diff --git a/Open_books_and_symplectic_cobordisms.tex b/Open_books_and_symplectic_cobordisms.tex index dccb4e05780941bb5b3e31febcbf11f04a7211fe..027adb7a79595f8d98c82af58e29173d919cd6e6 100755 --- a/Open_books_and_symplectic_cobordisms.tex +++ b/Open_books_and_symplectic_cobordisms.tex @@ -81,9 +81,9 @@ \newcommand{\com}[1]{\definecolor{Gray}{gray}{0.6}\noindent\textcolor{red}{\textit{#1}}} \newcommand{\note}[2]{ - \definecolor{Gray}{gray}{0.6} - \noindent - \textcolor{blue}{\textbf{#1} \textit{#2} } + \definecolor{Gray}{gray}{0.6} + \noindent + \textcolor{blue}{\textbf{#1} \textit{#2} } } \newcommand{\MNote}[1]{\definecolor{Gray}{gray}{0.6}\textcolor{Gray}{(!)}\marginpar{\textcolor{Gray}{Note: #1}}} @@ -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. @@ -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} @@ -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 $$\label{eqn:abstract open book} - M_{(\Sigma,\phi)} := \Sigma(\phi) \cup_{\id} (\partial \Sigma \times D^2), + M_{(\Sigma,\phi)} := \Sigma(\phi) \cup_{\id} (\partial \Sigma \times D^2),$$ 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 $[\theta, r\e^{i\pi\varphi}] \in @@ -408,14 +412,14 @@ The Liouville vector field$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}_{\geq 0} + \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)$. + \item$\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$. @@ -425,9 +429,9 @@ For some sufficiently small$\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$g'(y) \leq 0$, for each$y \in [0,\varepsilon]$. + \item$g(y) = 1$, for$y$near$0$, + \item$g(y) = 0$, for$y$near$\varepsilon$, + \item$g'(y) \leq 0$, 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 Theorem~\ref{cor:symplectic fibration}. @@ -438,17 +442,18 @@ 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 $- \Omega = \hat\omega + dx\wedge dy . + \Omega = \hat\omega + dx\wedge dy .$ Let$P = P_{a,b,c}$denote the subset of$\hat\Sigma \times \R^2$defined by $- P := \big\{(p,x,y)\co \heightOfSigma = 0, \ x^2 + y^2 \leq c^2, \ (x + P := \big\{(p,x,y)\co \heightOfSigma = 0, \ x^2 + y^2 \leq c^2, \ (x - b)^2 + y^2 \geq a^2 \ \mbox{and} \ (x - + b)^2 + y^2 \geq a^2 - \big\}, + + b)^2 + y^2 \geq a^2 + \big\},$ where$a,b,c \in \R$are some potentially very large constants satisfying$a < \tfrac{b}{2}$and$ b+a < c$. The final choice of these constants will later ensure that our @@ -456,17 +461,17 @@ desired (and yet to be defined) Liouville vector field$Z$will be transverse to Consider the vector field$Z'$on$\hat\Sigma \times \R^2$defined by $$\label{eqn:Z prime} - Z' = Y + X, + Z' = Y + X,$$ % 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$\lim_{{x \to \pm \infty}}f(x) = \pm \infty.$+% \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} where$Y$is the Liouville vector field of$(\hat\Sigma,\hat\omega)$defined at the beginning of the present section, $@@ -481,7 +486,7 @@ defined by F(x) = \begin{cases} s\cdot (x + b) & \text{for x \leq - \tfrac{b}{2},} \\ - s\cdot x & \text{for - \tfrac{b}{2} \leq x \leq \tfrac{b}{2}, and} \\ + - s\cdot x & \text{for - \tfrac{b}{2} \leq x \leq \tfrac{b}{2}, and} \\ s\cdot (x - b) & \text{for \tfrac{b}{2} \leq x .} \\ \end{cases}$ @@ -517,17 +522,17 @@$P_\pm := P \cap \{ \pm y \geq 0 \}$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 $- P(\phi_0,\phi_1) := ( P_+\sqcup P_- ) /_{\sim_\Phi}, + P(\phi_0,\phi_1) := ( P_+\sqcup P_- ) /_{\sim_\Phi},$ where we identify with respect to the map$\Phi\co P_0 \to P_0$given by $- \Phi( p, x,0) := - \begin{cases} - ( \phi_0(p), x,0 ) & \text{, for x < - b,}\\ - ( p, x,0 ) & \text{, for |x| < b,} \\ - ( \phi_1^{-1}(p), x,0 ) & \text{, for b < x.} - - \end{cases} + \Phi( p, x,0) := + \begin{cases} + ( \phi_0(p), x,0 ) & \text{, for x < - b,}\\ + ( p, x,0 ) & \text{, for |x| < b,} \\ + ( \phi_1^{-1}(p), x,0 ) & \text{, for b < x.} + + \end{cases}$ 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}. @@ -546,15 +551,15 @@ adding a constant. Hence we may assume that$\varphi_0$and$\varphi_1$vanish o To avoid confusing indices we will write $- \Phi^*\hat\beta - \hat\beta = d\varphi + \Phi^*\hat\beta - \hat\beta = d\varphi$ 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 $$\label{eqn:Z} - Z = \Big( g(y)\thinspace (T\Phi^{-1})(Y) + \big(1-g(y) \big) \thinspace Y \Big)+ - X + g'(y)\varphi(p)\thinspace\partial_x. + Z = \Big( g(y)\thinspace (T\Phi^{-1})(Y) + \big(1-g(y) \big) \thinspace Y \Big)+ + X + g'(y)\varphi(p)\thinspace\partial_x.$$ 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 @@ -575,23 +580,23 @@ Observe that, since$T\Phi(Z|_{P_0}) = Z'|_{P_0}$, we can extend$Z$over$P_-Z'$. 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 $- W' := \big\{ (p,x,y)\ | \ - \heightOfSigma^2 + (x + b)^2 + y^2 \geq a^2, \ + W' := \big\{ (p,x,y)\ | \ + \heightOfSigma^2 + (x + b)^2 + y^2 \geq a^2, \ \heightOfSigma^2 + (x - b)^2 + y^2 \geq a^2, \ - \ \ \mbox{and} \ \ - \heightOfSigma^2 + x^2 + y^2 \leq c^2 - \big\} + \ \ \mbox{and} \ \ + \heightOfSigma^2 + x^2 + y^2 \leq c^2 + \big\}$ and note that we have$P \subset W'$. Finally we define the symplectic cobordism$W = W_{a,b,c}$by $- W := ( W'\setminus P ) \cup P(\phi_0,\phi_1). + W := ( W'\setminus P ) \cup P(\phi_0,\phi_1).$ The boundary of$W$decomposes as$\partial W = \partial_-W \sqcup \partial_+W$, where we have $$\label{eqn:boundary of W minus} - \partial_-W = \{ \heightOfSigma^2 + (x + b)^2 + y^2 = a^2\} \cup \{\heightOfSigma^2 + (x - b)^2 + y^2 = a^2 \}, + \partial_-W = \{ \heightOfSigma^2 + (x + b)^2 + y^2 = a^2\} \cup \{\heightOfSigma^2 + (x - b)^2 + y^2 = a^2 \},$$ and @@ -628,12 +633,12 @@ We start with the negative boundary components of$W$. Denote by$(\partial_-W)_ the two distinct components of the negative boundary of $W$ (cf. Equation~\ref{eqn:boundary of W minus}). Set $- B_\pm := \{ x = \pm b, y = 0, \heightOfSigma = a \} \subset (\partial_-W)_{\pm} + B_\pm := \{ x = \pm b, y = 0, \heightOfSigma = a \} \subset (\partial_-W)_{\pm}$ 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 $- \pi_\pm (p,x,y) := \frac{(x \mp b,y)}{ \| (x\mp b,y) \| } \in S^1. + \pi_\pm (p,x,y) := \frac{(x \mp b,y)}{ \| (x\mp b,y) \| } \in S^1.$ 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 @@ -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 \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). 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 $\Sigma_{\heightOfSigma < a} \subset (\hat\Sigma, \hat\omega)$. @@ -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}). % @@ -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$, define an embedding $- S^0 \times N_\varepsilon(\Sigma) \hookrightarrow M_0 \sqcup M_1, + S^0 \times N_\varepsilon(\Sigma) \hookrightarrow M_0 \sqcup M_1,$ 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 @@ -753,21 +758,21 @@ In analogy to the proof of Theorem \ref{thm:stein-monoid} we consider the symplectic space$\Sigma' \times \R^2$with symplectic form $- \Omega = \omega' + dx \wedge dy. + \Omega = \omega' + dx \wedge dy.$ Over$\Sigma \times \R^2 \subset \Sigma' \times \R^2$we define the Liouville vector field $- Z' = Y + \tfrac{1}{2}\big(x\thinspace\partial_x + y\thinspace\partial_y \big). + Z' = Y + \tfrac{1}{2}\big(x\thinspace\partial_x + y\thinspace\partial_y \big).$ The final cobordism will depend on some potentially very large constants$a,b \in \mathbb{R}$satisfying$0 < a < b$. These constants will be fixed later to ensure the Liouville vector field is transverse to the boundary. Let$A = A_{a,b}$denote the subset of$\Sigma' \times \R^2$defined by $- A := \big\{ (p,x,y) \co \heightOfSigma = 0 - \ \mbox{and} \ - a^2 \leq x^2 + y^2 \leq b^2 \big\}. + A := \big\{ (p,x,y) \co \heightOfSigma = 0 + \ \mbox{and} \ + a^2 \leq x^2 + y^2 \leq b^2 \big\}.$ In analogy of the definition of$P(\phi_0,\phi_1)$in the proof of Theorem~\ref{thm:stein-monoid} we define$A(\phi)$as follows. @@ -779,22 +784,22 @@ We define \] where we identify with respect to the map$\Phi\co A_0 \to A_0$given by $- \Phi( p, x,0) := - \begin{cases} - ( \phi(p); x,0 ) & \text{, for x < 0,}\\ - ( p; x,0 ) & \text{, for x > 0.} - \end{cases} + \Phi( p, x,0) := + \begin{cases} + ( \phi(p); x,0 ) & \text{, for x < 0,}\\ + ( p; x,0 ) & \text{, for x > 0.} + \end{cases}$ Let$W' = W'_{a,b}$denote the subset of$\Sigma \times \R^2$defined by $- W' := \big\{ (p,x,y) \co \heightOfSigma^2 + x^2 + y^2 \geq {a^2} - \ \ \mbox{and} \ \ - x^2 + y^2 \leq b^2 \big\} + W' := \big\{ (p,x,y) \co \heightOfSigma^2 + x^2 + y^2 \geq {a^2} + \ \ \mbox{and} \ \ + x^2 + y^2 \leq b^2 \big\}$ and note that we have$A \subset W'$. Finally we define the symplectic cobordism$W$by $- W := ( W'\setminus A ) \cup A(\phi). + W := ( W'\setminus A ) \cup A(\phi).$ Observe that$\Omega$descends to a symplectic form on$W$. Furthermore we indeed have$\partial W = (-M) \cup M'$. @@ -809,9 +814,9 @@ Choose a sufficiently small$\delta>0$such that$Y$is still defined over$\Sig We define the Liouville vector field $Z$ on $W_{\heightOfSigma \leq 1 + \delta}$ by $- Z = \Big( g(y)\thinspace (T\Phi^{-1})(Y) + (1-g(y)) \thinspace Y \Big) - + \tfrac{1}{2}\big(x\thinspace\partial_x + y\thinspace\partial_y \big) - + g'(y)\varphi(p)\thinspace\partial_x. + Z = \Big( g(y)\thinspace (T\Phi^{-1})(Y) + (1-g(y)) \thinspace Y \Big) + + \tfrac{1}{2}\big(x\thinspace\partial_x + y\thinspace\partial_y \big) + + g'(y)\varphi(p)\thinspace\partial_x.$ As in the proof of Theorem~\ref{thm:stein-monoid} for sufficiently large constants $a,b$ the Liouville vector field $Z$ is @@ -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}. @@ -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 $\Omega$. @@ -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}), @@ -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 $g'(y)\varphi(p)\thinspace\partial_x$ @@ -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 $- (\Sigma \times S^1) \cong \Big( \Sigma \times [0,1] \sqcup \Sigma \times [2,3] \Big) /_\sim, + (\Sigma \times S^1) \cong \Big( \Sigma \times [0,1] \sqcup \Sigma \times [2,3] \Big) /_\sim,$ where we identify$(x,3)$with$(\phi(x),0)$and$(\phi(x),1))$with$(x,2))$for all$x \in \Sigma$. Consider the subsets$\Sigma \times (\tfrac{1}{2} - \varepsilon,\tfrac{1}{2} + \varepsilon )$and @@ -952,7 +957,7 @@$\Sigma \times (\tfrac{5}{2} - \varepsilon,\tfrac{5}{2} + \varepsilon )$of$(\Sigma \times S^1) \subset M_{(\Sigma,\tiny\id)}$. They define an embedding $- S^0 \times N(\Sigma) \hookrightarrow M_{(\Sigma,\tiny\id)}, + S^0 \times N(\Sigma) \hookrightarrow M_{(\Sigma,\tiny\id)},$ 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 @@ -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$\mathcal{N}_0$@@ -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 = \pm 1 \} $and$ N_0(\Sigma) \times \{ t = \pm 1 \}$respectively -- both viewed as subsets in$\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 $- -1 \leq g(p, z, t) \leq 1 + -1 \leq g(p, z, t) \leq 1$ 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 @@ -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$H_\Sigma$. @@ -1058,14 +1063,14 @@ with respect to the above embedding of$\mathcal{N}$in$(M,\xi = \ker \alpha )$continue to denote this map$M \setminus \mathcal{N}_0 \to \R^+$by$\mu$. Consider the symplectization$\big(\R \times M, d(e^r\alpha)\big)$and let$[0,\mu] \times (M\setminus \mathcal{N}_0)$denote the subset defined by $- [0,\mu] \times (M\setminus \mathcal{N}_0) := \big\{ (r,x) \co 0 \leq r \leq \mu(x) \big\}. + [0,\mu] \times (M\setminus \mathcal{N}_0) := \big\{ (r,x) \co 0 \leq r \leq \mu(x) \big\}.$ For any point$(0,x) \in \{0\} \times (M\setminus \mathcal{N}_0)$the time-$\mu(x)$map of the flow corresponding to the Liouville vector field$\partial_r$on$(\R \times M, d(e^r\alpha))$maps$(0,x)$to$(\mu (x),x)$. We define$W_{(M,\Sigma)}$as the quotient space $- W_{(M,\Sigma)} := \Big( \big( [0,\mu] \times (M\setminus \mathcal{N}_0) \big) \sqcup H_\Sigma \Big)/_\sim , + W_{(M,\Sigma)} := \Big( \big( [0,\mu] \times (M\setminus \mathcal{N}_0) \big) \sqcup H_\Sigma \Big)/_\sim ,$ 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$. @@ -1073,7 +1078,7 @@ This identification does respect the symplectic forms (cf.~\cite{MR2397738}*{Lem exact symplectic cobordism$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 $- \#_{H_{\Sigma}}(M,\xi) := (M,\xi) \setminus \big( S^0 \times N(\Sigma) \big) \cup_\partial \big(D^1 \times \partial \overline{N(\Sigma)}, \eta \big), + \#_{H_{\Sigma}}(M,\xi) := (M,\xi) \setminus \big( S^0 \times N(\Sigma) \big) \cup_\partial \big(D^1 \times \partial \overline{N(\Sigma)}, \eta \big),$ with the obvious identifications, and where$\eta$denotes the kernel of the contact form$i_Y\omega + dt$. We will refer to @@ -1090,10 +1095,10 @@ statement. % \begin{prop} \label{thm:extended sum} - There is an exact symplectic cobordism from$(M,\xi)$to$\#_{H_{\Sigma}}(M,\xi)$. - Furthermore if$(\Sigma,\omega)$is Weinstein, then so is the cobordism. - In particular we have the following. If$(M,\xi)$admits a symplectic - filling, then so does$\#_{H_{\Sigma}}(M,\xi)$. \qed + There is an exact symplectic cobordism from$(M,\xi)$to$\#_{H_{\Sigma}}(M,\xi)$. + Furthermore if$(\Sigma,\omega)$is Weinstein, then so is the cobordism. + In particular we have the following. If$(M,\xi)$admits a symplectic + filling, then so does$\#_{H_{\Sigma}}(M,\xi)\$. \qed \end{prop} % % @@ -1104,9 +1109,9 @@ statement. \begin{bibdiv} \pagestyle{myheadings} \markboth{\textsc{\textsl{BIBLIOGRAPHY}}}{} - \begin{biblist} - \bibselect{bib} - \end{biblist} + \begin{biblist} + \bibselect{bib} + \end{biblist} \end{bibdiv} %------------------------------------------------------------------------------------------ %------------------------------------------------------------------------------------------