1 Introduction
An open book decomposition of a closed connected oriented n-manifold M is a fibration
$\pi :M\setminus B\to S^1$
, where B is a codimension-two oriented submanifold of M with a trivial normal bundle. The submanifold B is called the binding of the open book. The closure of each fibre of
$\pi $
is called a page of the open book and each page is a codimension-one submanifold of M with boundary B. Alexander, in [Reference Alexander1], showed that every closed oriented
$3$
-manifold admits an open book. For more details on open books, we refer to the survey [Reference Etnyre2].
Open book decomposition on odd-dimensional manifolds is an important tool in studying contact structures on manifolds. By a contact structure on a closed oriented
$(2n+1)$
-manifold M, we mean a maximally nowhere integrable hyperplane field on
$M.$
Giroux, in [Reference Giroux4], showed that every co-oriented contact structure on a closed oriented
$(2n+1)$
-manifold is supported by an open book. In [Reference Thurston and Winkelnkemper11], Thurston and Winkelnkemper constructed contact structures on closed oriented
$3$
-manifolds using open books. In [Reference Giroux4], Giroux showed that there is a one-to-one correspondence between the isotopy classes of co-oriented contact structures on a closed oriented 3-manifold M and the open book decompositions of M up to positive stabilisations. By a positive stabilisation operation on an open book of
$ M$
, we mean a plumbing of a positive Hopf band to the page of the open book. Open book decomposition is a useful tool in studying
$3$
-manifolds. For instance, an open book decomposition of a manifold naturally gives a Heegaard splitting of the manifold, where the Heegaard surface is the closure of the union of two pages.
Open book embeddings of closed oriented
$3$
-manifolds into the open books of
$S^2\times S^3$
as well as into
$S^2 \,\tilde \times \, S^3$
with pages a
$2$
-disc bundle over
$S^2$
and monodromy the identity map are studied in [Reference Pancholi, Pandit and Saha9]. Results in [Reference Pancholi, Pandit and Saha9] are extended for nonorientable closed
$3$
-manifolds in [Reference Ghanwat, Pandit and A3]. We say that a smooth manifold M with a given open book decomposition admits an open book embedding in an open book decomposition of a smooth manifold N if there exists an embedding of M in N such that, as a submanifold of N, the given open book decomposition on M is compatible with the open book decomposition of N. For the precise definition of an open book embedding, we refer to Definition 2.3. However, the existence of an open book for a closed oriented
$3$
-manifold, such that its open book embeds in the trivial open book of the
$5$
-sphere
$S^5$
with pages the
$4$
-disk
$D^4$
and monodromy the identity map of
$D^4$
, is not known.
In this note, we study open book embeddings of closed oriented
$3$
-manifolds in the
$5$
-sphere
$S^5$
. We prove the following result.
Theorem 1.1. Let M be a closed connected oriented
$3$
-manifold. There exists a knot K in M such that the manifold
$M^{\prime }$
obtained from M, by performing an integer surgery, admits an open book decomposition and this open book embeds into the trivial open book of the
$5$
-sphere
$S^5.$
2 Preliminary
First, we recall the notions necessary for this note. All the manifolds and maps we consider are smooth.
Definition 2.1. An open book decomposition of a closed connected oriented manifold M is a pair
$(B, \pi ),$
with the following properties.
-
(1) B is an oriented codimension-two submanifold in M with a trivial normal bundle called the binding of the open book.
-
(2)
$ \pi : M \setminus B\rightarrow S^1$
is a locally trivial fibration such that the fibration
$\pi $
in a tubular neighbourhood 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
$D^2$
. -
(3) For each
$\theta \in S^1, \pi ^{-1}(\theta )$
is the interior of a compact codimension-one submanifold
$N_{\theta }\subset M$
and
$\partial N_\theta = B$
. The submanifold
$N=N_\theta $
, for any
$\theta $
, is called the page of the open book.
The fibration
$\pi :M\setminus B\to S^1$
with fibre N is determined by N and the monodromy
$\phi $
of the fibration up to conjugation by an orientation preserving diffeomorphism, that is, the fibre bundle
$M\setminus B$
is canonically isomorphic to the mapping torus
$$ \begin{align*}\mathcal{MT}(N,\phi)= N\times[0,1]/\sim,\end{align*} $$
where
$\sim $
is the equivalence relation identifying
$(x,1)$
with
$(\varphi (x),0)$
. From the above definition, we can see that M is diffeomorphic to
$$ \begin{align*}\mathcal{MT}(N, \phi) \cup_{id} B \times D^2.\end{align*} $$
Thus, an open book decomposition of
$M $
is determined, up to diffeomorphism of
$ M$
, by the topological type of the page N and the isotopy class of the monodromy which is an element of the mapping class group of N. The mapping class group of a manifold N with nonempty boundary is the group of isotopy classes of orientation-preserving diffeomorphisms of N which are the identity in a collar neighbourhood of the boundary.
This naturally leads to the notion called an abstract open book defined as follows.
Definition 2.2. An abstract open book associated with an n-manifold M is a pair
$(\Sigma , \phi ),$
where
$\Sigma $
is a compact connected oriented
$(n-1)$
-manifold with nonempty boundary and
$\phi $
is an orientation-preserving diffeomorphism of
$\Sigma $
such that M is diffeomorphic to
$$ \begin{align*}M_{(\Sigma,\phi)}=\mathcal{MT}(\Sigma, \phi) \cup_{id} \partial \Sigma \times D^2,\end{align*} $$
where
$id$
denotes the identity map of
$\partial \Sigma \times S^1.$
The map
$\phi $
in the above definition is called the monodromy of the abstract open book. Note that the mapping class of
$\phi $
determines M uniquely up to diffeomorphism. We will denote the manifold M with an abstract open book decomposition
$(\Sigma ,\phi )$
by
$ \mathcal {A}ob(\Sigma ,\phi )$
. One can easily see that given an abstract open book decomposition of M, we can clearly associate an open book decomposition of M with pages
$\Sigma $
and vice versa. We will not generally distinguish between open books and abstract open books. For more details on open books, see [Reference Etnyre2].
Let us recall the notion of an open book embedding.
Definition 2.3. Let
$M^k$
and
$N^l$
be manifolds with open book decompositions
$(B_1, \pi _1)$
and
$(B_2, \pi _2)$
, respectively. We say an embedding
$f: M \hookrightarrow N$
is an open book embedding of
$(B_1, \pi _1)$
into
$(B_2, \pi _2)$
if f embeds
$B_1$
into
$B_2$
such that
$\pi _2\circ f=\pi _1.$
Similarly, we can also define an abstract open book embedding.
Definition 2.4. Let
$M = \mathcal {A}ob (\Sigma _1, \phi _1)$
and
$N = \mathcal {A}ob( \Sigma _2, \phi _2 )$
be two abstract open books. We say that there exists an abstract open book embedding of M into N if there exists a proper embedding f of
$\Sigma _1$
into
$\Sigma _2$
such that
$\phi _1$
is isotopic to
$ f^{-1} \circ \phi _2 \circ f.$
It is clear from the definition that an abstract open book embedding produces an embedding for the associated open book and vice versa.
3 Proof of Theorem 1.1
In this section, we discuss our proof of Theorem 1.1. Recall that given an embedded circle c in the interior of a surface
$\Sigma ,$
a Dehn twist
$d_c$
(or
$d_c^{-1}$
) on
$\Sigma $
along the circle c is a diffeomorphism which is the identity outside a neighbourhood of c and is a full twist on an annular neighbourhood of c. We begin by stating the following lemma, which is proved in [Reference Etnyre2].
Lemma 3.1. Let M be a closed oriented
$3$
-manifold. Let
$M=\mathcal Aob(\Sigma , \phi )$
. Suppose L is a knot sitting on a page
$\Sigma $
of the open book. Then, the new manifold
$ M^{\prime }$
obtained by
$\pm 1$
surgery along L, with respect to the page framing, admits an open book
$\mathcal Aob(\Sigma ,\phi \circ d_L^{\mp 1})$
with pages
$\Sigma $
and monodromy
$\phi \circ d_L^{\mp 1}$
.
The following lemma will be needed in the proof of Theorem 1.1.
Lemma 3.2. Let M be a closed oriented
$3$
-manifold. Let
$M=\mathcal {A}ob(\Sigma , \phi )$
be an abstract open book of M. Suppose that K is a knot in M such that
$K=\{x\}\times [0,1]/(x,1)\sim (\phi (x),0)$
in M for some interior point
$x\in \Sigma $
. Let
$D_x$
be an open disc neighbourhood of x in
$ \Sigma $
such that
$\phi |_{D_x}=id$
and let c be the curve parallel to
$\partial D_x$
in
$\Sigma \setminus D_x$
. Then, an integer n surgery along K gives a new manifold
$M^{\prime }=\mathcal {A}ob(\Sigma ^{\prime },\phi ^\prime )$
with an open book decomposition having the surgery dual of K as one of the binding components, where the surface
$\Sigma ^\prime =\Sigma \setminus D_x$
is the page and the map
$\phi ^{\prime }=\phi |_{\Sigma ^{\prime }}\circ d_c^{-n}$
is the monodromy of the open book of
$M^{\prime }$
.
Proof. The knot K is transverse to each page of the open book
$\mathcal Aob(\Sigma ,\phi )$
and intersects each page exactly once (see Figure 1). Suppose
$n=0$
. Then, performing
$0$
surgery along K is equivalent to removing
$D_x$
from each page and filling the resulting boundary
$\partial D_x\times S^1$
of
$M\setminus \mathcal N(K)$
by
$\partial D_x\times D^2$
using the identity map. Here,
$\mathcal {N}(K)= D_x\times K$
denotes a tubular neighbourhood of K in
$M.$
From this, one can easily see that the manifold
$M^{\prime }$
, obtained from M by performing
$0$
surgery along K, admits an open book with pages
$\Sigma ^{\prime }=\Sigma \setminus D_x$
and monodromy
$\phi ^{\prime }=\phi |_{\Sigma ^{\prime }}.$
Blowing up operations preformed on the knot K.
Suppose that n is a positive integer. From blowing up operations (see Figure 1), one can see that performing integer n surgery along K is equivalent to performing
$-1$
surgery along n copies of a circle c parallel to
$\partial D_x$
lying in n distinct pages of the open book
$\mathcal {A}ob(\Sigma ,\phi )$
and performing
$0$
surgery along K. Using Lemma 3.1, one can see that the manifold
$M^{\prime }$
obtained from M by performing a positive integer n surgery along K admits an open book with pages
$\Sigma ^{\prime }=\Sigma \setminus D_x$
and monodromy
$\phi ^{\prime }=\phi |_{\Sigma ^{\prime }}\circ d_c^{-n}$
.
The Humphries generating curves
$b,a_1,\ldots ,a_{2g}$
on the surface
$\Sigma $
of genus g with connected boundary. The Dehn twist along these curves generates the mapping class group of
$\Sigma $
. The union of the regular neighbourhoods of the Humphries generating curves is a subsurface S of
$\Sigma $
with two boundary components. One of the boundary components of S is parallel to
$\partial \Sigma $
and the other boundary component bounds a disc D in
$\Sigma $
.
Similarly, when n is a negative integer, we can see that the surgered manifold
$M^{\prime }$
admits an open book with pages
$\Sigma ^{\prime }=\Sigma \setminus D_x$
and monodromy
$\phi ^{\prime }=\phi |_{\Sigma ^{\prime }}\circ d_c^{n}$
.
Remark 3.3. Suppose that
$\Sigma ^{\prime }$
has an arc
$\alpha $
joining the boundary component
$\partial D_x$
to some component of
$\partial \Sigma $
such that
$\phi $
fixes
$\alpha $
pointwise. Then, the open book
$\mathcal Aob(\Sigma ^{\prime }, \phi ^\prime )$
of
$M^\prime $
, obtained from M by
$\pm 1$
surgery along K, is just a stabilisation of
$\mathcal Aob(\Sigma ,\phi )$
. In this case,
$M^\prime $
is diffeomorphic to
$M.$
Let M be a closed connected oriented
$3$
-manifold. Recall that M admits an open book decomposition with connected binding (see [Reference Myers8]). Let
$M=\mathcal {A}ob(\Sigma ,\phi )$
, where
$\Sigma $
is a compact connected oriented surface with connected boundary. By considering certain stabilisations of the abstract open book of
$\mathcal Aob(\Sigma ,\phi )$
if required, we can assume that
$\Sigma $
has the genus
$g\geq 3$
. The monodromy
$\phi $
of the open book is an element in the mapping class group of
$\Sigma $
. Humphries [Reference Humphries and Fenn6] showed that the mapping class group of a closed surface
$\tilde \Sigma $
of genus
$g\geq 3$
is generated by the Dehn twist along
$2g+1$
closed curves
$b, a_1,a_2,\ldots ,a_{2g}$
, as shown in Figure 2, where the closed surface
$\tilde \Sigma $
is obtained from
$\Sigma $
by gluing a disc along the boundary. We call these curves
$b, a_1,a_2,\ldots ,a_{2g}$
in
$\Sigma $
the Humphries generating curves and the corresponding Dehn twists the Humphries generators of the mapping class group of
$\Sigma $
. Johnson [Reference Johnson7] showed that the
$2g+1$
Dehn twists about
$b,a_1,a_2,\ldots ,a_{2g}$
on
$\Sigma $
also generate the mapping class group of
$\Sigma $
.
Consider a regular neighbourhood of each Humphries generating curve in
$\Sigma $
. The union S of the regular neighbourhoods of the Humphries generating curves can be considered as a surface obtained by plumbing
$2g+1$
annuli, as shown in Figure 2. The surface S has two boundary components. One of the boundary components of S is parallel to
$\partial \Sigma $
, that is, they bound an annulus A in
$\Sigma $
, and the other boundary component of S bounds a disc
$D=\Sigma \setminus (S\cup A)$
in
$\Sigma $
, as shown in Figure 2. Note that
$\Sigma \setminus D$
is diffeomorphic to
$S.$
The monodromy
$\phi $
is supported in S and hence it is the identity on the disc D.
Let
$x\in D$
be an interior point in
$D\subset \Sigma .$
Consider the knot
$K=\{x\}\times [0,1]/(x,1)\sim (\phi (x),0)$
in M. As discussed in Lemma 3.2, we get an open book
$\mathcal Aob(\Sigma ^{\prime },\phi ^\prime )$
of
$M^\prime $
obtained by an integer n surgery along K in M. Here,
$\Sigma ^\prime =S$
and
$\phi ^\prime =\phi \circ d_c^{-n}$
, where c is a curve in
$\Sigma ^\prime $
parallel to
$\partial D.$
Now, we shall show that the open book
$M^{\prime }=\mathcal Aob(\Sigma ^\prime ,\phi ^\prime )$
embeds into the trivial open book of
$S^5$
. We need to recall the following notion.
Definition 3.4. A diffeomorphism
$\phi $
of a surface F is said to be a flexible diffeomorphism, with respect to a proper embedding f of
$\Sigma $
in a
$4$
-manifold X, if there exists a diffeomorphism
$\Phi $
of X isotopic to the identity map of X (also the identity near the boundary if X has nonempty boundary) such that
$\phi $
is isotopic to
$f^{-1} \circ \Phi \circ f$
.
One can easily observe that if a diffeomorphism
$\phi $
of F is isotopic to a diffeomorphism
$\psi $
of F and
$\phi $
is flexible with respect to an embedding f of F in X, then
$\psi $
is also flexible in X with respect to f. To prove the existence of an (abstract) open book embedding of
$\mathcal Aob(\Sigma ^{\prime },\phi ^{\prime })$
in the trivial open book of
$S^5$
, we need to find an embedding f of
$\Sigma ^{\prime }$
in
$D^4$
such that
$\phi ^{\prime }$
is flexible in
$D^4$
with respect to
$f.$
As
$\phi ^{\prime }$
is isotopic to a product of powers of Humphries generators and the Dehn twist
$d_c$
, it is enough to construct a proper embedding f of
$\Sigma ^{\prime }$
in
$D^4$
such that the Humphries generators and the Dehn twist
$d_c$
are flexible in
$D^4$
with respect to
$f.$
3.1 Construction of an embedding of
${\Sigma}^{\!\prime}$
in
$D^4$
A Dehn twist
$d_\gamma $
along an embedded circle
$\gamma $
in
$\Sigma ^{\prime }$
is supported in an annular neighbourhood of
$\gamma $
in
$\Sigma ^{\prime }$
. In [Reference Hirose and Yasuhara5], it is shown that there exists a (proper) embedding of an annulus
$A=S^1\times [0,1]$
in
$D^4$
such that the Dehn twist along the central curve of A is flexible and hence so is each power of this Dehn twist. This follows from the fact that there exists a flow
$\Phi _t$
on the
$3$
-sphere
$S^3$
associated to the open book decomposition of
$S^3$
with pages a Hopf annulus and monodromy the Dehn twist along the central curve of the Hopf annulus such that the time
$1$
map
$\Phi _1$
on
$S^3$
induces the Dehn twist along the central curve on each page of the open book of
$S^3$
. We can choose any Hopf annulus page
$\mathcal A$
as an embedding of the annulus A in
$S^3=\partial D^4\times \tfrac 12\subset \partial D^4\times [0,1]$
, where
$\partial D^4\times [0,1]$
is a collar of
$\partial D^4$
in
$D^4$
. Using the flow
$\Phi _t$
, we can construct a diffeomorphism of
$D^4$
which is isotopic to the identity and induces the Dehn twist along the central curve on the embedded Hopf annulus
$\mathcal A$
. For more details, see [Reference Hirose and Yasuhara5, Reference Pancholi, Pandit and Saha9]. Note that
$S^3$
admits an open book with pages a positive Hopf annulus as well as an open book with pages a negative Hopf annulus. So, we will construct a proper embedding of the surface
$\Sigma ^{\prime }$
in
$D^4$
such that each of the curves
$b, a_1,\ldots ,a_{2g}$
and the boundary parallel curve c admits either a positive or a negative Hopf annulus regular neighbourhood in the embedded
$\Sigma ^{\prime }\subset D^4$
.
Consider a collar
$S^3\times [0,1]$
of the
$4$
-ball
$D^4$
with
$\partial D^4=S^3\times 0.$
Consider an embedding
$S_1$
of the surface
$\Sigma ^{\prime }$
, which is obtained by plumbing
$2g$
positive Hopf annuli
$A_1,\ldots ,A_{2g}$
with central curves
$a_1,a_2,\ldots ,a_{2g}$
and one negative Hopf annulus with central curve b in
$S^3\times \tfrac 12$
, as shown in Figure 3. Recall that
$\Sigma ^{\prime }$
has two boundary components. We denote these components by
$\partial _1=\partial \Sigma $
and
$\partial _2=\partial D.$
We make this embedding a proper embedding f of
$\Sigma ^{\prime }$
by attaching cylinders
$\partial _i\times [0,\tfrac 12]$
to
$S_1$
along
$\partial _i$
, for
$i=1,2$
, and smoothing out the corners to get a smooth embedding. By construction, we can see that a regular neighbourhood of each of the Humphries generating curves
$a_1,\ldots ,a_{2g}$
in
$f(\Sigma ^{\prime })$
is a positive Hopf annulus in
$S^3\times \tfrac 12\subset D^4$
and a regular neighbourhood of the Humphries generating curve b in
$f(\Sigma ^{\prime })$
is a negative Hopf annulus
$S^3\times \tfrac 12$
. A regular neighbourhood of the curve c parallel to the boundary
$\partial _2$
in
$f(\Sigma ^{\prime })$
has two positive full twists and one negative full twist. Two of the twists cancel each other to get a positive Hopf annulus neighbourhood for the curve c in
$S^3\times \tfrac 12$
(see Figure 3). From this, one can see that the Dehn twists along the curves
$c,b,a_1,\ldots ,a_{2g}$
are flexible in
$D^4$
with respect to the embedding f of
$\Sigma ^{\prime }.$
For more details regarding the flexibility of the Dehn twists, see [Reference Pancholi, Pandit and Saha9]. An alternate argument for the flexibility of the Dehn twist along a simple closed curve in
$\Sigma ^{\prime }$
admitting a Hopf annulus regular neighbourhood in
$S^3\times \tfrac 12$
is given in [Reference Pandit and A10]. For the sake of completeness, we summarise it here.
An embedding of the surface
$\Sigma ^{\prime }=S$
in
$S^3$
, in which each Humphries generating curve as well as the boundary parallel curve c admits a Hopf annulus regular neighbourhood in
$S^3.$
Observe that each curve
$\alpha \in \{c, a_1,\ldots , a_{2g}\}$
bounds a disc
$D_\alpha $
in
$D^4$
which intersects
$f(\Sigma ^{\prime })$
in
$\alpha $
and a tubular neighbourhood
$\mathcal {N}(D_\alpha )=D^2\times D^2$
intersects
$f(\Sigma ^{\prime })$
in a regular neighbourhood
$\nu (\alpha )$
of
$\alpha $
in
$f(\Sigma ^{\prime })$
. We can choose coordinates
$(z_1,z_2)$
on
$\mathcal {N}(D_\alpha )$
such that
$\nu (\alpha )=g^{-1}(1)\cap f(\Sigma ^{\prime })$
, where
$g:\mathbb C^2\to \mathbb C$
is the map defined by
$g(z_1,z_2)=z_1z_2$
. The monodromy of the singular fibration
$g:\mathcal {N}(D_{\alpha })=\mathbb C^2\to \mathbb C= D^2$
with the singular point
$(0,0)$
is the positive Dehn twist
$d_{\alpha }$
along the curve
$\alpha $
which lies in the regular fibre
$g^{-1}(1)\cap \Sigma ^{\prime }=\nu (\alpha ).$
The isotopy of
$\mathcal {N}(D_{\alpha })$
, which produces the positive Dehn twist
$d_{\alpha }$
on
$g^{-1}(1)\cap \Sigma ^{\prime }$
, can be extended to an isotopy
$\Psi _s$
,
$s\in [0,1]$
, of
$D^4$
such that the isotopy is supported in the tubular neighbourhood
$\mathcal {N}(D_{\alpha })$
. Then, we have
$f^{-1}\circ \Psi _1 \circ f=d_{\alpha }$
up to isotopy. We have an isotopy of
$D^4$
supported in the tubular neighbourhood
$\mathcal {N}(D_{\alpha })$
of
$D_{\alpha }$
which produces the Dehn twist
$d_\alpha $
on
$\Sigma ^{\prime }.$
From this, it follows that the Dehn twist
$d_{\alpha }$
is flexible in
$ D^4$
with respect to the embedding
$f.$
The curve b bounds a disc
$D_b$
in
$D^4$
which intersects
$f(\Sigma ^{\prime })$
in b and a tubular neighbourhood
$\mathcal {N}(D_b)=D^2\times D^2$
intersects
$f(\Sigma ^{\prime })$
in a regular neighbourhood
$\nu (b)$
of b in
$f(\Sigma ^{\prime })$
. We can choose coordinates
$(z_1,z_2)$
on
$\mathcal {N}(D_b)$
such that
$\nu (b)=g^{-1}(1)\cap f(\Sigma ^{\prime })$
, where
$g:\mathbb C^2\to \mathbb C$
is the map defined by
$g(z_1,z_2)=z_1\bar {z}_2$
. The monodromy of the singular fibration
$g:\mathcal {N}(D_{b})=\mathbb C^2\to \mathbb C= D^2$
with the singular point
$(0,0)$
is the negative Dehn twist
$d_{b}^{-1}$
along the curve b which lies in the regular fibre
$g^{-1}(1)\cap \Sigma ^{\prime }=\nu (b).$
By arguments similar to those above, there is an isotopy of
$D^4$
supported in the tubular neighbourhood
$\mathcal {N}(D_{b})$
of
$D_{b}$
which produces the Dehn twist
$d_b^{-1}$
on
$\Sigma ^{\prime }.$
Both
$d_b$
and
$d_b^{-1}$
are flexible in
$D^4$
with respect to the embedding
$f.$
It follows that the monodromy
$\phi ^{\prime }$
is flexible in
$D^4$
with respect to the embedding f of
$\Sigma ^{\prime }$
, that is, there exists a diffeomorphism
$\Phi $
of
$D^4$
which is isotopic to the identity map of
$D^4$
and
$f^{-1}\circ \Phi \circ f$
is isotopic to
$\phi ^{\prime }$
. This gives an abstract open book embedding of
$M^{\prime }=\mathcal Aob(\Sigma ^{\prime },\phi ^{\prime })$
into
$S^5=\mathcal {A}ob(D^4,\Phi ).$
This completes the proof of Theorem 1.1.
