g,n}.$

Consider a collections $\mathcal{C}$ of curves $\gamma$ such that
$h(\gamma)=a_{j}$ for some $a_{j}\in\mathcal{A}$ and for some diffeomorphism
$h$ of $\Sigma_{g,n}$ which is a product of positive as well as negative
powers of the Dehn twists $d_{a_{i}}$ and $d_{b_{i}}$, i.e., $h$ belongs to
the subgroup of generated by the Dehn twists $d_{a_{i}}$ and $d_{b_{i}}$.
Similarly, we consider a collections $\mathcal{C}^{\prime}$ of curves $\gamma$
such that $h(\gamma)=b_{j}$ for some $b_{j}\in\mathcal{B}$ and for some
diffeomorphism $h$ of $\Sigma_{g,n}$ which is a product of positive as well as
negative powers of the Dehn twists $d_{a_{i}}$ and $d_{b_{i}}$. Note that such
a $h$ is flexible in $D^{4}.$

######  Theorem 29.

Suppose that $\pi:X\to D^{2}$ is a bordered achiral Lefschetz fibration with
the property that

  * •

the regular fiber of $\pi$ is the surface $\Sigma_{g,n}$ of genus $g$ with $n$
boundary components,

  * •

the vanishing cycle corresponding to each Lefschetz singularity is in the
collection $\mathcal{C}$,

  * •

the vanishing cycle corresponding to each achiral Lefschetz singularity is in
the collection $\mathcal{C}^{\prime}$.

Then, $\pi:X\to D^{2}$ admits a Lefschetz fibration embedding into the trivial
Lefschetz fibration $\tilde{\pi}:D^{4}\times D^{2}\to D^{2}.$

###### Proof.

Even though the preferred embedding $f_{pr}:\Sigma_{g,n}\to D^{4}$ is not a
flexible embedding, we can use the flexibility of Dehn twits
$d_{a_{i}},d_{a_{i}}^{-1},d_{b_{i}},d_{b_{i}}^{-1}$ and any $h$ which is
product of positive and negative powers of the Dehn twists $d_{a_{i}}$ and
$d_{b_{i}}$ together with the arguments given in the proof of Theorem 6 and
13, to obtain the required embedding.

∎

######  Remark 30.

Similar result to Theorem 29 can be proved when $\pi:X\to D^{2}$ is an achiral
Lefschetz fibration with closed fibers.

Using the arguments given for Theorem 29, we can show the following result.

######  Theorem 31.

Suppose that $\pi:X\to S^{2}$ is an achiral Lefschetz fibration with the
property that

  * •

the vanishing cycle corresponding to each Lefschetz singularity is in the
collection $\mathcal{C}$,

  * •

the vanishing cycle corresponding to each achiral Lefschetz singularity is in
the collection $\mathcal{C}^{\prime}$.

Then, $\pi:X\to S^{2}$ admits a Lefschetz fibration embedding into the trivial
Lefschetz fibration $\tilde{\pi}:D^{4}\times S^{2}\to S^{2}$ and into
$\tilde{\pi}:S^{4}\times S^{2}\to S^{2}$.

Recall that the mapping class group of a compact orientable surface
$\Sigma_{g,1}$ of genus $g$ with connected boundary is generated by the
Humphries generating curves

$\gamma_{1},\gamma_{2},\alpha_{1},\dots,\alpha_{g},\beta_{1},\dots,\beta_{g-1}$

given in Figure 15. Let
$\mathcal{K}=\\{\gamma_{1},\alpha_{1},\dots,\alpha_{g},\beta_{1},\dots,\beta_{g-1}\\}$.
Consider a collections $\mathcal{H}$ of curves $\gamma$ such that
$h(\gamma)=c$ for some $c\in\mathcal{K}$ and for some diffeomorphism $h$ of
$\Sigma_{g,n}$ which is a product of positive as well as negative powers of
the Dehn twists along the curves in $\mathcal{K}.$ Now, we have the following
theorem.

######  Theorem 32.

Let $\pi:X\to D^{2}$ be a bordered Lefschetz fibration, where $X$ is a
compact, connected, orientable $4$–manifold with non-empty boundary such that
for each regular value, the fiber is a compact, connected, orientable surface
$\Sigma_{g,1}$ of genus $g\geq 1$ with connected boundary. Suppose that all
the vanishing cycles corresponding to the Lefschetz fibration are in the set
$\mathcal{H}$, then there exists a Lefschetz fibration embedding of $X$ into a
trivial Lefschetz fibration $\tilde{\pi}:D^{4}\times D^{2}\to D^{2}.$

###### Proof.

Figure 15. A surface of genus $g$ having connected boundary with the Humphries
generating curves
$\gamma_{1},\gamma_{2},\alpha_{1},\dots,\alpha_{g},\beta_{1},\dots,\beta_{g-1}$.
Figure 16. An embedding of a surface of genus $g>0$ having connected boundary
with the Humphries generating curves
$\gamma_{1},\alpha_{1},\dots,\alpha_{g},\beta_{1},\dots,\beta_{g-1}$. Figure
17. An embedding of a surface of genus $g>0$ with non-empty boundary
constructed using Hopf annuli together with the Humphries generating curves
$\gamma_{1},\alpha_{1},\dots,\alpha_{g},\beta_{1},\dots,\beta_{g-1}$.

This construction is from [17]. We can construct an embedding of the surface
$\Sigma_{g,1}$ in $S^{3}$ as follow: Consider $2g$ mutually disjoint and
unlinked standardly embedded annuli $A_{1},A_{2},\dots,A_{2g}$ in $S^{3}$,
i.e. the boundary components of each $A_{i}$ are unknots and unlinked. Let
$A_{i}=C_{i}\times[-1,1]$, where $C_{i}=S^{1}$, for each $i$. Consider two
properly embedded disjoint arcs $a_{i}=\theta_{i}\times[-1,1]$ and
$a_{i}^{\prime}=\theta_{i}^{\prime}\times[-1,1]$ in $A_{i}$, for each $i$. Let
$R_{i}=a_{i}\times[-1,1]$ and $R_{i}^{\prime}=a_{i}^{\prime}\times[-1,1]$ be
two disjoint rectangular neighborhoods of $a_{i}$ and $a_{i}^{\prime}$ in
$A_{i}$. Now, we glue $A_{i}$ to $A_{i+1}$ along $R^{\prime}_{i}$ and
$R_{i+1}$ so that $a_{i}^{\prime}\times\\{-1,1\\}=\partial
a_{i+1}\times[-1,1]$ to get an embedding of a surface $\Sigma_{g,1}$ of genus
$g$ with connected boundary, see Figure 16. One can observe that the Humphries
generating curves $\alpha_{i}$’s, $\beta_{j}$’s and $\gamma_{1}$ correspond to
the central curves of the distinct annuli $A_{j}$’s as shown in Figure 16.

In the above construction, we replace all the annuli $A_{i}$ by the positive
Hopf annuli $H_{i}^{+}$ see Figure 17. Thus, we get an embedding $f$ of
$\Sigma_{g,1}$ in $S^{3}$ such that each of the Humphries generating curves
$\alpha_{i}$’s, $\beta_{j}$’s and $\gamma_{1}$ admit a positive Hopf annulus
regular neighbourhood. We can consider this $S^{3}$ as
$S^{3}\times\frac{1}{2}$, where $S^{3}\times[0,1]$ is a collar of the boundary
$\partial D^{4}=S^{3}\times 0$ in $D^{4}.$ We can see that this embedding can
be made a proper embedding $f_{pr}$ in $D^{4}$ such that the curves
$\gamma_{1},\alpha_{1},\dots,\alpha_{g},\beta_{1},\dots,\beta_{g-1}$ given in
Figure 15 are in the standard position with respect to the embedding $f_{pr}$.
Note that the embedding $f_{pr}$ of $\Sigma_{g,1}$ in $D^{4}$ is not flexible
in $D^{4}$ but the Dehn twists along the curves
$\gamma_{1},\alpha_{1},\dots,\alpha_{g},\beta_{1},\dots,\beta_{g-1}$ are
flexible in $D^{4}$.

Now, we can use the flexibility of Dehn twits $d_{c},d_{c}^{-1}$ where
$c\in\mathcal{K}$ and any $h$ which is product of positive and negative powers
of the Dehn twists $d_{c^{\prime}},c^{\prime}\in\mathcal{K}$ together with the
arguments given in the proof of Theorem 6 and 13, to obtain the required
embedding.

∎

######  Remark 33.

Note that by plumbing a negative Hopf annuli instead of a positive Hopf annuli
in the proof of Theorem 32 for appropriate vanishing cycle,s we can produce
embeddings of certain bordered achiral Lefschetz fibration into a trivial
Lefschetz fibration $\tilde{\pi}:D^{4}\times D^{2}\to D^{2}.$ In fact, one can
consider various cases.

###  6.2. Embedding of closed orientable $4$–manifolds in $S^{4}\times S^{2}$
and $S^{4}\tilde{\times}S^{2}$

Now, we shall show that every closed, connected, orientable $4$–manifold
admits a smooth embedding in $S^{4}\times S^{2}$ as well as in
$S^{4}\tilde{\times}S^{2}.$ We recall the following result from [4]

######  Theorem 34 (Etnyre–Fuller).

Let $X$ be a closed, connected, orientable $4$–manifold. Then, we may write
$X=Y_{1}\cup Y_{2}$, where each $Y_{i}$ is a $2$–handlebody which admits an
achiral Lefschetz fibration over $D^{2}$ with bounded fibers of the same
genus, and with the induced open books on $\partial Y_{1}=-\partial Y_{2}$
coinciding.

######  Remark 35.

We can assume that vanishing cycles of each Lefschetz fibration described in
the above result to be non-nullhomologus.

Let $f_{1}:Y_{1}\to D^{2}$ and $f_{2}:Y_{2}\to D^{2}$ be two achiral Lefschetz
fibrations with bounded fibers such that each $f_{i}$ induces an open book
structure on the common boundary $\partial Y_{1}=-\partial Y_{2}$ and the open
book structures on $\partial Y_{1}=-\partial Y_{2}$ coincide. We can stabilize
the achiral Lefschetz fibrations so that each has fibers with connected
boundary. We can reconstruct $X$ from the pieces $Y_{1}$ and $Y_{2}$ as
follows: We first glue along the pages of the open books, by forming

$W=Y_{1}\cup_{f_{1}^{-1}(\partial D^{2})=f_{2}^{-1}(\partial D^{2})}Y_{2}.$

Now, we have an achiral Lefschetz fibration with bounded fibers $f_{1}\cup
f_{2}:W\to S^{2}$. Note that each fiber of these achiral Lefschetz fibrations
has connected boundary. Therefore, $\partial W=S^{1}\times S^{2}=S^{1}\times
D_{1}^{2}\cup S^{1}\times D^{2}_{2},$ where $S^{1}\times D^{2}_{i}$ is
$\partial Y_{i}\setminus f^{-1}_{i}(\partial D^{2})$. Hence, if we glue
$S^{1}\times D^{2}_{1}$ to $S^{1}\times D^{2}_{2}$ in $\partial W$, we get the
$4$-manifold $X$. Note that gluing an $S^{1}\times D^{3}$ to $W$ will produce
the same result as gluing $S^{1}\times D^{2}_{1}$ to $S^{1}\times D^{2}_{2}$.
Hence, we have $X=W\cup S^{1}\times D^{3}$.

######  Theorem 36.

Every closed, connected, orientable $4$-manifold $X$ admits a smooth embedding
into $S^{4}\times S^{2}$ as well as into $S^{4}\tilde{\times}S^{2}$.

###### Proof.

Let $X=Y_{1}\cup Y_{2}$ be a closed, connected, orientable smooth
$4$–manifold, where each $Y_{i}$ is a $2$–handlebody which admits an allowable
achiral Lefschetz fibrations $f_{1}:Y_{1}\to D^{2}$ and $f_{2}:Y_{2}\to D^{2}$
with fibers $\Sigma$ having the same genus and the connected boundary, and
with the induced open books on $\partial Y_{1}=-\partial Y_{2}$ coinciding.
Existence of such a decomposition follows from the Theorem 34.

Now, we construct the proper Lefschetz fibration embeddings $g_{i}$ of the
Lefschetz fibrations $f_{1}:Y_{1}\to D^{2}$ and $f_{2}:Y_{2}\to D^{2}$ into
two different copies of trivial Lefschetz fibrations
$\tilde{\pi}_{i}:Q_{i}=\mathcal{D}E(m)\times D^{2}\to D^{2}$, $i=1,2$, as
described in Theorem 27. The Lefschetz fibration embedding of $f_{i}:Y_{i}\to
D^{2}$ into $\tilde{\pi}_{i}:\mathcal{D}E(m)\times D^{2}\to D^{2}$ gives an
open book embedding of the open book of $M_{i}=\partial Y_{i}$ into the open
book of $N_{i}=\partial(\mathcal{D}E(m)\times D^{2})$. Note that

$N=\partial(\mathcal{D}E(m)\times D^{2})=\begin{cases}\begin{split}S^{2}\times
S^{3}\end{split}&\text{when m is even,}\\\
\begin{split}S^{2}\tilde{\times}S^{3}\end{split}&\text{when m is
odd.}\end{cases}$

We have $N_{1}=-N_{2}$ and the open books on $N_{1}$ and $N_{2}$ coincides.
Note that ${f_{i}}_{|_{M_{i}}}:M_{i}\setminus(\partial\Sigma\times
D^{2})\to\partial D^{2}=S^{1}=\\{e^{i\theta}:0\leq\theta\leq 2\pi\\}$ and
$\tilde{\pi}_{i|_{N_{i}}}:N_{i}\setminus(\partial\mathcal{D}E(m)\times
D^{2})\to S^{1}$ are locally trivial fiber bundle over $S^{1}$ such that
$\tilde{\pi}_{i|_{N_{i}}}\circ g_{i|_{M_{i}}}=f_{i|_{M_{i}}}.$ Let
${f_{i}}^{-1}_{|_{M_{i}}}(0)=\Sigma_{0}=f_{pr}(\Sigma)$ be the page of the
open book on $M_{i}$ and
$\tilde{\pi}^{-1}_{i|_{N_{i}}}(0)=\mathcal{D}E(m)_{pr}$ be the page of the
open book on $N_{i}$. Let $\phi_{i}$ and $\Phi_{i}(\sim id)$ be the monodromy
on the open books of $M_{i}$ and $N_{i}$ respectively. Then, we have
$f_{pr}^{-1}\circ\Phi_{i}\circ f_{pr}=\phi_{i}$. As $\Phi_{i}$ is isotopic to
the identity map of $\mathcal{D}E(m)$, we have
$\tilde{\pi}_{i|_{\tilde{\pi}_{i}^{-1}(\partial
D^{2})}}:\tilde{\pi}_{i}^{-1}(\partial D^{2})\to S^{1}$ is a trivial
$\mathcal{D}E(m)$ bundle over $S^{1}.$

Suppose that $h:\Sigma\to\Sigma$ is the identification of the page
$\Sigma_{0}$ of $M_{1}$ and $M_{2}$ which gives identification on the pages of
the open books on $M_{1}$ and $M_{2}.$ Since $f_{pr}:\Sigma\to\mathcal{D}E(m)$
is a flexible embedding, there exists a diffeomorphism
$\Phi:\mathcal{D}E(m)\to\mathcal{D}E(m)$ which is isotopic to the identity map
of $\mathcal{D}E(m)$ such that $f_{pr}^{-1}\circ\Phi\circ f_{pr}=h.$ We use
$\Phi$ to identify the page
$\tilde{\pi}^{-1}_{i|_{N_{i}}}(0)=\mathcal{D}E(m)_{pr}$ of the open books of
$N_{1}$ and $N_{2}$ which will use to identify the pages of the open books on
$N_{i}.$

Now, we glue $Q_{1}$ and $Q_{2}$ along the pages of open books of $N_{1}$ and
$N_{2}$ using the identification $\Phi$ to get a trivial Lefschetz fibration
$\tilde{\pi}_{1}\cup\tilde{\pi}_{2}:\mathcal{D}E(m)\times S^{2}\to S^{2}$.
Note that $\mathcal{D}E(m)\times S^{2}$ contains the proper embedding of
$X\setminus S^{1}\times D^{3}$ of $X$. Now, we glue
$\partial\mathcal{D}E(m)\times D^{3}$ to $\mathcal{D}E(m)\times S^{2}$ such
that we get an embedding of $X$ in $Y=\mathcal{D}E(m)\times
S^{2}\cup\partial\mathcal{D}E(m)\times D^{3}$

Note that $Y=\partial(\mathcal{D}E(m)\times
D^{3})=\partial\mathcal{D}E(m)\times D^{3}\cup\mathcal{D}E(m)\times S^{2}$.
But, $\mathcal{D}E(m)\times D^{3}$ is a ($D^{2}$–bundle over $S^{2}$) $\times
D^{3}=\mathcal{D}E(m)\bigoplus(S^{2}\times D^{3})$ which is a $D^{5}$–bundle
over $S^{2}$. Hence, $Y$ is the boundary of a $D^{5}-$ bundle over $S^{2}$,
i.e., $Y$ is a $S^{4}-$bundle over $S^{2}.$ But a $S^{4}-$bundle over $S^{2}$
is completely determined by $\pi_{1}(SO(5))=Z_{2}$. From this one can see
that,

$Y=\begin{cases}\begin{split}S^{4}\times S^{2}\end{split}&\text{when m is
even,}\\\ \begin{split}S^{4}\tilde{\times}S^{2}\end{split}&\text{when m is
odd.}\end{cases}$

This completes the proof.

∎

Now, we give another proof of a theorem of Hirsch [9] which states that every
closed orientable $4$-manifold embeds in $\mathbb{R}^{7}$.

######  Corollary 37.

Every closed, connected, orientable $4$-manifold is smoothly embedded in
$\mathbb{R}^{7}$.

###### Proof.

By Theorem 36, every closed, connected, orientable smooth $4$-manifold
smoothly embeds in $S^{4}\times S^{2}$. As $S^{4}$ sits inside
$\mathbb{R}^{7}$ with the trivial normal bundle $S^{4}\times D^{3}$, we have
smooth embedding of $S^{4}\times S^{2}$ in $\mathbb{R}^{7}$. Hence, every
closed, connected, orientable $4$-manifold smoothly embeds in
$\mathbb{R}^{7}.$ ∎

##  7\. Lefschetz fibration on non-orientable $4$–manifolds

Recently, in [14] Miller and Ozbagci discuss the Lefschetz fibration on a
compact non-orientable $4$–manifold $X$ with non-empty boundary which is a
handlebody without $3$– and $4$–handles. They showed that any non-orientable
$4$–dimensional handlebody without $3-$ and $4-$handles admits a Lefschetz
fibration over the $2-$disc $D^{2}$ whose regular fibers are compact non-
orientable surface with non-empty boundary. We will review Lefschetz fibration
on non-orientable $4$–manifolds.

###  7.1. Lefschetz fibration

Now, we discuss the notion of Lefschetz fibration.

######  Definition 38.

(Lefschetz fibration on non-orientable $4$–manifold) Let $X$ be a compact,
connected, non-orientable smooth $4$–manifold with non-empty boundary. A
smooth map $\pi:X\to D^{2}$ is said to be a Lefschetz fibration if

  1. (1)

the map $\pi$ has finitely many critical (or singular) points $q_{i},$ all lie
in the interior of $X$ with critical (or singular) values $f(q_{i})=p_{i}$,
$1\leq i\leq n$ and $p_{i}\neq p_{j}$, for $i\neq j$,

  2. (2)

for each critical point $q_{i}$ and its corresponding critical value $p_{i}$,
there exist coordinate charts $U_{q_{i}}=\mathbb{C}^{2}$ around $q_{i}=(0,0)$
and $V_{p_{i}}=\mathbb{C}$ around $p_{i}=0$ in $X$ and $D^{2}$ respectively
such that the restriction of $\pi$ on $U_{q_{i}}$ is the map $\pi:U_{q_{i}}\to
V_{p_{i}}$ defined as $\pi(z_{1},z_{2})=z_{1}z_{2}.$ Such a singularity is
called Lefschetz singularity.

  3. (3)

for a regular value $p$, the fiber is a compact connected non-orientable
surface with non-empty boundary.

Now, let us discuss the $2$–handle description near a singular point of a
Lefschetz fibration $\pi:X^{4}\to D^{2}$ of a non-orientable $4$–manifold $X$.
Suppose that $q_{1},q_{2},...,q_{n}$ and $p_{1},p_{2},...,p_{n}$ are the
critical points and the critical values of $\pi$ respectively, such that
$\pi(q_{i})=p_{i}$. First, consider the trivial Lefschetz fibration
$\pi:D^{2}\times\Sigma\to D^{2}$, where $\Sigma$ is a compact non-orientable
surface with non-empty boundary. We attach $2$–handles along the $2$–sided
simple closed curves $\gamma_{i}$ – one for each critical point $q_{i}$ –
embedded in the distinct regular fibers along the boundary
$\partial(D^{2}\times\Sigma)$ of $D^{2}\times\Sigma$. The framing of each
$2$–handle is given by the $\pm 1$ framing with respect to the surface
framing. Note that there is no canonical orientation on the regular
neighbourhood of the attaching curve. We will choose some orientation on the
regular neighbourhood of the attaching curve $\gamma_{i}$ and we will consider
the framing of the attaching $2-$handle $\pm 1$ accordingly. The local
monodromy around the critical value is a Dehn twist along attaching curve
$\gamma_{i}$ which we call the _vanishing cycle_ corresponding to the critical
value $p_{i}$. The choice of the right(positive) handed Dehn twist and the
left(negative) handed Dehn twist depends on the choice of the orientation on
the regular neighbourhood of the vanishing cycle, i.e., for a fixed
orientation of the regular neighbourhood of a $2$–sided curve if we attach a
$2$–handle with $-1$ (respectively, $+1$) framing, then we can assume the
local monodromy corresponds to the positive (respectively, the negative) Dehn
twist along the vanishing cycle in the surface $\Sigma$. If a vanishing cycle
$\gamma$ appears twice corresponding to two distinct critical values and if
the $2$–handles are attached along the vanishing cycle $\gamma$ – lying in the
different regular fibers in $\partial(D^{2}\times\Sigma)$ – with framing $+1$
and $-1$ with respect to the choice of the surface framing in the regular
neighbourhood of $\gamma$ on $\Sigma$, then the Dehn twists $d_{\gamma}$ and
$d_{\gamma}^{-1}$ will appear in the boundary monodromy description of the
Lefschetz fibration. Here onwards, we will fix some orientation on the regular
neighbourhood of each vanishing cycle on $\Sigma.$

Kobayashi and Omori in [11] proved that $\mathcal{M}CG(\Sigma)$ is generated
by the Dehn twists along the $2$–sided simple closed curves $\alpha_{i}$,
$\beta_{i}$, $\alpha_{ij}$, $\rho_{ij}$, $\sigma_{ij}$,
$\sigma^{\prime}_{ij}$, $\sigma$ as shown in Figure 18 and the $Y$–
homeomorphism obtained by sliding $\mu_{1}$ along $\alpha_{1}$ once on the
$Y$–space.

Figure 18.  The non-orientable surface $\Sigma$ of genus $g$ with $n$ boundary
components with the generators of the mapping class group.

Now, we prove the following theorem.

######  Theorem 39.

Let $\pi:X\to D^{2}$ be a bordered Lefschetz fibration, where $X$ is a
compact, connected, non-orientable $4$–manifold with non-empty boundary such
that the handle decomposition of $X$ does not admit $3$– and $4$–handles.
Suppose that for each regular value, the fiber is a compact, connected, non-
orientable surface $\Sigma_{g,n}$ of genus $g\geq 1$ with $n$ boundary
components. If the boundary monodromy is a product of only positive powers of
Dehn twists along the vanishing cycles, then there exists a Lefschetz
fibration embedding of $X$ into a trivial Lefschetz fibration
$\tilde{\pi}:\mathcal{D}E(m)\times D^{2}\to D^{2}.$

First, we define the following notion.

Any simple closed curve $\gamma$ in $\Sigma_{g}$ is said to be in _the achiral
standard position_ with respect to the embedding $f_{pr}$ if $f_{pr}(\gamma)$
can be isotoped to a curve $\gamma^{\prime}$ in $f_{pr}(\Sigma_{g})\cap
S^{3}\times 1$ such that $\gamma^{\prime}$ admits a regular neighbourhood
$\nu(\gamma^{\prime})$ in $f_{pr}(\Sigma_{g})$ which is a negative Hopf
annulus in $S^{3}\times 1$. Any such curve will satisfy the following
conditions.

  * •

the curve $\gamma^{\prime}$ admits a regular neighbourhood $\nu(\gamma)$ in
$f_{pr}(\Sigma_{g})$ which is a Hopf annulus in $S^{3}\times 1$,

  * •

$\gamma^{\prime}$ bounds the disc in $\mathcal{D}E(m)$ which intersects
$f_{pr}(\Sigma_{g})$ only in $\gamma^{\prime}$,

  * •

a tubular neighbourhood $\mathcal{N}(D)=D^{2}\times D^{2}$ intersects
$f_{pr}(\Sigma_{g})$ in a regular neighbourhood $\nu(\gamma^{\prime})$ of
$\gamma^{\prime}$ in $f_{pr}(\Sigma_{g})$,

  * •

we can choose coordinates $(z_{1},z_{2})$ on $\mathcal{N}(D)$ such that
$\nu(\gamma^{\prime})=g^{-1}(1)\cap f_{pr}(\Sigma_{g})$, where
$g:\mathbb{C}^{2}\to\mathbb{C}$ is a map defined by
$g(z_{1},z_{2})=z_{1}z_{2}$ and the plane
$\mathbb{R}^{2}=\\{(z_{1},z_{2})\in\mathbb{C}^{2}:z_{1}=x+iy,z_{2}=x-iy\\}$ in
$\mathbb{C}^{2}$ intersects $\mathcal{N}(D)$ in the disc $D$ and the surface
$\Sigma$ in $\gamma^{\prime}.$ Note that $\gamma^{\prime}$ is the circle
$x^{2}+y^{2}=1$ and the disc $D$ is $x^{2}+y^{2}\leq 1$ in $\mathbb{R}^{2}.$

Note that Remark 11 also true in this setup.

###  7.2. Flexible embedding of a non-orientable surface in $\mathcal{D}E(m)$

We describe the non-orientable surface $\Sigma_{g,n}$ of genus $g$ with $n$
boundary components as follows: Consider a surface $S_{g+n}$ obtained from a
$2$–sphere $S$ by removing the interiors of $g+n$ disjointly embedded discs,
say $A_{1},A_{2},\dots,A_{g},D_{1},\dots,D_{n}$. If we attach $g$ disjoint
Möbius bands $\mu_{1},\dots,\mu_{g}$ to $\partial A_{1},\dots,\partial A_{g}$
respectively, then we get the non-orientable surface $\Sigma$ of genus $g$
with $n$ boundary components as shown in Figure 18. Equivalently, $\Sigma$ can
obtained from $S_{g+n}$ by antipode identification on the $\partial A_{i}$,
for each $i$.

Figure 19. An embedding of the non-orientable surface $\Sigma$ of genus $g$
with $n$ boundary components with the curve $\eta$ in $\mathcal{D}E(m)$.
$\alpha_{1}\\#_{b}\eta$ is an isotopy of $\alpha_{1}$ along the disc in the
handle of $\mathcal{D}E(m).$

Now, we describe a proper flexible embedding of the non-orientable surface
$\Sigma$ of genus $g$ with $n$ boundary components in $\mathcal{D}E(m)$. This
construction is from [3]. Consider a $4$–ball $D^{4}$. Consider a $2$-sphere
$S$ which is the boundary of a $3$–ball in $S^{3}\times\frac{1}{2}$ in the
collar $S^{3}\times[0,1]$ of $\partial D^{4}$ with $S^{3}\times 1=\partial
D^{4}$. We remove $g+n$ disjointly embedded $2$–discs, say
$A_{1},A_{2},\dots,A_{g},D_{1},\dots,D_{n}$ from $S$. We attach $g$ right
half-twisted bands to $\partial A_{1},\dots,\partial A_{g}$ and a right (or
left) full-twisted band to the $\partial D_{1}$ in $S^{3}\times\frac{1}{2}$.
We denote the new boundary components by
$b_{1},\dots,b_{g},l^{\prime}_{1},l_{1},l_{2},\dots,l_{n}$. We attach disjoint
discs $D_{1}^{\prime},\dots,D_{g}^{\prime}$ to $b_{1},\dots b_{g}$,
respectively such that $D_{i}^{\prime}\subset
S^{3}\times[\frac{1}{2},\frac{3}{4}]$ and $D_{i}^{\prime}\cap
S^{3}\times\frac{1}{2}=b_{i}$. We denote the new surface obtained above by
$S^{\prime}$ whose boundary components are $l^{\prime}_{1},l_{1},\dots,l_{n}$.
Consider the surface $S^{\prime\prime}$ which is obtained from $S^{\prime}$ by
considering the union of $S^{\prime}$ with $n+1$ cylinders, namely
$l^{\prime}_{1}\times[\frac{1}{2},1]$ and $l_{i}\times[\frac{1}{2},1]$ for
each $i=1,2,\dots,n$. Now, we attach a $2$–handle $H^{2}$ to $D^{4}$ along
$l^{\prime}_{1}\times 1$ with framing an integer $m$ to obtain
$\mathcal{D}E(m)$. Note that $l^{\prime}_{1}\times 1$ bounds a disc $D$ in the
$2$–handle such that the interior of $D$ is disjoint from $S^{\prime\prime}$.
We attach this disc to $S^{\prime\prime}$ to obtain a proper embedding $f$ of
$\Sigma$ with boundary components $l_{1},l_{2},\dots,l_{n}$ in
$\mathcal{D}E(m)$. We perturb this embedding slightly to make it into a smooth
and proper embedding. For flexibility of this embedding, refer [3].

We call this embedding as the preferred embedding $f_{pr}$ of $\Sigma$ in
$\mathcal{D}E(m).$

Note that we have a preferred embedding of $\Sigma$ in $\mathcal{D}E(m)$ such
that

  1. (1)

each of the curves $\alpha_{1},\dots,\alpha_{g-1},\alpha_{ij}$’s has a Hopf
annulus as a regular neighborhood and is in the standard position with respect
to $f_{pr}$,

  2. (2)

each of the curves $\beta_{1},\dots,\beta_{n-1},\sigma_{ij}$’s, $\rho_{ij}$’s
has a regular neighborhood a standardly embedded annulus,

  3. (3)

each of the curves $\sigma^{\prime}_{ij}$’s, $\sigma$ has a regular
neighborhood a two full-twisted annulus.

Note that we can isotope the curve
$\beta\in\\{\beta_{1},\dots,\beta_{n-1},\sigma_{ij},\rho_{ij}\\}$ to
$\beta\\#_{b}\eta$ so that regular neighborhood of $\beta\\#_{b}\eta$ is a
Hopf annulus as shown in the Figure 19. Here, $\eta$ is the curve which goes
over the the full twisted band exactly once and does not meet any Möbius band
$\mu_{i}$ and $\beta\\#_{b}\eta$ denotes the band connected sum of $\beta$ and
$\eta$. We will choose a band which does not meet any Möbius band $\mu_{i}$ of
the surface. Moreover, each curve $\beta$ is in the standard position with
respect to the embedding $f_{pr}$. Note that the orientations on $\nu(\beta)$
and $\nu(\eta)$ will induce an orientation on $\nu(\beta\\#_{b}\eta)$.

Now, while constructing the embedding of $\Sigma$ in $\mathcal{D}E(m)$, we can
attach a left full-twisted band to the boundary component $\partial D_{1}$ as
shown the Figure 18 so that the regular neighborhood of each of the curves
$\sigma^{\prime}_{ij}\\#_{b}\eta$ and $\sigma\\#_{b}\eta$ is a Hopf annulus.
For, these regular neighborhoods will have three full-twists but two of them
cancel each other to get a Hopf annulus neighborhood. Note that any $2$-sided
separating curve shown in Figure 18 and any curve which does not meet any
Möbius band $\mu_{i}$ can be isotoped to be in the standard position with
respect to $f_{pr}$ by considering band connected sum with the curve $\eta$.

Figure 20.  Separating curves on the non-orientable surface $\Sigma^{\prime}$
of genus $g$

If a $2-$sided simple closed curve $\gamma$ in $\Sigma$ is not in the standard
position with respect to $f_{pr}$, then we can construct an embedding of
$\Sigma$ as described in the proof of Theorem 6 such that $\gamma$ is in the
standard position with respect to that embedding, according to whether the
given curve is separating or non-separating. The following fact is also true
for non-orientable surfaces: Given any two $2$-sided non-separating simple
closed curves, there exists a diffeomorphism of a non-orientable surface
$\Sigma$ which maps one to another and preserves the given orientations of the
regular neighbourhoods. Also, for any two $2$-sided separating simple closed
curves, there exists a diffeomorphism of a non-orientable surface $\Sigma$
which maps one to another and preserves the given orientations of the regular
neighbourhoods, provided they separate the surface $\Sigma$ into surfaces with
the same genera and the same boundary components. So, if the given curve
$\gamma$ is non-separating, then we shall consider a diffeomorphism of
$\Sigma$ which maps $\gamma$ to say, $\alpha_{1}$ shown in Figure 18. If the
given curve $\gamma$ is separating, then we shall consider a diffeomorphism of
$\Sigma$ which maps $\gamma$ to a separating curve which does not meet any
Möbius band $\mu_{i}$ shown in Figure 18 and separates the surface $\Sigma$
into the surfaces with the same genera and the same number of boundary
components. As the embedding $f_{pr}$ is flexible, we can get a new embedding,
say $f_{1}$ of $\Sigma$ in $\mathcal{D}E(m)$ such that the curve $\gamma$ is
in the standard position with respect to the embedding $f_{1}$.

###  7.3. Lefschetz fibration embedding for non-orientable $4$–manifolds

Suppose $q_{1},q_{2},...,q_{k}$ be the critical points of the Lefschetz
fibration $\pi:X\to D^{2}$ and $p_{1},p_{2},...,p_{k}$ be the corresponding
critical values in $D^{2}$ such that $\pi(q_{i})=p_{i}$. Suppose that the
regular fiber at $p\in D^{2}$ is identified with the compact, connected non-
orientable surface $\Sigma=\Sigma_{g,n}$ of genus $g$ and with $n$ boundary
components. Let $\gamma_{1},\gamma_{2},...,\gamma_{k}$ be the vanishing cycles
corresponding to the critical values $p_{1},p_{2},...,p_{k}$ respectively. We
shall fix the orientation of the regular neighbourhood $\nu(\gamma_{i})$ of
each vanishing cycle $\gamma_{i}$ in $\Sigma$ as well as on the regular
neighbourhood $\nu(\eta)$ of the curve $\eta$.

Note that in the handle description of $X$ corresponding to the Lefschetz
fibration $\pi:X\to D^{2}$, the $2$–handles are attached to the vanishing
cycle $\gamma$ with framing $+1$ with respect to the surface framing. This
will corresponds to the Dehn twist $d_{\gamma}$ in the boundary monodromy of
the Lefschetz fibration which is described as a product of Dehn twist along
the vanishing cycles. Then, using the flexibility of $f_{pr}$ as well as the
flexibility of embeddings with respect to which the vanishing cycles are in
the standard position, we can proceed as the arguments given in the proof of
Theorem 6, to get the Lefschetz fibration embedding of the Lefschetz fibration
$\pi:X\to D^{2}$ into the trivial Lefschetz fibration
$\tilde{\pi}:\mathcal{D}E(m)\times D^{2}\to D^{2}.$ This proves Theorem 39.

######  Theorem 40.

Let $\pi:X\to D^{2}$ be a bordered Lefschetz fibration, where $X$ is a compact
connected non-orientable $4$–manifold with non-empty boundary without $3$– and
$4$–handles in the handle decomposition such that for each regular value the
fiber is a compact connected non-orientable genus $g>2$ surface $\Sigma$ with
connected boundary. If the vanishing cycle corresponding to the Lefschetz
singularity is not the boundary parallel, then there exists a Lefschetz
fibration embedding of $X$ into a trivial Lefschetz fibration
$\tilde{\pi}:\mathcal{D}E(m)\times D^{2}\to D^{2}.$

Proof of Theorem 40. We can consider the setup as in the last theorem. The
result follows from the last theorem if in the handle description of $X$
corresponding to the Lefschetz fibration $\pi:X\to D^{2}$, the $2$–handles are
attached to the vanishing cycle $\gamma$ with framing $+1$ with respect to the
surface framing.

Now, we shall consider the case where in the handle description of $X$
corresponding to the Lefschetz fibration $\pi:X\to D^{2}$, the $2$-handles are
attached to the vanishing cycles with framing $-1$ with respect to the surface
framing. Let $\gamma$ be such a vanishing cycle in $\Sigma.$ This corresponds
to the Dehn twist $d_{\gamma}^{-1}$ in the boundary monodromy of the Lefschetz
fibration. Then, we proceed as in the following two steps.

Step -I:

If $\gamma$ is non-separating, we can consider a diffeomorphism $h_{1}$ which
maps $\gamma$ to one of the non-separating generating curve $\alpha_{i}$. In
particular, we can send $\gamma$ to the curve $\alpha_{1}$. If $\gamma$ is
separating, we can consider a diffeomorphism $h_{1}$ which maps $\gamma$ to
one of the separating curve which does not meet any Möbius band $\mu_{i}.$

Figure 21. Left of the figure shows a separating curve $c$ bounding a
subsurface $\Sigma^{\prime}$ and a Möbius band $\mu_{j}$. Right of the figure
shows the result of the slide diffeomorphism of $\Sigma$ which slides
$\Sigma^{\prime}$ along the core of the Möbius band $\mu_{j}$. Note that the
orientation of the regular neighbourhood of $c$ changes by this slide
diffeomorphism.

Step -II:

Now, we need to consider a diffeomorphism $h_{2}$ of the surface $\Sigma$
which will map $h_{1}(\gamma)$ to $h_{2}(h_{1}(\gamma))$ such that
$h_{2}h_{1}\circ(d_{\gamma})^{-1}\circ(h_{2}h_{1})^{-1}=d_{h_{2}h_{1}(\gamma)}.$
This means, we need a diffeomorphism $h_{2}$ such that $h_{2}\circ h_{1}$
should change the orientation when restricted to the regular neighbourhoods of
$\gamma$ and $h_{2}(h_{1}(\gamma))$ and is in the standard position with
respect to $f_{pr}$. Then, using the flexibility of the diffeomorphism
$h_{2}\circ h_{1}$, we get a flexible embedding $f_{2}$ of $\Sigma$ such that
$\gamma$ is in the standard position with respect to $f_{2}.$

Now, we will define the diffeomorphism $h_{2}$. First, we define the notion of
a slide diffeomorphism of $\Sigma$.

Slide diffeomorphism: Consider a Möbius band $\mu_{j}$ with core $\lambda$ as
shown in Figure 18 and consider a connected, compact subsurface
$\Sigma^{\prime}$ disjoint from $\mu_{j}$ and bounded by a curve $C$ in
$f_{pr}(\Sigma)$ which does not meet any Möbius band $\mu_{i}$. Let $B$ be a
$3$–ball in $S^{3}\times\frac{1}{2}$ such that $C\subset\partial B$ and $B\cap
f_{pr}(\Sigma)$ is the subsurface $\Sigma^{\prime}$. Now, there is an isotopy
$\rho_{t}$ of $S^{3}\times\frac{1}{2}$ which slides $B$ along the core curve
of the Möbius band $\mu_{j}$ and come back to its original position. Note that
this isotopy of $\mathcal{D}E(m)$ induces a diffeomorphism on $\Sigma$, for
more details refer [7] and [3]. We call this diffeomorphism a _slide
diffeomorphism_ of the surface $\Sigma$ along $\mu_{j}$ which slides only the
subsurface $\Sigma^{\prime}$ in $f_{pr}(\Sigma)$.

Note that the slide diffeomorphism will not disturb the curves inside the
surface $\Sigma^{\prime}$ but changes the orientation on the regular
neighbourhoods of the curves, for example, refer Figure 21. Also, note that
the slide diffeomorphisms are flexible in $f_{pr}$.

Suppose that $h_{1}(\gamma)$ is separating and $h_{1}(\gamma)$ bounds a
subsurface $\Sigma^{\prime}$ which does not contain a Möbius band say,
$\mu_{j}$ in $\Sigma$. Consider the slide diffeomorphism $h_{2}$ of the
surface which is obtained by sliding the subsurface $\Sigma^{\prime}$ once
along the core curve of the Möbius band $\mu_{j}$, refer Figure 21. Note that
the slide diffeomorphism $h_{2}$ preserves the curve $h_{1}(\gamma)$ but
changes the orientation of the regular neighbourhood of $h_{1}(\gamma).$ Now,
we can isotope $h_{2}(h_{1}(\gamma))$ along the Hopf band to get
$h_{2}(h_{1}(\gamma))\\#_{b}\eta$ which admits the required Hopf annulus
regular neighbourhood. Now, using the flexibility of $h_{2}\circ h_{1},$ we
get an embedding $f_{2}$ of $\Sigma$ in $\mathcal{D}E(m)$ such that $\gamma$
is in the standard position with respect to this embedding $f_{2}$.

Suppose that $h_{1}(\gamma)$ is non-separating. With out loss of generality,
we can assume that $h_{1}(\gamma)=\alpha_{1}$. As the genus $g>2,$ we can use
the same idea described in the last paragraph to get $h_{2}$. We shall
consider the separating curve which bounds the subsurface $\Sigma^{\prime}$
such that $\Sigma^{\prime}$ contains only the Möbius bands $\mu_{1}$ and
$\mu_{2}$ and no other Möbius bands $\mu_{j}$. Then, we consider the
diffeomorphism $h_{2}$ which slides $\Sigma^{\prime}$ over some Möbius band
$\mu_{j},\ j\neq 1,2$. This choice of $h_{2}$ will provide us the required
orientation of the regular neighbourhood of $\alpha_{1}.$

Thus, corresponding to each vanishing cycle, we can construct a proper
flexible embedding $f$ of $\Sigma$ in $\mathcal{D}E(m)$ such that the
vanishing cycle is in the standard position with respect to $f$.

Now, we can use the arguments given in Theorem 6, to get the Lefschetz
fibration embedding of the Lefschetz fibration $\pi:X\to D^{2}$ into the
trivial Lefschetz fibration $\tilde{\pi}:\mathcal{D}E(m)\times D^{2}\to
D^{2}.$ This proves Theorem 40.

######  Remark 41.

Similar to Theorem 32, one can show that a huge collection of non-orientable
bordered Lefschetz fibrations Lefschetz fibration embeds into the trivial
Lefschetz fibration $\tilde{\pi}:D^{4}\times D^{2}\to D^{2}.$

###  7.4. Embedding of closed non-orientable $4-$manifolds

In [14], Miller and Ozbagci proved that if $X$ is a closed, non-orientable,
connected $4$–manifold which admits a Lefschetz fibration over $S^{2}$
equipped with a section of square $\pm 1$, then $X$ admits a decomposition
$X=W\cup_{\partial}V$, where both $W$ and $V$ admit bordered Lefschetz
fibrations over $D^{2}$ such that the induced open books coincide on their
common boundary $\partial W=\partial V$, refer the proof of Theorem 1.10 in
[14].

We have the following result.

######  Theorem 42.

Let $X$ be a closed non-orientable $4$–manifold. Suppose that $X$ admits a
decomposition $X=W\cup_{\partial}V$, where both $W$ and $V$ admit bordered
Lefschetz fibrations over $D^{2}$ so that the induced open books coincide on
their common boundary $\partial W=\partial V$. If both $V$ and $W$ satisfy the
hypothesis of either Theorem 39 or Theorem 40, then $X$ admits a smooth
embedding into $S^{4}\times S^{2}$ as well as into $S^{4}\tilde{\times}S^{2}.$

###### Proof.

The proof follows from the similar arguments given in the proof of Theorem 36.

∎

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