Commit 2e62bc74 authored by Mirko KLUKAS's avatar Mirko KLUKAS

More Changes

parent 35819ec9
......@@ -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}
......@@ -426,7 +430,7 @@ $g \colon\thinspace [0,\varepsilon] \to \R$ 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 $g(y) = 0$, for $y$ near $\varepsilon$,
\item $g'(y) \leq 0$, 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
......@@ -438,6 +442,7 @@ Theorem~\ref{cor:symplectic fibration}.
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Let $(\Sigma, \omega)$ be the page of the open books for which we construct the desired cobordism $W$.
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
\[
......@@ -481,7 +486,7 @@ defined by
F(x) =
\begin{cases}
s\cdot (x + b) & \text{for $x \leq - \tfrac{b}{2}$,} \\
s\cdot x & \text{for $- \tfrac{b}{2} \leq x \leq \tfrac{b}{2}$, and} \\
- s\cdot x & \text{for $- \tfrac{b}{2} \leq x \leq \tfrac{b}{2}$, and} \\
s\cdot (x - b) & \text{for $\tfrac{b}{2} \leq x $.} \\
\end{cases}
\]
......@@ -1045,7 +1050,7 @@ to work with such a perturbed handle.
\label{sec:1 handle result}
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Let $(M,\xi = \ker \alpha)$ be a $(2n + 1)$-dimensional contact manifold.
Let $(M,\xi = \ker \alpha)$ be a (not necessarily connected) $(2n + 1)$-dimensional contact manifold.
Suppose we are given a strict contact embedding of $\mathcal{N}$, endowed with the contact structure
induced by $i_Z\Omega$, into $(M,\xi = \ker \alpha)$.
In the following we will describe the symplectic cobordism $W_{(M,\Sigma)}$ associated to the attachment of the handle $H_\Sigma$.
......
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