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}.

%

\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.

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.

%

\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.

%

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

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 $M$.

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 $B$.

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}_{\geq0}

$

be a smooth function on $\hat\Sigma$,

satisfying the following properties:

\begin{list}{$\bullet$}{}

\item$\heightOfSigma\equiv0$ over $\hat\Sigma\setminus\big((-\varepsilon , \infty)\times\partial\Sigma\big)$,

\item$\frac{\partial\heightOfSigma}{\partial_r} > 0$ and $\frac{\partial\heightOfSigma}{\partial_x}\equiv0$ 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$g'(y)\leq0$, 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

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\leq0, \ 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

$P_\pm := P \cap\{\pm y \geq0\}$ 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)

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

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}).