Commit 19e84313 authored by Mirko KLUKAS's avatar Mirko KLUKAS

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\begin{document}
\title[]{Open books and exact symplectic cobordisms}
\author{Mirko Klukas}
\address{Mathematisches Institut, University of Cologne, Weyertal 86--90,
D-50931 Cologne, Germany}
\email{mklukas@math.uni-koeln.de}
\date{\today}
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\maketitle
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\begin{abstract}
Given two open books with equal pages we show the existence of an exact symplectic cobordism
whose negative end equals the disjoint union of the contact manifolds associated to the given open books,
and whose positive end induces the contact manifold associated to the open book with the same
page and concatenated monodromy.
Using similar methods we show the existence of strong fillings for contact manifolds
associated with \textit{doubled open books}, a certain class of fiber bundles over the circle obtained
by performing the binding sum of two open books with equal pages and inverse
monodromies. From this we conclude, following an outline by Wendl, that the
complement of the binding of an open book cannot contain any local filling obstruction.
Given a contact $3$-manifold, according to Eliashberg there is a symplectic cobordism to a
fibration over the circle with symplectic fibers. We extend this result to higher
dimensions recovering a recent result by D\"orner--Geiges--Zehmisch. Our cobordisms can also be
thought of as the result of the attachment of a generalized symplectic $1$-handle.
\end{abstract}
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\begin{center}
{\small\textsc{Mathematics Subject Classification 2000:} 53D05, 53D10, 57R17, 57R65.}
\end{center}
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\section*{Introduction}
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Let $\Sigma$ denote a compact, $2n$-dimensional manifold admitting an exact symplectic form
$\omega = d\beta$ and let $Y$ denote the Liouville vector field defined by
$\iota_Y \omega = \beta$. Suppose that $Y$ is transverse to the boundary $\partial \Sigma$,
pointing outwards. These properties are precisely the ones requested
for $\Sigma$ to be a page of an abstract open book in the contact setting.
Given a symplectomorphism $\phi$ of $(\Sigma,\omega)$, equal to the identity
near $\partial\Sigma$, one can, following a construction of Thurston
and Winkelnkemper \cite{MR0375366} or rather its adaption to higher
dimensions by Giroux \cite{MR1957051},
associate a $(2n+1)$-dimensional contact manifold
$M_{(\Sigma,\omega,\phi)}$ to the data $(\Sigma,\omega,\phi)$.
\parskip 0pt
The main result of the present paper is part of the author's thesis
\cite{Klukas-Thesis}.
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\begin{thm}
\label{thm:stein-monoid}
Given two symplectomorphisms $\phi_0$ and $\phi_1$ of $(\Sigma,\omega)$,
equal to the identity near the boundary $\partial\Sigma$, there
is an exact symplectic cobordism whose negative end equals the disjoint union of the contact manifolds $M_{(\Sigma,\omega,\phi_0)}$ and $M_{(\Sigma,\omega,\phi_1)}$, and whose positive end
equals $M_{(\Sigma,\omega,\phi_0 \circ \phi_1)}$. If, in addition, the page $(\Sigma,\omega)$ is a Weinstein manifold, then so is the cobordism.
\end{thm}
%
For $n=1$ the above statement is due to Baker--Etnyre--van Horn-Morris~\cite{2010arXiv1005.1978B} and,
independently, Baldwin~\cite{MR2923174}. The general case of Theorem~\ref{thm:stein-monoid} was
independently obtained by Avdek in \cite{2012arXiv1204.3145}, where the cobordism is associated with
a so-called \textit{Liouville connected sum}. In \cite{Klukas-Thesis} I observed that the cobordism
in Theorem~\ref{thm:stein-monoid} can also be understood as result of the attachment of a
\textit{generalized symplectic $1$-handle} of the form $D^1 \times N(\Sigma)$, where $N(\Sigma)$ denotes a
vertically invariant neighborhood of the symplectic hypersurface $\Sigma$.
I will shed some more light on this in \S\ref{sec:1 handle}.
\parskip 0pt
From the methods introduced in the proof of Theorem~\ref{thm:stein-monoid} we can deduce some
further applications such as the following.
We show the existence of strong fillings for contact manifolds associated with \textit{doubled open books},
a certain class of fibre bundles over the circle obtained
by performing the binding sum of two open books with equal pages and inverse
monodromies (cf.~\S\ref{sec:sym open books}).
%
\begin{thm}
\label{thm:balanced fibration}
Any contact manifold associated to a doubled open book admits an exact symplectic filling.
\end{thm}
%
In dimension $3$ the above statement appeared in \cite{MR3102479},
though details of the argument have not been carried out.
As outlined by Wendl in \cite{MR3102479}*{Remark 4.1} this statement has the following consequence
for \textit{local filling obstructions}, i.e.\ subsets in the likes of overtwisted discs in dimension $3$ that inhibit the existence of a symplectic filling, in arbitrary dimensions.
Similar results in dimension $3$, concerning planar and Giroux torsion,
are presented in \cite{MR3102479} and \cite{MR2737776}.
%
\begin{cor}
\label{cor:filling obstr}
Let $(B,\pi)$ be an open book decomposition of a $(2n + 1)$-dimensional
contact manifold $(M,\xi)$ and let $\mathcal{O} \subset (M,\xi)$
be any local filling obstruction,
then $B$ must intersect $\mathcal{O}$ non-trivially.
\end{cor}
%
Our final result will be the following.
Let $(M,\xi)$ be a closed, oriented, $(2n+1)$-dimensional contact manifold supported by an open book with page $(\Sigma,\omega)$ and monodromy $\phi$.
Suppose further that $(\Sigma,\omega)$ symplectically embeds into a second $2n$-dimensional (not necessarily closed) symplectic manifold $(\Sigma',\omega')$, i.e.
\[
(\Sigma,\omega) \subset (\Sigma',\omega').
\]
Let $M'$ be the symplectic fibration over the circle with fibre $(\Sigma',\omega')$ and monodromy equal to
$\phi$ over $\Sigma \subset \Sigma'$ and equal to the identity elsewhere.
The following theorem has previously been proved by
D\"orner--Geiges--Zehmisch \cite{DGZ}. The proof in the present paper uses slightly different methods (cf.~\S\ref{sec:sympl fibration}).
%
\begin{thm}
\label{cor:symplectic fibration}
There is a smooth manifold $W$ with $\partial W = (-M) \sqcup M'$ and a symplectic
form $\Omega$ on $W$ for which $(M,\xi)$ is a concave boundary component,
and $\Omega$ induces $\omega'$ on the fibers of the fibration $M' \to S^1$.
\end{thm}
%
For $n=1$ we could, for example, choose $\Sigma'$ to be the closed surface obtained by capping
off the boundary components of $\Sigma$. Then Theorem~\ref{cor:symplectic fibration} would
recover one of the main results (Theorem 1.1) in \cite{MR2023279}. The low-dimensional case ($n=1$)
of Theorem~\ref{cor:symplectic fibration} was, using different methods, already carried out in \cite{MR3128981}.
One may think of Theorem~\ref{cor:symplectic fibration} as an extension of the result in \cite{MR2023279}, or \cite{MR3128981} respectively, to higher dimensions.
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\subsection*{Acknowledgements}
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The main results of the present paper are part of my thesis~\cite{Klukas-Thesis}.
I want to thank my advisor Hansj\"org Geiges for introducing me to the world of contact topology.
Furthermore I thank Chris Wendl for perceptive
comments on an earlier version of the paper, in particular for suggesting to add
Corollary~\ref{cor:filling obstr}.
Finally I thank Max D\"orner for bringing Theorem~\ref{cor:symplectic fibration} to my attention and
for useful comments that helped to improve the exposition.
The author was supported by the DFG (German Research Foundation) as fellow of the
graduate training program \textit{Global structures in geometry and analysis}
at the Mathematics Department of the University of Cologne, Germany, and by grant GE 1245/2-1 to H. Geiges.
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\section{Preliminaries}
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\subsection{Symplectic cobordisms}
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Suppose we are given a symplectic $2n$-manifold $(X,\omega)$, oriented by the volume form
$\omega^n$, such that the oriented boundary $\partial X$ decomposes as $\partial
X = (- M_-) \sqcup M_+$, where $-M_-$ stands for $M_-$ with reversed
orientation. Suppose further that in a neighborhood of $\partial X$ there is a
Liouville vector field $Y$ for $\omega$, transverse to the boundary and pointing
outwards along $M_+$, inwards along $M_-$. The $1$-form $\alpha = i_Y\omega$
restricts to $TM_\pm$ as a contact form defining cooriented contact structures
$\xi_\pm$.
\parskip 0pt
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We call $(X,\omega)$ a \textbf{(strong) symplectic cobordism}
from $(M_-,\xi_-)$ to $(M_+,\xi_+)$, with \textbf{convex} boundary $M_+$ and
\textbf{concave} boundary $M_-$. In case $(M_-,\xi_-)$ is empty $(X,\omega)$ is called a
\textbf{(strong) symplectic filling} of $(M_+,\xi_+)$.
If the Liouville vector field is defined not only in a neighborhood of $\partial X$ but
everywhere on $X$ we call the cobordism or the filling respectively \textbf{exact}.
\parskip 0pt
A \textbf{Stein manifold} is an affine complex manifold, i.e.~a complex manifold
that admits a proper holomorphic embedding into $\C^N$ for some large integer $N$.
By work of Grauert \cite{MR0098847} a complex manifold $(X,J)$ is Stein if and only if it
admits an exhausting plurisubharmonic function $\rho\co X \to \R$.
Eliashberg and Gromov's symplectic
counterparts of Stein manifolds are \textit{Weinstein manifolds}.
\parskip 0pt
A \textbf{Weinstein manifold} is a quadruple $(X,\omega,Z,\varphi)$, see
\cite{eliGro-convex_symplectic_manifolds}, where $(X,\omega)$ is an exact
symplectic manifold, $Z$ is a complete globally defined Liouville vector field,
and $\varphi\co X \to \R$ is an exhausting (i.e.~proper and bounded below)
Morse function for which $Z$ is gradient-like.
Suppose $(X,\omega)$ is an exact symplectic cobordism with boundary $\partial X
= (- M_-) \sqcup M_+$ and with Liouville vector field $Z$. We call $(X,\omega)$
\textbf{Weinstein cobordism} if there exists a Morse function $\varphi\co X \to
\R$ which is constant on $M_-$ and on $M_+$, has no boundary critical points on $M_-$ and on $M_+$,
and for which $Z$ is gradient-like.
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\subsection{Open books}
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An \textbf{open book decomposition} of an $n$-dimensional manifold $M$ is a pair
$(B,\pi)$, where
$B$ is a co-dimension $2$ submanifold in $M$, called the
\textbf{binding} of the open book and
$\pi\co M\setminus B \to S^1$ is a (smooth, locally trivial)
fibration such that
each fibre $\pi^{-1}(\varphi)$, $\varphi\in S^1$, corresponds to the interior
of a compact hypersurface $\Sigma_\varphi \subset M$ with
$\partial\Sigma_\varphi = B$.
The hypersurfaces $\Sigma_\varphi$, $\varphi \in S^1$, are called the
\textbf{pages} of the open book.
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\parskip 0pt
In some cases we are not interested in the exact position of the binding or the pages of an open book decomposition inside the ambient space. Therefore, given an open book decomposition $(B,\pi)$ of an $n$-manifold $M$, we could ask for the relevant data to remodel the ambient space $M$ and its underlying open books structure $(B,\pi)$, say up to diffemorphism. This leads us to the following notion.
\parskip 0pt
An \textbf{abstract open books} is a pair $(\Sigma,\phi)$, where $\Sigma$ is a compact hypersurface with non-empty boundary $\partial \Sigma$, called the \textbf{page} and $\phi\co\thinspace \Sigma \to \Sigma$ is a diffeomorphism equal to the identity near $\partial \Sigma$, called the \textbf{monodromy} of the open book.
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Let $\Sigma(\phi)$ denote the mapping torus of $\phi$, that is, the quotient space obtained from $\Sigma \times [0,1]$ by identifying $(x,1)$ with $(\phi(x),0)$ for each $x \in \Sigma$. Then the pair $(\Sigma,\phi)$ determines a closed manifold $M_{(\Sigma,\phi)}$ defined by
\begin{equation}
\label{eqn:abstract open book}
M_{(\Sigma,\phi)} := \Sigma(\phi) \cup_{\id} (\partial \Sigma \times D^2),
\end{equation}
where we identify $\partial \Sigma(\phi) = \partial \Sigma \times S^1$ with $\partial (\partial \Sigma \times D^2)$ using the identity map.
%
Let $B \subset M_{(\Sigma,\phi)}$ denote the embedded submanifold $\partial \Sigma \times \{0\}$. Then we can define a fibration $\pi\co M_{(\Sigma,\phi)}\setminus B \to S^1$ by
\[
\left.
\begin{array}{l}
\lbrack x,\varphi \rbrack \\
\lbrack \theta, r\e^{i\pi\varphi} \rbrack
\end{array} \right\}
\mapsto [\varphi],
\]
where we understand $M_{(\Sigma,\phi)}\setminus B$ as decomposed as in Equation~\ref{eqn:abstract open
book} and $[x,\varphi] \in \Sigma(\phi)$ or $ [\theta, r\e^{i\pi\varphi}] \in
\partial\Sigma \times D^2 \subset \partial\Sigma \times \C$ respectively. Clearly $(B,\pi)$ defines an open book decomposition of $M_{(\Sigma,\phi)}$.
\parskip 0pt
On the other hand, an open book decomposition $(B,\pi)$ of some $n$-manifold
$M$ defines an abstract open book as follows: identify a neighborhood of $B$
with $B \times D^2$ such that $B = B\times\{0\}$ and such that the fibration on
this neighborhood is given by the angular coordinate, $\varphi$ say, on the
$D^2$-factor. We can define a $1$-form $\alpha$ on the complement $M \setminus (B \times
D^2)$ by pulling back $d\varphi$ under the fibration $\pi$, where this time we
understand $\varphi$ as the coordinate on the target space of $\pi$.
The vector field $\partial \varphi$ on $\partial\big(M \setminus (B \times D^2)
\big)$ extends to a nowhere vanishing vector field $X$ which we normalize by
demanding it to satisfy $\alpha(X)=1$. Let $\phi$ denote the time-$1$ map of the
flow of $X$. Then the pair $(\Sigma,\phi)$, with $\Sigma = \overline{\pi^{-1}(0)}$, defines an abstract open book such that$M_{(\Sigma,\phi)}$ is diffeomorphic to $M$.
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\subsection{Compatibility}
\label{sec:compatibility}
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Let $\Sigma$ denote a compact, $2n$-dimensional manifold admitting an exact symplectic form
$\omega = d\beta$ and let $Y$ denote the Liouville vector field defined by
$\iota_Y \omega = \beta$. Suppose that $Y$ is transverse to the boundary $\partial \Sigma$,
pointing outwards.
Given such a triple $(\Sigma,\omega,\phi)$ a construction of Giroux
\cite{MR1957051}, cf.\ also \cite{MR2397738}*{\S 7.3},
produces a contact manifold $M_{(\Sigma,\omega,\phi)}$ whose contact structure is
adapted to the open book in the following sense.
A positive contact structure $\xi = \ker \alpha$ and an open book decomposition $(B,\pi)$ of an $(2n + 1)$-dimensional manifold $M$ are said to be \textbf{compatible}, if
the $2$-form $d\alpha$ induces a symplectic form on the interior $\pi^{-1}(\varphi)$ of each page, defining
its positive orientation, and the $1$-form $\alpha$ induces a positive contact form on $B$.
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\section{Concatenation of open books and symplectic fibrations}
\label{sec:concatenation}
\label{sec:sympl fibration}
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\newcommand{\heightOfSigma}{\boldsymbol{r}}
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The following definitions will turn up in the proofs of all our main results:
let $\Sigma$ denote a compact, $2n$-dimensional manifold admitting an exact symplectic form
$\omega = d\beta$ and let $Y$ denote the Liouville vector field defined by
$\iota_Y \omega = \beta$. Suppose that $Y$ is transverse to the boundary $\partial \Sigma$,
pointing outwards. Denote by $(r,x)$ coordinates on a collar neighborhood
$(-\varepsilon,0] \times \partial \Sigma$ induced by the negative
flow corresponding to the Liouville vector field $Y$.
Let $(\hat{\Sigma},\hat\omega = d\hat{\beta})$ denote the completion of $(\Sigma, \omega)$,
obtained by attaching the positive half
$\big([0,\infty) \times \partial\Sigma, d(\text{e}^r\beta_0) \big)$ of the
symplectization of $(\partial\Sigma, \beta_0 = \beta|_{\partial\Sigma})$.
The Liouville vector field $Y$ extends over $\hat\Sigma$ by $\partial_r$ and we will continue
to denote the extended vector field by $Y$.
Let
$
\heightOfSigma \co \hat\Sigma \to \mathbb{R}_{\geq 0}
$
be a smooth function on $\hat\Sigma$,
satisfying the following properties:
\begin{list}{$\bullet$}{}
\item $\heightOfSigma \equiv 0$ over $\hat\Sigma \setminus \big( (-\varepsilon , \infty ) \times \partial\Sigma \big)$,
\item $\frac{\partial\heightOfSigma}{\partial_r} > 0$ and $\frac{\partial\heightOfSigma}{\partial_x} \equiv 0$ over $\big( (-\varepsilon , \infty ) \times \partial\Sigma \big)$ with coordinates $(r,x)$, and
\item $\heightOfSigma(r,x) = r + 1$ over $\big( [0 , \infty ) \times \partial\Sigma \big)$.
\end{list}
Note that over the collar neighborhood $(-\varepsilon,\infty) \times \partial \Sigma$ the vector field $Y$ is
gradient-like for $\heightOfSigma$.
In order to define the desired Liouville vector fields in our proofs we will need the following ingredient.
For some sufficiently small $\delta >0$ let
$g \colon\thinspace [0,\delta] \to \R$ be a functions satisfying the
following properties:
\begin{list}{$\bullet$}{}
\item $g(y) = 1$, for $y$ near $0$,
\item $g(y) = 0$, for $y$ near $\delta$,
\item $g'(y) \leq 0$, for each $y \in [0,\delta]$.
\end{list}
We are now ready to construct the desired exact symplectic cobordisms of Theorem~\ref{thm:stein-monoid} as well as of
Theorem~\ref{cor:symplectic fibration}.
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\begin{proof}[Proof of Theorem \ref{thm:stein-monoid}]
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The starting point for the desired cobordism will be
the space $\hat\Sigma \times \R^2$ with coordinates $(p, x,y)$. This space is symplectic with symplectic form
\[
\Omega = \hat\omega + dx\wedge dy .
\]
Let $P = P_{a,b,c}$ denote the subset of $\hat\Sigma \times \R^2$ defined by
\[
P := \big\{(p,x,y)\co \heightOfSigma \leq 0, \ x^2 + y^2 \leq c^2 \ \mbox{and} \ (x
\pm b)^2 + y^2 \geq a^2
\big\},
\]
where $a,b,c \in \R$ are some potentially very large constants satisfying $a<b<b+a<c$.
The final choice of these constants will later ensure that our
desired (and yet to be defined) Liouville vector field $Z$ will be transverse to the boundary of the cobordism.
Consider the vector field $Z'=Z'_{b}$ on $\hat\Sigma \times \R^2$ defined by
\begin{equation}
\label{eqn:Z prime}
Z' = Y + X,
\end{equation}
where $X = \big(1 - f'(x)\big)y\thinspace\partial_y +
f(x)\thinspace\partial_x$ and $f:\R \to \R$ is a function satisfying the
following properties:
%
\begin{list}{$\bullet$}{}
\item $f(\pm b) = f(0)=0$,
\item $f'$ has exactly two zeros $\pm x_0$ with $0 < |x_0| < b$,
\item $|f'(x)| < 1$ for each $x\in \R$, and
\item $\lim_{{x \to \pm \infty}}f(x) = \pm \infty.$
\end{list}
%
An easy computation shows that $X$ is a Liouville vector field on
$(\R^2,dx\wedge dy)$ for any function $f$. Hence $Z'$ defines a Liouville
vector field on $\big(\hat\Sigma \times \R^2,\Omega \big)$. Note that $Z'$ is transverse to the boundary of $P$, cf. Figure~\ref{fig:vf X}.
%
\begin{figure}[h]
\center
\includegraphics{P_phi_0_1.pdf}
\put (-193,38){$x$}
\put (-248,88){$y$}
\put (-93,72){$P_+$}
\put (-93,22){$P_-$}
\put (-132,46){$\phi_0$}
\put (-56,46){$\phi_1^{-1}$}
\caption{Left: Flow lines of the Liouville vector field $X$. Right: Construction of $P(\phi_0,\phi_1)$}
\label{fig:vf X}
\end{figure}
%
\parskip 0pt
We are now ready to define the desired exact symplectic cobordism $(W, \Omega ,Z)$.
Start by cutting $P$ along $\{y=0\}$ and re-glue
with respect to $\phi_0$ and $\phi_1$ as follows. Set
$P_\pm := P \cap \{ \pm y \geq 0 \}$ and $P_0 = P \cap \{y=0\}$.
Obviously $P_0$ can be understood as part of the boundary of $P_+$ as well as of $P_-$.
Now consider
\[
P(\phi_0,\phi_1) := ( P_+\sqcup P_- ) /_{\sim_\Phi},
\]
where we identify with respect to the map $\Phi\co P_0 \to P_0$ given by
\[
\Phi( p, x,0) :=
\begin{cases}
( \phi_0(p), x,0 ) & \text{, for $x < - b$,}\\
( p, x,0 ) & \text{, for $|x| < b$,} \\
( \phi_1^{-1}(p), x,0 ) & \text{, for $b < x$.}
\end{cases}
\]
Here we understand the domain of definition of $\Phi$ as part of the boundary of $P_+$
and the target space as part of $P_-$, cf. also Figure~\ref{fig:vf X}.
Note that, since $\phi_0$ and $\phi_1$ are
symplectomorphisms of $(\Sigma,\omega)$, and can be assumed to equal the identity over $(-\varepsilon,0]\times \partial\Sigma$, they extend trivially over $\hat\Sigma$. Furthermore $\Phi$ keeps the $x$-coordinates
fixed, and hence $\Omega$ descends to a symplectic
form on $P(\phi_0,\phi_1)$ which we will continue to denote by $\Omega$.
We are now going to define the Liouville vector field $Z$ on $P(\phi_0,\phi_1)$.
Without any loss of generality
the symplectomorphisms $\phi_0$ and $\phi_1^{-1}$ can be chosen to be exact (cf.~\cite{MR2397738}),
i.e.~we have $\phi_0^*\hat\beta - \hat\beta = d\varphi_0$ and $(\phi_1^{-1})^*\hat\beta - \hat\beta = d\varphi_1$
defining functions $\varphi_0$ and $\varphi_1$ on $\Sigma$, unique up to
adding a constant. Hence we may assume that $\varphi_0$ and $\varphi_1$ vanish over $(-\varepsilon,\infty) \times \partial \Sigma$.
To avoid confusing
indices we will write
\[
\Phi^*\hat\beta - \hat\beta = d\varphi
\]
to summarize these facts. Let $g \colon\thinspace [0, \varepsilon] \to \R$ be the function
as defined at the beginning of the present section.
Over $P_+$ we define $Z = Z_b$ to be given as
\begin{equation}
\label{eqn:Z}
Z = \Big( g(y)\thinspace (T\Phi^{-1})(Y) + \big(1-g(y) \big) \thinspace Y \Big)+
X + g'(y)\varphi(p)\thinspace\partial_x.
\end{equation}
To show that $Z$ is indeed a Liouville vector field we have to take a look
at the Lie derivative of $\Omega$ along $Z$. With the help of the Cartan
formula we compute
\begin{eqnarray*}
\mathcal{L}_{Z}\Omega
&=&
d \big( g\thinspace \Phi^*\hat\beta + (1-g) \thinspace \hat\beta \big)
+ dx \wedge dy
+ d(g'\varphi\thinspace dy) \\
&=&
\big( dg\wedge(\Phi^*\hat\beta) - dg \wedge \hat\beta +
g\thinspace(\Phi^*\hat\omega) + (1-g)\thinspace\hat\omega \big) + dx \wedge dy +
g'\thinspace d\varphi\wedge dy \\
&=&
\big( g'\thinspace dy\wedge(\Phi^*\hat\beta) - g' \thinspace dy \wedge \hat\beta +
g\thinspace \hat\omega + (1-g)\thinspace\hat\omega \big) + dx \wedge dy +
g'\thinspace d\varphi\wedge dy \\
&=&
\big( g'\thinspace dy\wedge d\varphi+
\hat\omega \big) + dx \wedge dy -
g'\thinspace dy \wedge d\varphi \\
&=&
\hat\omega + dx \wedge dy \\ &=& \Omega.
\end{eqnarray*}
Observe that, since $T\Phi(Z|_{P_0}) = Z'|_{P_0}$, we can extend $Z$ over $P_-$ by
$Z'$. In particular $Z$ descends to a vector field on $P(\phi_0,\phi_1)$.
Let $W' = W'_{a,b,c}$ denote the subset of $\hat\Sigma \times \R^2$ defined by
\[
W' := \big\{ (p,x,y)\ | \
\heightOfSigma^2 + (x \pm b)^2 + y^2 \geq a^2
\ \ \mbox{and} \ \
\heightOfSigma^2 + x^2 + y^2 \leq c^2
\big\}
\]
and note that we have $P \subset W'$. Finally we define the symplectic
cobordism $W = W_{a,b,c}$ by
\[
W := ( W'\setminus P ) \cup P(\phi_0,\phi_1).
\]
The boundary of $W$ decomposes as $\partial W = \partial_-W
\sqcup \partial_+W$, where we have
\begin{equation}
\label{eqn:boundary of W}
\partial_-W = \heightOfSigma^2 + (x \pm b)^2 + y^2 = a^2 \} \ \ \mbox{and} \ \
\partial_+W = \{ \heightOfSigma^2 + x^2 + y^2 = c^2 \} .
\end{equation}
We do not have to worry about the well-definedness of the function
$\boldsymbol{r}$ on $P(\phi_0,\phi_1) \subset W$ since $\phi_0$ and $\phi_1$ can be assumed to equal the
identity over $(-\varepsilon, \infty) \times \partial\Sigma$, which is the only region where $\heightOfSigma$ is
non-trivial.
Observe that yet we cannot fully ensure that
the Liouville vector field $Z$ is transverse to $\partial W$ pointing inwards
along $\partial_-W$ and outwards along $\partial_+W$. However the only problem is the
last term in $Z$, namely the term $g'(y)\varphi(p)\thinspace\partial_x$.
We tame this deviation as follows:
up to this point we have not fixed the constants $a,b,c$ yet. By choosing $a,b,c$ sufficiently
large the deviation induced by $g'(y)\varphi(p)\thinspace\partial_x$ becomes non-essential and
the Liouville vector field $Z$ becomes transverse to $\partial W$ pointing inwards
along $\partial_-W$ and outwards along $\partial_+W$.
It remains to show that
we indeed have $\partial_-W = M_{(\Sigma,\omega,\phi_0)} \sqcup
M_{(\Sigma,\omega,\phi_1)}$ and $\partial_+W = M_{(\Sigma,\omega,\phi_0 \circ
\phi_1)}$.
We start with the negative boundary components of $W$. Denote by $(\partial_-W)_{\pm} \subset \partial_-W $ the two distinct components of the negative boundary of $W$ (cf. Equation~\ref{eqn:boundary of W}).
Set
\[
B_\pm := \{ x = \pm b, y = 0 \} \subset (\partial_-W)_{\pm}
\]
and note that $B_\pm$ has trivial normal bundle. Further note that
the complement $(\partial_-W)_{\pm}\setminus B_\pm$ admits a fibration over the circle defined by
\[
\pi_\pm (p,x,y) := \frac{(x \mp b,y)}{ \| (x\mp b,y) \| } \in S^1.
\]
This definition is compatible with the gluing induced by $\Phi$ and
defines an open book decomposition of $(\partial_-W)_{\pm}$ with pages diffeomorphic to $\Sigma$ and
monodromy given by $\phi_0$ and $\phi_1$ respectively. In the definition of $X$ (cf. Equation~\ref{eqn:Z prime}) we can choose
the underlying function $f$ such that in a neighborhood of $\pm b$ it is given by $f(x) = \frac{1}{2}x$.
Therefore in a neighborhood of the binding $B_\pm \subset (\partial_-W)_{\pm}$ the $1$-form $\iota_Y \Omega$ is given by
\begin{equation}
\label{eqn:induced form is standard}
\iota_Z \Omega = \beta + \tfrac{1}{2}\big(x \thinspace dy - y \thinspace dx\big).
\end{equation}
In addition, pulling back $d (\iota_Z \Omega) = \Omega$ to a fiber $(\pi_\pm)^{-1}(\theta)$, $\theta \in S^1$, yields the given symplectic form $\omega$.
The projection on the $p$-coordinate yields a symplectomorphism of each page $(\pi_\pm)^{-1}(\theta)$ endowed with the symplectic form induced by $\Omega$ to the subset $\Sigma_{\heightOfSigma < a} \subset (\hat\Sigma, \hat\omega)$.
Note that, since we are just interested in the induced contact structure on $\partial_-W$ (not the whole cobordism $W$), we are allowed to set $a=1$.
This shows that the contact structure on $(\partial_-W)_{\pm}$ induced by $\iota_Z \Omega$ is the same as given by the generalized Thurston-Winkelnkemper construction (cf.~\cite{MaxThesis}).
The argument for $\partial_+W$ is almost similar -- except for the fact that in a neighborhood of the binding the vector field $X$ (cf. Equation~\ref{eqn:Z prime}), or rather the underlying function $f$, is not of the right form. However since we are just interested in the induced contact structure on $\partial_+W$ (not the whole cobordism $W$) we are allowed to change $f$ accordingly: choose an isotopy $(f_t)_{t\in[0,1]}$ with $f_0\equiv f$, $f_1(x)=\frac{1}{2}x$ and such that for each $f_t$ the induced vector field $Z$ (cf. Equation~\ref{eqn:Z}) stays transverse to $\partial_+W$. The we obtain a contact structure for each $t \in [0,1]$ on $\partial_+W$ all of which are contactomorphic by Gray stability. Set
\[
B := \{ x = y = 0 \} \subset \partial_+W
\]
and note that $B$ has trivial normal bundle. Further note that
the complement $( \partial_+W )\setminus B$ admits a fibration over the circle defined by
\[
\pi(p,x,y) := \frac{(x,y)}{ \| (x,y) \| } \in S^1.
\]
Following the same line of arguments as for $\partial_-W$ one concludes that the contact structure on $\partial_+W$ induced by $\iota_Z \Omega$ is the same as given by the generalized Thurston-Winkelnkemper construction (cf.~\cite{MaxThesis}).
%
\begin{figure}[h]
\center
\includegraphics{higher_dims.pdf}
\put(-180,100){$\heightOfSigma$}
\put(-95,18){$x$}
\put(-171,55){$y$}
\put(-117,61){$M_{(\Sigma,\omega,\phi_1)}$}
\put(-287,61){$M_{(\Sigma,\omega,\phi_0)}$}
\put(-300,92){$M_{(\Sigma,\omega,\phi_0 \circ \phi_1)}$}
\caption{Schematic picture of the symplectic cobordism constructed in Theorem
\ref{thm:stein-monoid}.}
\label{fig:the cobordism}
\end{figure}
%
\end{proof}
%
As mentioned above we will now briefly sketch an alternative approach to Theorem~\ref{thm:stein-monoid} utilizing a
\textit{generalized symplectic $1$-handle} as defined in \S\ref{sec:1 handle}.
A similar approach was independently followed by Avdek in \cite{2012arXiv1204.3145}.
%
%
%------------------------------------------------------------------------------------------
\begin{proof}[Sketch of the alternative approach]
%------------------------------------------------------------------------------------------
%
%
Suppose we are given two $(2n +1 )$-dimensional contact manifolds $(M_0,\xi_0)$ and $(M_1,\xi_1)$.
Suppose further that they are associated with compatible open books $(\Sigma, \omega, \phi_0)$
and $(\Sigma, \omega, \phi_1)$ with equal pages.
For $i=0,1$ let $\pi_i\colon\thinspace M_i \setminus B_i \to S^1$ denote the induced fibrations.
Note that the subsets $ \pi_i^{-1}\big((-\varepsilon,\varepsilon)\big) \setminus (B_i \times D^2_\varepsilon) \subset (M_i,\xi_i)$, $i=0,1$,
define an embedding
\[
S^0 \times N_\varepsilon(\Sigma) \hookrightarrow M_0 \sqcup M_1,
\]
where $N_\varepsilon(\Sigma)$ denotes a neighborhood of $\Sigma$ as described in \S\ref{sec:1 handle}.
We can understand this as the attaching region $\mathcal{N}$ of a
generalized $1$-handle $H_\Sigma$ as described in \S\ref{sec:1 handle}.
Attaching $H_\Sigma$ to the positive end of a symplectization of $M_0 \sqcup M_1$ one can show that
we end up with a cobordism whose positive end equals the contact manifold associated to
$(\Sigma, \omega, \phi_0 \circ \phi_1)$, cf.\ Figure~\ref{fig:concatenation}.
In the proof of Theorem~\ref{thm:stein-monoid} the subset $W_{|x|<b} \subset W$ can be understood
as a perturbed instance of a handle $H_\Sigma$.
%
\begin{figure}[h]
\center
\includegraphics{mon_concatenation.pdf}
\put (-190,56){$(\Sigma, \phi_0)$}
\put (-295,56){$(\Sigma, \phi_1)$}
\put (-292,32){\small{$\phi_0$}}
\put (-174,32){\small