### More Changes

parent 35819ec9
 ... ... @@ -213,7 +213,11 @@ D\"orner--Geiges--Zehmisch \cite{DGZ}. The proof in the present paper uses sligh % \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}. %------------------------------------------------------------------------------------------ % % 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} %------------------------------------------------------------------------------------------ % 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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