@@ -204,7 +204,7 @@ 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').

(\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.

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@@ -213,7 +213,11 @@ D\"orner--Geiges--Zehmisch \cite{DGZ}. The proof in the present paper uses sligh

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\begin{thm}[cf. \cite{DGZ}*{Theorem 4.1}]

\label{cor:symplectic fibration}

There is a smooth manifold $W$ with $\partial W =(-M)\sqcup M'$ and a symplectic

Let $(M,\xi)$ be a contact manifold supported by an open book with

page $(\Sigma,\omega)$ and monodromy $\phi$,

and let $M'=\textup{sf}(\Sigma',\omega')$ be a symplectic fibration over

the circle as described above.

Then 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}

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@@ -320,18 +324,18 @@ An \textbf{abstract open books} is a pair $(\Sigma,\phi)$, where $\Sigma$ is a c

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],

\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

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@@ -408,14 +412,14 @@ The Liouville vector field $Y$ extends over $\hat\Sigma$ by $\partial_r$ and we

to denote the extended vector field by $Y$.

Let

$

\heightOfSigma\co\hat\Sigma\to\mathbb{R}_{\geq0}

\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 $\big([0 , \infty)\times\partial\Sigma\big)$.

\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 $\heightOfSigma$.

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@@ -425,9 +429,9 @@ For some sufficiently small $\varepsilon >0$ let

$g \colon\thinspace[0,\varepsilon]\to\R$ be a cut-off function 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 $y \in[0,\varepsilon]$.

\item$g(y)=1$, for $y$ near $0$,

\item$g(y)=0$, for $y$ near $\varepsilon$,

\item$g'(y)\leq0$, for each $y \in[0,\varepsilon]$.

\end{list}

We are now ready to construct the desired exact symplectic cobordisms of Theorem~\ref{thm:stein-monoid} as well as of

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

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@@ -642,7 +647,7 @@ the underlying function $f$ such that in a neighborhood of $\pm b$ it is given b

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).

\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 $\Sigma_{\heightOfSigma < a}\subset(\hat\Sigma, \hat\omega)$.

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@@ -652,12 +657,12 @@ This shows that the contact structure on $(\partial_-W)_{\pm}$ induced by $\iota

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. We have $f(x)=-\frac{1}{2}x$ and we have to deal with the minus sign. Luckily we are just interested in the induced contact structure on $\partial_+W$ (not the whole cobordism $W$), and 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 $\partial_+W$. Then we obtain a contact structure for each $t \in[0,1]$ on the closed manifold $\partial_+W$ all of which are contactomorphic by Gray stability. Set

\[

B :=\{ x = y =0\}\subset\partial_+W

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.

\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{Max_Thesis}).

%

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@@ -695,7 +700,7 @@ For $i=0,1$ let $\pi_i\colon\thinspace M_i \setminus B_i \to S^1$ denote the ind

Note that the subsets $\pi_i^{-1}\big((-\varepsilon,\varepsilon)\big)\setminus(B_i \times D^2_\varepsilon)\subset(M_i,\xi_i)$, $i=0,1$,

As in the proof of Theorem~\ref{thm:stein-monoid} for sufficiently large

constants $a,b$ the Liouville vector field $Z$ is

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@@ -858,18 +863,18 @@ result by $(M',\xi')$, i.e.\ denoting by $\Psi$ the fibre orientation reversing

diffeomorphism of $B \times D^2\subset B \times\C$ sending $(b,z)$ to $(b,\bar z)$,

using the notation in \cite{MR2397738}*{\S 7.4}, we define

\[

(M',\xi') :=(M_0,\xi_0)\#_\Psi(M_1,\xi_1).

(M',\xi') :=(M_0,\xi_0)\#_\Psi(M_1,\xi_1).

\]

The result $(M',\xi')$ defines a fibration over the circle with fibre given by

\[

\Sigma' =(-\Sigma)\cup_B \Sigma.

\Sigma' =(-\Sigma)\cup_B \Sigma.

\]

Note that each fibre $\Sigma'$ defines a convex hypersurface, i.e.\

there is a contact vector field $X$ on $(M',\xi')$ which is transverse to the fibers.

Furthermore for each fibre $\Sigma'$ the contact vector field $X$ is tangent to the contact structure exactly over $B$.

We will refer to $(M',\xi')$ as a \textbf{doubled open book} and will sometimes denote it by

\[

(\Sigma,\phi)\boxplus(\Sigma,\phi^{-1}).

(\Sigma,\phi)\boxplus(\Sigma,\phi^{-1}).

\]

Before we show the existence of the desired symplectic filling in the above statement,

we show how it can be utilized to prove Corollary~\ref{cor:filling obstr}.

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@@ -894,7 +899,7 @@ It remains to show the existence of an exact symplectic filling for any doubled

Let $\heightOfSigma\co\hat\Sigma\to[0,\infty)$ be a $C^\infty$-function on the completion $(\hat\Sigma,\hat\omega)$ of $(\Sigma,\omega)$ as defined at the beginning of \S\ref{sec:concatenation}.

Consider the symplectic space $\hat\Sigma\times\R\times[0, 2\pi]$ with coordinates $(p,x,y)$ and symplectic form $\Omega=\hat\omega+ dx\wedge dy$. Set

\[

\hat A :=(\hat\Sigma\times\R\times[0, 2\pi])/_\sim,

\hat A :=(\hat\Sigma\times\R\times[0, 2\pi])/_\sim,

\]

where we identify with respect to the map defined by $\Phi\co(p,x,0)\mapsto(\phi^{-1}(p),x, 2\pi)$.

Since $\phi$ is a symplectomorphism of $(\Sigma,\omega)$, equal to the identity in a sufficiently small neighborhood of the boundary $\partial\Sigma$, the symplectic form $\Omega$ on $\hat\Sigma\times\R\times[0, 2\pi]$ descends to a symplectic form on $\hat A$ which we continue to denote by $\Omega$.

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@@ -903,7 +908,7 @@ Note that the fibers of the projection $\hat A \to \R$ on the $x$-coordinate are

Let $W = W_a \subset\hat A$, for some constant $a > 0$, denote the subset defined by

\[

W :=\{(p,x,y)\co\heightOfSigma^2+ x^2\leq a^2\}.

W :=\{(p,x,y)\co\heightOfSigma^2+ x^2\leq a^2\}.

\]

We follow the line of reasoning in the proof of Theorem \ref{thm:stein-monoid}.

The symplectomorphism $\phi$ can be chosen to be exact (cf.~\cite{MR2397738}),

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@@ -912,9 +917,9 @@ adding a constant. Hence we may assume that $\varphi$ vanishes over a neighborho

With $g\co[0, \varepsilon]\to\R$ as defined at the beginning of \S\ref{sec:concatenation}

we define a Liouville vector field $Z$ on $W$ by

\[

Z =\big( g(y)\thinspace(T\Phi^{-1})(Y)+(1- g(y))\thinspace Y \big)

+ x\thinspace\partial_x

+ g'(y)\varphi(p)\thinspace\partial_x.

Z =\big( g(y)\thinspace(T\Phi^{-1})(Y)+(1- g(y))\thinspace Y \big)

+ x\thinspace\partial_x

+ g'(y)\varphi(p)\thinspace\partial_x.

\]

Recall that $g'(y)$ vanishes outside an $\varepsilon$-neighbourhood of the origin, and note that

for sufficiently large $a > 0$ the term $x\thinspace\partial_x$ dominates $g'(y)\varphi(p)\thinspace\partial_x$

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@@ -944,7 +949,7 @@ Recall that $M_{(\Sigma,\tiny\id)}$ is given by $(\Sigma \times S^1) \cup_{\id}(

For any symplectomorphism $\phi$ of $(\Sigma,\omega)$, equal to the identity near

$\partial\Sigma$, the part $(\Sigma\times S^1)$ can be described as

where $N(\Sigma)$ 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

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@@ -1003,19 +1008,19 @@ Consider $\homeOfHandle$ with coordinates $(p,z,t)$ and symplectic form

$\Omega=\omega+ dz\wedge dt$, where $\hat\Sigma$ denotes the completion

of $(\Sigma, \beta)$ defined at the beginning of \S\ref{sec:concatenation}. The vector field

\[

Z = Y +2z\thinspace\partial_z - t\thinspace\partial_t

Z = Y +2z\thinspace\partial_z - t\thinspace\partial_t

\]

defines a Liouville vector field for $\Omega$.

Notice that $Z$ is gradient like for the function on $\homeOfHandle$ defined by

\[

g(p,z,t):=\heightOfSigma^2+ z^2-\frac{1}{2}t^2,

g(p,z,t):=\heightOfSigma^2+ z^2-\frac{1}{2}t^2,

\]

where $\heightOfSigma\co\hat\Sigma\to[0,\infty)$ is a $C^\infty$-function on $\hat\Sigma$

as specified at the beginning of \S\ref{sec:concatenation}.

In particular the Liouville vector field $Z$ is transverse to the non-degenerate level sets of $g$ and hence induces contact structures on them.

Denote by $N(\Sigma),N_0(\Sigma)\subset\hat\Sigma\times\R$ the subsets defined by

\[

N(\Sigma) := N_\delta(\Sigma) :=\{\heightOfSigma < \delta, z < 1\}\ \ \mbox{ and }\ \ N_0(\Sigma) :=\{\heightOfSigma= z =0\}

N(\Sigma) := N_\delta(\Sigma) :=\{\heightOfSigma < \delta, z < 1\}\ \ \mbox{ and }\ \ N_0(\Sigma) :=\{\heightOfSigma= z =0\}

\]

endowed with the contact structure induced by $\iota_Z\omega$.

Let $\mathcal{N}=\mathcal{N}_\delta$ and $\mathcal{N}_0$

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@@ -1023,12 +1028,12 @@ denote the set of points $(p,z, t) \subset g^{-1}(-1)$ which lie on a

flow line of $Z$ through $ N(\Sigma)\times\{ t =\pm1\}$ and $ N_0(\Sigma)\times\{ t =\pm1\}$ respectively -- both viewed as subsets in $\homeOfHandle$. The set $\mathcal{N}$ is going to play the role of the lower boundary.

We now define our \textbf{generalized symplectic $1$-handle}$H_\Sigma$ as the locus of points $(p,z, t)\in\homeOfHandle$ satisfying the inequality

\[

-1\leq g(p, z, t)\leq1

-1\leq g(p, z, t)\leq1

\]

and lying on a flow line of $Z$ through a point of $\mathcal{N}$.

Since the Liouville vector field $Y$ is transverse to the level sets of $g$, the $1$-form

\[

\alpha=\iota_Z\Omega=\beta+2z \thinspace dt + t \thinspace dz

\alpha=\iota_Z\Omega=\beta+2z \thinspace dt + t \thinspace dz

\]

induces a contact structure on the lower and upper boundary of $H_\Sigma$.

\parskip 0pt

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@@ -1045,7 +1050,7 @@ to work with such a perturbed handle.

where we identify $(r,x)\in[0,\mu]\times(\mathcal{N}\setminus\mathcal{N}_0)$ with

the image $\psi^Z_r(x)\in H_\Sigma$ of $x \in\mathcal{N}\setminus\mathcal{N}_0$ -- understood as sitting in $H_\Sigma$ -- under the time-$r$ map of the flow corresponding to the Liouville vector field $Z$.

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@@ -1073,7 +1078,7 @@ This identification does respect the symplectic forms (cf.~\cite{MR2397738}*{Lem

exact symplectic cobordism $W_{(M,\Sigma)}$. The concave boundary component $\partial_- W_{(M,\Sigma)}$ is equal to $(M,\xi)$

whereas the convex component $\partial_+ W_{(M,\Sigma)}$ equals