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Mirko KLUKAS
openbooksandsymplecticcorbordisms
Commits
2e62bc74
Commit
2e62bc74
authored
Apr 12, 2017
by
Mirko KLUKAS
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Open_books_and_symplectic_cobordisms.tex
Open_books_and_symplectic_cobordisms.tex
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Open_books_and_symplectic_cobordisms.tex
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2e62bc74
...
...
@@ 213,7 +213,11 @@ D\"ornerGeigesZehmisch \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 cutoff 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:steinmonoid
}
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
$
(
2
n
+
1
)
$
dimensional contact manifold.
Let
$
(
M,
\xi
=
\ker
\alpha
)
$
be a
(not necessarily connected)
$
(
2
n
+
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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