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

More Changes

parent 35819ec9
......@@ -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
\begin{equation}
\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),
\end{equation}
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
\begin{equation}
\label{eqn:Z prime}
Z' = Y + X,
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 $\lim_{{x \to \pm \infty}}f(x) = \pm \infty.$
% \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
\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.
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
......@@ -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
\begin{equation}
\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 \},
\end{equation}
and
\begin{equation}
......@@ -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
\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 $\Sigma_{\heightOfSigma < a} \subset (\hat\Sigma, \hat\omega)$.
......@@ -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}
%------------------------------------------------------------------------------------------
%------------------------------------------------------------------------------------------
......
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment