text:
\section{Introduction\label{sec:intro}}

By the isotropic neighbourhood theorem
a neighbourhood of a closed isotropic submanifold $Q$
in a given contact manifold is determined
by the diffeomorphism type of $Q$
and by the isomorphism class of the conformally
symplectic normal bundle $\mathrm{CSN}(Q)$ of $Q$,
cf.\ \cite[Theorem 2.5.8]{gei08}.
For instance if $\mathrm{CSN}(Q)$ is trivial,
a trivialisation of $\mathrm{CSN}(Q)$ determines
a local model given by
a neighbourhood of $Q$ in the contactisation
of $T^*Q\times\mathbb C^{n-d}$,
$d=\dim Q\leq n$,
cf.\ \cite[Section 3.1]{vkoe18}.
On the other hand,
the restriction of any defining contact form
to the tangent bundle of a compact hypersurface
determines the germ of the contact structure,
see Ding--Geiges \cite[Proposition 6.4]{dg12}.
In particular,
this applies to the boundary
of a disc-like neighbourhood of $Q$.
Combined with local contact inversion
(see Proposition \ref{prop:contact-inversion})
it turns out that there is no canonical distinction
between in- or outside for this hypersurface
if $\mathrm{CSN}(Q)$ is trivial.

In this work we consider the case where $Q$ is a torus $T^d$
and where the complement $M$ of a tubular neighbourhood of $Q$
is compact.
Assuming $n>d$
we investigate to which extent a choice of a contact form
on $M$ that is of model type near the boundary
determines the topology of $M$.
As demonstrated by Eliashberg--Hofer \cite{eh94}
if $\dim M=3$ and
Geiges--Zehmisch \cite{gz16b} for $\dim M\geq5$,
$M$ is diffeomorphic to an $(2n+1)$-dimensional ball
whenever $M$ does not have any
short contractible periodic Reeb orbits
and $\partial M$ is a sphere.
This situation corresponds to $Q=*$.
In fact,
Eliashberg--Hofer \cite{eh94} proved
a global Darboux theorem in the absence of
short periodic Reeb orbits if $\dim M=3$.
In contrast,
Geiges--R\"ottgen--Zehmisch \cite{grz14}
constructed an aperiodic Reeb flow with trapped orbits
on $\mathbb R^{2n+1}$, $n\geq2$,
that is standard outside a compact set.
In order to find a relation between the topology
of the isotropic knot complement $M$
and the existence of short periodic Reeb orbits on $M$
we will utilise holomorphic curves as it is typical
in symplectic dynamics as propagated by
Bramham--Hofer \cite{bh12}.

\subsection{Main result\label{subsec:mainres}}

Let us assume that $Q$ is the $d$-dimensional torus
\[
T^d:=\mathbb R^d/2\pi\mathbb Z^d\,.
\]
We consider a compact, connected $(2n+1)$-dimensional
strict contact manifold $(M,\alpha)$ with boundary
\[
\partial M=S\big(T^*Q\oplus\underline{\mathbb R}^{2n+1-2d}\big)
\]
equal to the unit sphere bundle
of the stabilised cotangent bundle
$T^*Q\oplus\underline{\mathbb R}^{2n+1-2d}$ of $T^*Q$.
In particular, the boundary of $M$ is diffeomorphic to
\[
\partial M=T^d\times S^{2n-d}\,.
\]
The aim of this work is to give a criterion for $M$
to be diffeomorphic to the unit disc bundle
\[
D\big(T^*Q\oplus\underline{\mathbb R}^{2n+1-2d}\big)
=T^d\times D^{2n+1-d}
\]
in terms of the infimum $\inf_0(\alpha)$ of all positive periods of
{\it contractible} closed Reeb orbits of the Reeb vector field of $\alpha$
and the following {\it embeddability condition}:

Write
\[
Z:=\mathbb R\times T^*T^d\times D^2\times\mathbb C^{n-1-d}
\]
for the model neighbourhood of an isotropic torus
$T^d$ with trivial conformally symplectic normal bundle
inside a contact manifold
equipped with the contact form
\[
\alpha_Z:=
\mathrm d b
+\sum_{j=1}^dp_j\mathrm d q_j
+\frac12\big(x_0\mathrm d y_0-y_0\mathrm d x_0\big)
-\sum_{j=1}^{n-1-d}y_j\mathrm d x_j\,,
\]
where $b\in\mathbb R$,
$p_j,q_j$ are coordinates on the cotangent bundle $T^*T^d$,
$x_0,y_0$ are coordinates on the closed unit disc $D^2$,
and $x_j+\mathrm i y_j$ are coordinates on $\mathbb C^{n-1-d}$.
We will use the following short form
\[
\alpha_Z=
\mathrm d b
+\mathbf p\!\;\mathrm d\mathbf q
+\frac12\big(x_0\mathrm d y_0-y_0\mathrm d x_0\big)
-\mathbf y\mathrm d\mathbf x\,,
\]
of the contact form during the text.

We say that $\partial M$ admits a {\bf contact embedding}
into $(Z,\alpha_Z)$ if there exists
a strict contact embedding $\varphi$
of a collar neighbourhood $U$ of $\partial M\subset M$
into the interior of $Z$
in the sense that  $\varphi^*\alpha_Z=\alpha$
such that
\begin{itemize}
\item
each flow line of the Reeb vector field $\partial_b$
intersects $\varphi(\partial M)\subset Z$ in at most two points,
\item
the image $\varphi(U)$ is
contained in the bounded component of
$Z\setminus\varphi(\partial M)$, and
\item
$\varphi(\partial M)$
is smoothly isotopic to
$S\big(T^*T^d\oplus\underline{\mathbb R}^{2n+1-2d}\big)$
inside $Z$.
\end{itemize}

\begin{thmintr}
\label{thm:mainthm}
 Let $(M,\alpha)$ be a strict contact manifold
 as described above such that $\partial M$
 has a contact embedding into $(Z,\alpha_Z)$
 with $n>d$.
 If $\inf_0(\alpha)\geq\pi$,
 then $M$ and $T^d\times D^{2n+1-d}$
 are homotopy equivalent if $n=2$
 and diffeomorphic otherwise.
\end{thmintr}

Observe the standing assumption $n\geq d$.
The extra condition we require in the theorem is $n\neq d$
meaning that $Q$ is {\bf subcritically isotropic}.
This assumption allows us the use of holomorphic discs
as a $\mathbb C$-factor can be split off in the model situation.

The case $n=1$ is covered by the work of Eliashberg--Hofer \cite{eh94};
the critical case $n=1$ and $d=1$,
without any shortness assumption on $\inf_0(\alpha)$,
is proved by Kegel--Schneider--Zehmisch \cite{kschz}
using a different method.
Therefore,
we assume $n\geq2$ (and $n>d$) throughout the article.
The case $d=0$ is due to Geiges--Zehmisch \cite{gz16b}
in which even being diffeomorphic can be concluded.

Performing contact connected sum
of $Z$ with any contact manifold
one obtains a periodic Reeb orbit
of period strictly less than
(but arbitrarily close to) $\pi$
contained in the belt sphere,
cf.\ \cite[Remark 1.3.(1)]{gz16b}.
Hence,
the bound $\pi$ in Theorem \ref{thm:mainthm} is optimal.
Moreover,
the contrapositive of Theorem \ref{thm:mainthm}
can be used to prove existence
of periodic Reeb orbits on non-compact manifolds.
Consider a compact contact manifold $(M,\xi)$
whose boundary has precisely two connected components
each admitting a contact embedding into $(Z,\alpha_Z)$
individually.
Using the Reeb flow on the model $(Z,\alpha_Z)$
the images can be assumed to be not nested
so that a gluing of $(M,\xi)$ to $(Z,\alpha_Z)$
along the boundary is possible.
The gluing result can not be homotopy equivalent to
$T^d\times D^{2n+1-d}$
so that any contact form on $(M,\xi)$
standard near the boundary possesses
a contractible periodic Reeb orbit,
cf.\ \cite[Remark 1.3.(4)]{gz16b}.

For existence results of periodic Reeb orbits
on non-compact contact manifolds
with asymptotic and periodic boundary conditions
we refer to the work of
Suhr--Zehmisch \cite{sz16}
and Bae--Wiegand--Zehmisch \cite{bwz},
respectively.

\subsection{Filling by holomorphic discs\label{subsec:filling}}

The basic idea of the proof of Theorem \ref{thm:mainthm}
is the same as described in \cite[Section 1.2]{gz16b}
invoking {\it filling by holomorphic discs}
techniques as worked out in
\cite{kbdiss,gz10,gz13,gz16a,gz16b,zeh16}.
Using the contact embedding $\varphi$ of $\partial M$
into $(Z,\alpha_Z)$ we form a new strict contact manifold
$(\hat{M},\hat{\alpha})$ by replacing the bounded component
of $\mathbb R\times T^*T^d\times\mathbb C^{n-d}\setminus\varphi(\partial M)$
by $M$.
The contact form $\hat{\alpha}$ equals $\alpha$
on $M$ and
\[
\mathrm d b
+\mathbf p\!\;\mathrm d\mathbf q
+\frac12\big(x_0\mathrm d y_0-y_0\mathrm d x_0\big)
-\mathbf y\mathrm d\mathbf x
\]
on the unbounded component of
$\mathbb R\times T^*T^d\times\mathbb C^{n-d}\setminus\varphi(\partial M)$.
Observe that the latter contact form
coincides with $\alpha_Z$ on $Z$.
We define $\hat{Z}$ similarly by gluing $M$ into $Z$.

Further, we will consider holomorphic maps
\[
u=(a,f)\colon\thinspace\mathbb D\longrightarrow W
\]
defined on the closed unit disc $\mathbb D\subset\mathbb C$
and taking values in the symplectisation
$W$ of $(\hat{M},\hat{\alpha})$
subject to varying Lagrangian boundary conditions.
The moduli space $\mathcal W$ of all such holomorphic discs
carries an evaluation map
\[
 \begin{array}{rccc}
  \ev\colon\thinspace & \mathcal W\times\mathbb D     & \longrightarrow & \hat{Z}\\
            & \bigl( (a,f),z\bigr) & \longmapsto     & f(z)\,.
 \end{array}
\]
{\it A priori} $\ev$ takes values in $\hat{M}$,
but we will show that $\ev$ indeed takes values
in the smaller set $\hat{Z}\subset\hat{M}$.
It will turn out that
either the evaluation map $\ev$ is proper and surjective of degree one,
in which case we can draw conclusions with the $s$-cobordism theorem
as in the work of Barth--Geiges--Zehmisch \cite{bgz},
or the moduli space $\mathcal W$ is not locally compact
in the sense that there will be breaking off of finite energy planes.
By a result of Hofer \cite{hof93,hof99}
this in turn results in the existence of short
contractible periodic Reeb orbits of $\alpha$
as the Reeb flow of $\alpha_Z$ is linear,
given by $\partial_b$.

Observe that a contact embedding
of $\partial M$ into $(Z,\alpha_Z)$
yields an embedding of $\partial M$
into $\mathbb R\times T^*T^d\times D^2_r\times\mathbb C^{n-1-d}$
for some slightly smaller radius $r\in(0,1)$.
The proof of Theorem \ref{thm:mainthm}
that we are going to present in this work
will show that being short for a contractible
periodic Reeb orbit should mean to have period
less than or equal to $\pi r^2$.
Therefore, the above mentioned second alternative
will be excluded by requiring $\inf_0(\alpha)>\pi r^2$
as an Arzel\`a--Ascoli argument shows.
For ease of notation we will assume $r=1$
so that we assume the stronger condition
$\inf_0(\alpha)>\pi$
during the proof.

\subsection{Relevance of the torus\label{subsec:reltorus}}

Large parts of the argument
work under considerably weaker assumptions --
mainly the topological part,
which is similar to \cite{bgz}.
In order to set up
the holomorphic disc analysis
we use a foliation by Lagrangian submanifolds of $T^*Q$
as parametrised boundary condition.
Moreover,
we use a choice of strictly plurisubharmonic potential
for the Liouville form on $T^*Q$
that together with the maximum principle
ensures $C^0$-bounds in the compactness argument,
see Section \ref{subsubsec:maxprinc}.
This together with the Niederkr\"uger map,
which we use to construct holomorphic discs,
works particularly well in global (periodic) coordinates on $T^*Q$.
It is not clear how to change the setup
to enlarge the class of examples.

\section{Standard holomorphic discs\label{sec:stholdiscs}}

The model contact manifold $(Z,\alpha_Z)$
is the contactisation of the Liouville manifold
\[
(V,\lambda_V):=
\Big(
T^*T^d\times D^2\times\mathbb C^{n-1-d},\,
\mathbf p\!\;\mathrm d\mathbf q
+\frac12\big(x_0\mathrm d y_0-y_0\mathrm d x_0\big)
-\mathbf y\mathrm d\mathbf x
\Big)
\,,
\]
which contains the holomorphic discs
$\{\mathbf w\}\times D^2\times\{\mathbf s+\mathrm i\mathbf t\}$.
The aim of this section is to describe a lift
of these holomorphic discs to the symplectisation
of $(Z,\alpha_Z)$.
These holomorphic discs will appear
as the standard discs of the moduli space $\mathcal W$
and serve as a description of the end of $\mathcal W$.
In order to lift we proceed in two steps.
The first will be a lift to
$\mathbb C\times T^*T^d\times D^2\times\mathbb C^{n-1-d}$;
the second is a transformation along a biholomorphic map
$\Phi$ from $\mathbb R\times\mathbb R\times T^*T^d\times D^2\times\mathbb C^{n-1-d}$ to
$\mathbb C\times T^*T^d\times D^2\times\mathbb C^{n-1-d}$,
the Niederkr\"uger map from \cite[Proposition 5]{nie06}.

\subsection{The contactisation\label{subsec:contactisation}}

Following the explanations from \cite[Section 2]{gz16b}
we denote the Liouville manifold form
the beginning of Section \ref{sec:stholdiscs}
by $(V,\lambda_V)$
so that its contactisation
$(\mathbb R\times V,\mathrm d b+\lambda_V)$
is equal to the strict contact manifold $(Z,\alpha_Z)$.
The corresponding contact structure $\xi_Z$
is given by the set of tangent vectors
$v-\lambda_V(v)\partial_b$ for all $v\in TV$.

\subsection{Liouville manifold and K\"ahler potential\label{subsec:liouandpot}}

The Liouville manifold $(V,\lambda_V)$
admits a complex structure
\[
J_V:=(-\mathrm i)\oplus\mathrm i\oplus\mathrm i\,,
\]
where $-\mathrm i$ is meant to be
the negative of the complex structure on $T^d$
obtained by the quotient of $\mathbb R^d$ by $2\pi\mathbb Z^d$
and $T^*\mathbb R^d\equiv\mathbb R^{2n}\equiv\mathbb C^d$
such that $-\mathrm i$ is an almost complex structure on $T^*T^d$
compatible with $\mathrm d\mathbf p\wedge\mathrm d\mathbf q$,
cf.\ \cite[Appendix B]{nie06}.
A strictly plurisubharmonic potential $\psi$
in the sense of \cite[Section 3.1]{gz12}
so that $J_V$ is compatible
with the symplectic form $\mathrm d\lambda_V$
and $\lambda_V=-\mathrm d\psi\circ J_V$
is given by
\[
\psi\big(\mathbf w,z_0,\mathbf z\big):=
\frac12\sum_{j=1}^dp_j^2+
\frac14|z_0|^2+
\frac12\sum_{j=1}^{n-1-d}y_j^2
\,,
\]
where the point $\mathbf w\in T^*T^d$
is written in coordinates as
$(q_1,p_1,\ldots,q_d,p_d)$ and
$\mathbf z=z_1,\ldots,z_{n-1-d}$,
where $z_j=x_j+\mathrm i y_j$, $j=0,1,\ldots,n-1-d$,
denote coordinates
on $D^2\times\mathbb C^{n-1-d}$.

\subsection{The symplectisation\label{subsec:symplectisation}}

For any positive, strictly increasing smooth function
$\tau\equiv\tau(a)$ on $\mathbb R$ the symplectisation of $(Z,\alpha_Z)$
is defined to be the symplectic manifold
\[
\big(\mathbb R\times Z, \mathrm d(\tau\alpha_Z)\big)\,.
\]
A compatible and translation invariant
almost complex structure
that preserves the contact hyperplanes $\xi_Z$
on all slices $\{a\}\times Z$
is determined by $\partial_a\mapsto\partial_b$
and the requirement that for all $v\in TV$ the tangent vectors
$v-\lambda_V(v)\partial_b$ get mapped to
$J_Vv-\lambda_V(J_Vv)\partial_b$.
With that choice of an almost complex structure
on $\mathbb R\times\mathbb R\times V$
the {\bf Niederkr\"uger map}
\[
\Phi(a,b\,;p)=\big(a-\psi(p)+\mathrm i b,p\big)
\]
is a biholomorphism onto $\mathbb C\times V$
equipped with the almost complex structure
$\mathrm i\oplus J_V$.

\subsection{The Niederkr\"uger transform\label{subsec:niedtrans}}

The resulting holomorphic discs maps
\[
\mathbb D\longrightarrow
\mathbb R\times\mathbb R\times T^*T^d\times D^2\times\mathbb C^{n-1-d}
\,,
\]
to which we refer as being {\bf standard},
can be parametrised by
\[
u_{\mathbf s,b}^{\mathbf t,\mathbf w}(z)=
\Big(
\tfrac14\big(|z|^2-1\big),b\,;\mathbf w,z,\mathbf s+\mathrm i\mathbf t
\Big)
\]
for parameters $b\in\mathbb R$, $\mathbf w\in T^*T^d$,
and $\mathbf s,\mathbf t\in\mathbb R^{n-1-d}$,
cf.\ \cite[Section 2.2]{gz16b}.
Natural Lagrangian boundary conditions for the restrictions
of the standard holomorphic disc maps to $\partial\mathbb D$
are given by Lagrangian cylinders
\[
L^{\mathbf t}_{\mathbf p}:=
\{0\}\times\mathbb R\times T^d\times\{\mathbf p\}\times\partial D^2
\times\mathbb R^{n-1-d}\times\{\mathbf t\}
\]
parametrised by $\mathbf t\in\mathbb R^{n-1-d}$ and $\mathbf p\in\mathbb R^d$,
which foliate $\{0\}\times\partial Z$.
In order to verify $L^{\mathbf t}_{\mathbf p}$ to be Lagrangian
observe that the restriction of $\mathrm d(\tau\alpha_Z)$
to the tangent bundle of $\{0\}\times Z$ equals
$\tau(0)\mathrm d\alpha_Z$,
which is a positive multiple of
\[
\mathrm d\mathbf p\wedge\mathrm d\mathbf q
+\mathrm d x_0\wedge\mathrm d y_0
+\mathrm d\mathbf x\wedge\mathrm d\mathbf y
\,,
\]
and that $L^{\mathbf t}_{\mathbf p}$ is of dimension $n+1$.

\section{A boundary value problem\label{sec:abdryvalprob}}

Let $(W,\omega)$ be the symplectisation
\[
(W,\omega):=\big(\mathbb R\times\hat{M}, \mathrm d(\tau\hat{\alpha})\big)
\]
of the glued
strict contact manifold $(\hat{M},\hat{\alpha})$
introduced in Section \ref{subsec:filling},
where $\tau$ is a positive,
strictly increasing smooth function on $\mathbb R$
such that $\tau(a)=\mathrm e^a$ for all $a\geq0$.

\subsection{An almost complex structure\label{subsec:analmcpxstr}}

Let $J$ be a compatible
almost complex structure on $(W,\omega)$
that is invariant under translations in $\mathbb R$-direction,
sends the coordinate vector field $\partial_a$
to the Reeb vector field of the contact from $\hat{\alpha}$,
and restricts to a compatible complex bundle structure on
$\big(\hat{\xi},\mathrm d\hat{\alpha}\big)$,
where $\hat{\xi}$ denotes the contact structure
defined by $\hat{\alpha}$.
Observe that the required conditions for $J$
are satisfied simultaneously for all admissible $\tau$.

We would like to specify a choice
of almost complex structure $J$
in order to deal with the non-compactness of $\hat{M}$.
For positive real numbers $b_0,r,R$
we define the {\bf box} by
\[
B
:=[-b_0,b_0]
\times D_RT^*T^d
\times D^2_r
\times D^{2n-2-2d}_R
\,,
\]
where $D^{2\ell}_{\rho}\subset\mathbb C^{\ell}$ denotes
the closed $2\ell$-disc of radius $\rho$
and $D_{\rho}T^*T^d$ is the closed $\rho$-disc
subbundle of $T^*T^d$.
We choose $r<1$ so that the box is contained in $Z$
and require that the interior of the box
contains $\varphi(\partial M)$,
i.e.\
\[
\varphi(\partial M)\subset \Int(B)\subset Z\,.
\]
Similarly to the use of the symbols $\hat{M}$ and $\hat{Z}$
we write $\hat{B}$ for the result of gluing $M$ into $B$.
Observe the chain of strict inclusions
\[
M\subset\hat{B}\subset\hat{Z}\subset\hat{M}
\,.
\]
On the complement of $\mathbb R\times\Int(\hat{B})$
we require the almost complex structure $J$
to be the one defined in Section \ref{sec:stholdiscs}
with the obvious modification of the construction
by taking the contactisation
of $\mathbb R\times T^*T^d\times\mathbb C\times\mathbb C^{n-1-d}$
instead of $Z$.
On $\mathbb R\times\Int(\hat{B})$ the choice of $J$
will be subject to genericity considerations
specified in Section \ref{subsubsec:lincrop}.

\subsection{The moduli space\label{subsec:themodspace}}

Let $\mathcal W$ be the {\bf moduli space}
of all holomorphic discs
\[
u=(a,f)\colon\thinspace\mathbb D\longrightarrow (W,J)
\]
for which there exists a {\bf level}
$(\mathbf p,\mathbf t)\in\mathbb R^d\times\mathbb R^{n-1-d}$
selecting the Lagrangian boundary cylinder
$L^{\mathbf t}_{\mathbf p}$ in $\{0\}\times\partial Z$
such that the following boundary condition
is satisfied:
\[
u(\partial\mathbb D)\subset L^{\mathbf t}_{\mathbf p}
\,.
\]
In particular,
the map $\Psi\circ f$,
which is defined in a neighbourhood of $\partial\mathbb D$,
is constant along $\partial\mathbb D$
setting $\Psi(b,\,.\,)=\psi$ for all $b\in\mathbb R$.
Additionally, we require that for all $u\in\mathcal W$
there exist sufficiently large
parameters $b\in\mathbb R$, $\mathbf w=(\mathbf q,\mathbf p)\in T^*T^d$,
and $\mathbf s\in\mathbb R^{n-1-d}$ such that the standard disc
$u_{\mathbf s,b}^{\mathbf t,\mathbf w}$ -- $(\mathbf p,\mathbf t)$ being the level of $u$ --
can be regarded as holomorphic disc in $(W,J)$
and is homologous to $u$ in $W$ relative $L^{\mathbf t}_{\mathbf p}$,
i.e.\
\[
[u]=[u_{\mathbf s,b}^{\mathbf t,\mathbf w}]
\qquad\text{in}\quad
H_2(W,L^{\mathbf t}_{\mathbf p})
\,.
\]
Because of $n\geq2$
all standard holomorphic discs of the same level
are homotopic relative boundary
so that the homological condition is well posed.
The reparametrisation group
-- the group of biholomorphic diffeomorphisms of $\mathbb D$ --
is divided out by the requirement
\[
f(\mathrm i^k)\in
\mathbb R\times T^d\times\{\mathbf p\}\times\{\mathrm i^k\}
\times\mathbb R^{n-1-d}\times\{\mathbf t\}
\qquad\text{for}\quad
k=0,1,2
\,,
\]
i.e.\ $u$ is required to map
the marked points $1,\mathrm i,-1$
to the characteristic leaves
$L^{\mathbf t}_{\mathbf p}\cap\{z_0=1\}$,
$L^{\mathbf t}_{\mathbf p}\cap\{z_0=\mathrm i\}$,
and $L^{\mathbf t}_{\mathbf p}\cap\{z_0=-1\}$,
respectively.

\subsection{Convergence\label{subsec:convergence}}

We will study $C^{\infty}$-compactness
properties holomorphic discs in $\mathcal W$.
In the following we list
elementary properties
that all $u=(a,f)\in\mathcal W$ share:

\subsubsection{Uniform energy bounds\label{subsubsec:energybounds}}

 The $L^2$-norm of the gradient is uniformly bounded
 in the sense that the {\bf symplectic energy}
 $\int_{\mathbb D}u^*\omega$,
 which is equal to the {\bf action}
 $\int_{\partial\mathbb D}f^*\hat{\alpha}$
 of the boundary circle,
 is equal to $\pi$.
 This follows as in \cite[Lemma 3.2]{gz16b} because $u$
 is homologous to a certain standard disc.

\subsubsection{$C^0$-bounds and maximum principle\label{subsubsec:maxprinc}}

 As it is the case for any holomorphic curve
 $u=(a,f)$ in symplectisations
 the function $a$ is subharmonic,
 cf.\ \cite[Lemma 3.6.(i)]{gz16b}.
 In the situation at hand we conclude
 with the arguments from \cite[Lemma 3.6.(i)]{gz16b}
 that $a<0$ on $\Int(\mathbb D)$ for all $u=(a,f)\in\mathcal W$.

 In order to describe the behaviour
 of $u=(a,f)\in\mathcal W$
 in the direction of $\hat{M}$
 we denote by $G$ the $f$-preimage
 of $\hat{M}\setminus\Int(\hat{B})$.
 Namely,
 on $G$ we can introduce coordinate functions
 \[
 f=(b,\mathbf w,h_0,\mathbf h)
\qquad\text{on}\quad
\mathbb R\times T^*T^d\times\mathbb C\times\mathbb C^{n-1-d}
 \]
 according to the indicated splitting.
 By the properties of the Niederkr\"uger map
 the coordinate function $b$ is harmonic
 and the $\mathbf w,h_0,\mathbf h$ are holomorphic.

 If $G=\mathbb D$,
 then $u$ will be one of the standard discs
 $u_{\mathbf s,b}^{\mathbf t,\mathbf w}$
 sitting in the complement of $\mathbb R\times\Int(\hat{B})$.
 The argument for that is the same as for
 \cite[Lemma 3.7]{gz16b} with the following
 additional observation:
 The holomorphic map $\mathbf w\colon\thinspace\mathbb D\rightarrow T^*T^d$
 lifts to a holomorphic map to the universal cover
 resulting into an anti-holomorphic disc map into $\mathbb C^d$
 with boundary circle $\partial\mathbb D$
 mapped into a totally real affine plane $\mathbb R^d\times\{\mathbf p\}$.
 Hence, as in \cite[Lemma 3.7]{gz16b}
 or by Schwarz reflection $\mathbf w$ must be constant.
 An alternative argument is based on the fact
 that the symplectic energy of $\mathbf w$,
 which is equal to the Dirichlet energy, vanishes,
 so that -- again -- $\mathbf w$ must be constant.
 Denoting the level of $u$ by $(\mathbf p,\mathbf t)$
 we use a retraction of $T^*T^d$ to $T^d\times\{\mathbf p\}$
 to homotope the disc $\mathbf w$ into
 the Lagrangian submanifold $T^d\times\{\mathbf p\}$
 relative boundary.
 As the homotoped disc has vanishing symplectic energy
 the symplectic energy of $\mathbf w$ vanishes
 by Stoke's theorem too.

 In the situation that $G$ is a proper subset of $\mathbb D$
 we will make the following observations:
 By construction,
 $G$ contains a neighbourhood of $\partial\mathbb D$
 so that the strong maximum principle
 and the boundary lemma of E.\ Hopf
 apply to $h_0$.
 Indeed, as in \cite[Lemma 3.6.(ii)]{gz16b}
 we conclude that $f(\Int(\mathbb D))$ is contained in $\Int(\hat{Z})$.
 Moreover, by the comments on \cite[p.~669 and p.~671]{gz16b}
 we see that $h_0$ restricts to an immersion on $\partial\mathbb D$
 so that $u(\partial\mathbb D)$ is positively transverse
 to each of the characteristic leaves
 $L^{\mathbf t}_{\mathbf p}\cap\{z_0=\mathrm e^{\mathrm i\theta}\}$,
 $\theta\in[0,2\pi)$, denoting the level of $u$ by $(\mathbf p,\mathbf t)$.
 By the homological condition
 posed by the boundary value problem for $u=(a,f)\in\mathcal W$
 we infer that $h_0$ restrict in fact to an embedding on $\partial\mathbb D$.

 Continuing the discussions on the case $G\neq\mathbb D$
 we observe that by the arguments in \cite[Lemma 3.8]{gz16b}
 the coordinate function $b$ of $u$,
 as $u$ can not be a standard disc,
 takes values in $[-b_0,b_0]$.
 For that recall that $\Psi\circ f$
 is constant along $\partial\mathbb D$.
 Similarly, there exists a real number $R_0>R$
 such that the intersection of the non-standard disc
 $f(\mathbb D)$ with
 \[
 \mathbb R
 \times\Big(T^*T^d\setminus D_{R_0}T^*T^d\Big)
 \times\mathbb C
 \times\Big(\mathbb C^{n-1-d}\setminus D^{2n-2-2d}_{R_0}\Big)
 \]
 is empty.
 The projection $\mathbf h$ to the $\mathbb C^{n-1-d}$-factor
 can be treated similarly to \cite[Lemma 3.9]{gz16b}.
 For the cotangent factor notice
 that a composition of local lifts of $\mathbf w$
 with complex conjugation on the universal cover
 results in local holomorphic maps w.r.t.\
 the standard complex structure on $\mathbb C^d$.
 Therefore, the maximum principle implies then
 that $\mathbf w$ takes values in the codisc bundle
 of radius $|\mathbf p|$.
 In view of the uniform energy bounds
 stated in Section \ref{subsubsec:energybounds}
 the monotonicity lemma gives an upper bound
 on the level $|\mathbf p|$ of $u$ analogously
 to the arguments on \cite[p.~674]{gz16b}
 (by possibly using several geodesic balls of the same radius
 less than or equal to $\pi$
 to cut area out of the holomorphic disc $\mathbf w$).

 In conclusion,
 we obtain uniform $C^0$-bounds
 on the $\hat{M}$-part of all non-standard
 holomorphic discs in $\mathcal W$.

\subsubsection{Compactness\label{subsubsec:compactness}}

The space of non-standard holomorphic discs
in $\mathcal W$ is $C^{\infty}$-compact under the assumption
that all contractible periodic Reeb orbits
of $\hat{\alpha}$ have action greater than $\pi$.
This follows with the arguments in \cite[Section 4]{gz16b}
and \cite{fr08,fz15},
and an identification of $T^*T^d$ with $T^d\times\mathbb R^d$
so that the variation of the boundary condition $T^d\times\{\mathbf p\}$
in the cotangent factor of $L^{\mathbf t}_{\mathbf p}$
can be described with help of translations in $\mathbb R^d$.
What remains to show is the following
indecomposability statement:

\begin{lem}
 \label{lem:indecomposable}
 The homology class $[u]$ of all $u\in\mathcal W$
 in $H_2(W,L^{\mathbf t}_{\mathbf p})$, where $(\mathbf p,\mathbf t)$
 denotes the level of $u$, is $J$-indecomposable.
\end{lem}

\begin{proof}
 If not, we could find non-constant holomorphic discs
 $u^1,\ldots,u^N$ with boundary on $L^{\mathbf t}_{\mathbf p}$
 such that $[u]$ can be decomposed into the sum
 \[
 [u]=m_1[u^1]+\ldots+m_N[u^N]
 \]
 for natural numbers $N$ and $m_1,\ldots,m_N$,
 where at least one of them is greater than $1$.
 By exactness of the symplectic form $\omega$
 none of the holomorphic maps $u^1,\ldots,u^N$
 can be defined on a sphere.
 Denote the restrictions to the boundary by
 $\gamma$ and $\gamma^1,\ldots,\gamma^N$,
 and observe that
 \[
 [\gamma]=m_1[\gamma^1]+\ldots+m_N[\gamma^N]
 \]
 in $H_1(L^{\mathbf t}_{\mathbf p})$.
 As $L^{\mathbf t}_{\mathbf p}$ is contained in the complement
 of $\Int(\hat{B})$ the boundary loops admit a splitting
 w.r.t.\
 \[
 \{0\}\times\mathbb R\times T^d\times\{\mathbf p\}\times\partial D^2
 \times\mathbb R^{n-1-d}\times\{\mathbf t\}
 \,.
 \]
 The symplectic energy of holomorphic discs
 in $(W,\omega)$ is equal to the action
 of the boundary loops so that by
 positive transversality w.r.t.\ the characteristic leaves
 mentioned in Section \ref{subsubsec:maxprinc}
 we obtain
 \[
 \pi=
 \sum_{j=1}^Nm_j[\gamma^j_{T^d}]+
 \sum_{j=1}^Nm_jn_j\pi
 \]
 for natural numbers $n_1,\ldots,n_N$.
 The first summand equals the total action
 of the projections of $u^j|_{\partial\mathbb D}$ to $T^d\times\{\mathbf p\}$
 and so is in turn equal to the action of the corresponding projection
 $\gamma_{T^d}$ of $\gamma$.
 Because $[u]$ can be represented
 by a standard disc $u_{\mathbf s,b}^{\mathbf t,\mathbf w}$, $\mathbf w=(\mathbf q,\mathbf p)$,
 whose boundary map equals
 \[
 \mathrm e^{\mathrm i\theta}\longmapsto
 \Big(
 0,b\,;\mathbf w,\mathrm e^{\mathrm i\theta},\mathbf s+\mathrm i\mathbf t
 \Big)\,,
 \]
 the action of $\gamma_{T^d}$ vanishes.
 In total, we reach the inequality $\pi\geq N\pi$.
 Therefore, $N=1$ proving $J$-indecomposability.
\end{proof}

\subsection{Transversality\label{subsec:transversality}}

Each standard holomorphic disc in $\mathcal W$
admits a neighbourhood that can be parametrised
by
\[
\Big(
b\,;\mathbf w,\mathbf s+\mathrm i\mathbf t
\Big)
\in
\mathbb R\times T^*T^d\times\mathbb C^{n-1-d}
\,.
\]
We will show that a similar parametrisation
near each of the non-standard holomorphic discs
in $\mathcal W$ exists so that $\mathcal W$
will be a smooth manifold of dimension $2n-1$.

\subsubsection{Maslov index\label{subsubsec:maslovindex2}}

The Maslov index of all $u\in\mathcal W$ is equal to $2$.
By the considerations in \cite[Lemma 3.1]{gz16b}
it is enough to compute
the Maslov index for all standard discs $u\in\mathcal W$,
which lift up to standard holomorphic discs in
\[
\mathbb R\times\mathbb R\times T^*\mathbb R^d\times D^2\times\mathbb C^{n-1-d}
\equiv
\mathbb R\times\mathbb R\times D^2\times\mathbb C^{n-1}
\]
in the sense of \cite{gz16b}.

\subsubsection{Simplicity\label{subsubsec:simplicity}}

Using Lemma \ref{lem:indecomposable}
and \cite[Theorem A]{lazz11}
we see that all holomorphic discs
$u\in\mathcal W$ are simple,
cf.\ \cite[Lemma 3.4]{gz16b}.
Based on that one shows as in
\cite[Lemma 3.5]{gz16b}
that for all $u=(a,f)\in\mathcal W$
the set of all $f$-injective points
is open and dense in $\mathbb D$.
One only has to observe
that the projection $h_0$ of $f$
to the $D^2$-factor is an embedding
along $\partial\mathbb D$ so that $u|_{\partial\mathbb D}$
is positively transverse to the characteristic leaves
-- as done in Section \ref{subsubsec:maxprinc}.

\subsubsection{Linearised Cauchy--Riemann operator\label{subsubsec:lincrop}}

Based on Section \ref{subsubsec:simplicity}
one chooses a regular almost complex structure $J$
as in \cite[Section 5.2]{gz16b}
so that the linearised Cauchy--Riemann operator $D_u$
is onto for all $u\in\mathcal W$.
Taking variations of the level parameters
$(\mathbf p,\mathbf t)\in\mathbb R^d\times\mathbb R^{n-1-d}$
for the Lagrangian boundary conditions
$L^{\mathbf t}_{\mathbf p}$ induced by translations
similarly to \cite[Section 4.1]{gz16b}
one computes
using Section \ref{subsubsec:maslovindex2}
the Fredholm index of $D_u$
to be $2n+2$ as in \cite[Section 5.1]{gz16b}.
Subtracting $3$ for the marked points
fixed by three characteristic leaves yields $2n-1$,
which turns out to be the dimension of $\mathcal W$.
In fact, $\mathcal W$ is a smooth manifold
that admits a natural orientation
obtained by the orientation of $D_u$
described in \cite[Section 5.3]{gz16b}
-- observe that $L^{\mathbf t}_{\mathbf p}$
admits a canonical parallelization and
and the above mentioned variations
of the Lagrangian boundary conditions $L^{\mathbf t}_{\mathbf p}$.

\section{The homotopy type\label{sec:thehomotopytype}}

In Section \ref{sec:abdryvalprob} we showed
that the moduli space $\mathcal W$ is a smooth,
naturally oriented manifold of dimension $2n-1$
such that the evaluation map
\[
 \begin{array}{rccc}
  \ev\colon\thinspace & \mathcal W\times\mathbb D     & \longrightarrow & \hat{Z}\\
            & \bigl( (a,f),z\bigr) & \longmapsto     & f(z)
 \end{array}
\]
is proper and of degree one.
In this section we will use the evaluation map
in order to draw conclusions
on the homotopy type of the contact manifold $M$.

\subsection{Homology type and fundamental group\label{subsec:homologytypefundamantal}}

With the argumentation used in
\cite[Sections 2.3 and 2.5]{bgz}
and \cite[Section 6]{gz16b}
we obtain that the evaluation map
\[
\ev\colon\thinspace\mathcal W\times\mathbb D\longrightarrow\hat{Z}
\]
is surjective in homology and $\pi_1$-surjective.
The restriction to $z_0=1$ is given by
\[
\ev\colon\thinspace\mathcal W\times\{1\}
\longrightarrow
\bigcup_{(\mathbf p,\mathbf t)\in\mathbb R^d\times\mathbb R^{n-1-d}}
L^{\mathbf t}_{\mathbf p}\cap\{z_0=1\}
\]
the target being equal to
\[
\mathbb R\times T^*T^d\times\{1\}\times\mathbb C^{n-1-d}
\,.
\]
Both evaluation maps
complete to a commutative square
\begin{diagram}
\mathcal W \times \mathbb D &
	\rTo^{\qquad\qquad\ev} &
	\hat{Z} \\
\uTo_\subset &
	&
	\uTo_\subset \\
\mathcal W\times\{1\} &
	\rTo^{\quad\quad\ev\quad\quad} &
	\mathbb R\times T^*T^d\times\{1\}\times\mathbb C^{n-1-d} \\
\end{diagram}
via the homotopy equivalence
\[
\mathcal W\times\{1\}
\subset
\mathcal W\times\mathbb D
\]
and the inclusion
\[
\mathbb R\times T^*T^d\times\{1\}\times\mathbb C^{n-1-d}
\subset\hat{Z}
\,.
\]
Therefore, the induced map
\[
T^d\subset\hat{Z}\simeq M
\]
is surjective in homology and $\pi_1$-surjective too.

Similarly to \cite[Section 2.4]{bgz} one shows
-- by simply replacing $2n$ by $2n+1$ --
that $H_kM=0$ for all higher degrees
$k\geq d+1$ and that the inclusion of
\[
\partial M\longrightarrow M
\]
induces an isomorphism for the homology groups $H_k$
of low degree
\[
k=0,1,\ldots,2n-1-d
\,.
\]
Therefore,
\[
H_*M=H_*\Big(T^d\times D^{2n+1-d}\Big)
\,.
\]
Because the fundamental group of $\partial M$
is abelian we infer with \cite[Section 2.5]{bgz}
that the inclusion $\partial M\subset M$
is $\pi_1$-isomorphic.
In particular,
\[
\pi_1M=\pi_1\Big(T^d\times D^{2n+1-d}\Big)
\,.
\]

\subsection{A cobordism\label{subsec:acobordism}}

Recall that $\hat{Z}$
is obtained by removing the bounded component
of the complement of $\varphi(\partial M)\subset\Int(Z)$ in $Z$
and gluing with $M$ along the boundaries via $\varphi$.
By assumption,
$\varphi(\partial M)$ is isotopic to the sphere bundle
$S\big(T^*T^d\oplus\underline{\mathbb R}^{2n+1-2d}\big)$
inside $Z$ viewed as a subset of
$\mathbb R\times T^*T^d\times D^2\times\mathbb C^{n-1-d}$.
Choosing a suitable bundle metric for
$T^*T^d\oplus\underline{\mathbb R}^{2n+1-2d}$
we assume that $\varphi(\partial M)$ is contained
in the interior of the corresponding disc bundle
$D\big(T^*T^d\oplus\underline{\mathbb R}^{2n+1-2d}\big)$.
After gluing $M$ to the unbounded component
in the total space we obtain
a manifold denoted by $M_1$,
which deformation retracts onto $M$
and is homotopy equivalent to $\hat{Z}$.
Furthermore,
we denote by $M_0$
a possibly rescaled copy of
$D\big(T^*T^d\oplus\underline{\mathbb R}^{2n+1-2d}\big)$
to which a nowhere vanishing section is added
-- before gluing --
so that $M_0$ is contained in $M_1\setminus M$ -- after gluing.
In the following we will study
homotopical properties
of the cobordism
\[
X:=M_1\setminus\Int M_0
\,.
\]

Denote by $D_0\subset\partial M_0$
and $D_1\subset\partial M_1$
in $M_1$ isotopic copies of the disc bundle of
$\mathbb R\times T^*T^d\times\{1\}\times\mathbb C^{n-1-d}$,
which strongly deformation retracts to $T^d$.
The isotopy between $D_0$ and $D_1$
can be chosen to be the restriction of
the above used shift and rescaling
isotopy between $\partial M_0$ and $\partial M_1$,
and extends to an isotopy between
$M_0$ and $M_1$.
Therefore, we obtain the following
homotopy commutative diagram:
\begin{diagram}
\HmeetV  &
	\rDashto &
	&
	&
	&
	&
	X \\
&
	&
	&
	&
	&
	\ldTo_{\substack{\textrm{gen.}\\\textrm{pos.}}} &
	\\
&
	&
	M_0 &
	\rTo^{\textrm{time-1 map}}_{\textrm{of isotopy}} &
	M_1 &
	&
	\\
\uDash&
	\ruTo_{\substack{\textrm{gen.}\\\textrm{pos.}}} &
	&
	&
	&
	\luDoubleto &
	\uDashto\\
\partial M_0 &
	&
	\uTo_{\simeq} &
	&
	&
	&
	\partial M_1 \\
\dLine &
	\luTo &
	 &
	&
	\uTo &
	\ruTo&
	\\
&
	&
	D_0 &
	\rTo^{\textrm{time-1 map}}_{\textrm{of isotopy}} &
	D_1 &
	&
	\uTo \\
\HmeetV &
	&
	&
	\rLine^{\substack{\\ \\ \\ \textrm{time-1 map}}}_{\textrm{of former isotopy}} &
	&
	&
	\HmeetV \\
\end{diagram}
where all indicated maps are obtained by inclusion
with the exception of the diffeomorphisms
$D_0\rightarrow D_1$ and $\partial M_0\rightarrow\partial M_1$.
The arrow $M_0\rightarrow M_1$ can be alternatively understood
to mean the time-$1$ map of the described isotopy
besides the meaning of the inclusion.

As in \cite[Lemma 5.1 and 5.2]{bgz} one shows
that the inclusions
\[
\partial M_0, \partial M_1\longrightarrow X
\]
are $\pi_1$- and $H_*$-isomorphic.
Indeed,
in low degrees $k=0,1,\ldots,2n-1-d$
this follows with
the results stated in
Section \ref{subsec:homologytypefundamantal}
and
general position arguments,
which are available whenever $k+d<2n+1$,
using $n>d$ and $n\geq 2$;
the arrows labeled by {\it gen.\ pos.} are
homotopy (resp. homology) isomorphisms in those degrees
in view of the induced long exact sequences.
In higher degrees $k\geq d+1$,
in which the homology groups of $M_0$ and $M_1$
vanish by Section \ref{subsec:homologytypefundamantal},
this follows with the induced long exact sequence
of the pair $(M_1,M_0)$, excision and
Poincar\'e duality
applied to the compact cobordism $X$
in combination with the universal coefficient theorem.

\begin{rem}
 \label{rem:isincliso?}
 The above arguments show
 that the inclusion of $M_0$ into $M_1$
 induces an isomorphism in homology.
 This is {\it a priori} not clear even if the involved
 homology groups are isomorphic.
\end{rem}

\subsection{Being an $h$-cobordism\label{subsec:bnhcob}}

In order to prove that the inclusions
$\partial M_0, \partial M_1\rightarrow X$
are in fact homotopy equivalences
one shows that the topological pairs
$(X,\partial M_0)$ and $(X,\partial M_1)$
are homotopically trivial.
As these are homotopically trivial
an application of the relative Hurewicz theorem
shows that the quotients of
the relative homotopy groups
by the action of the fundamental group of $\partial M$,
which is isomorphic to $\mathbb Z^d$,
are trivial.

To conclude with the vanishing of the relative homotopy groups
we will employ universal coverings as done in the argumentation
in \cite[Section 6]{bgz}.
The following proposition is based on the triviality
of $\pi_1(X,\partial M_0)$ and $\pi_1(X,\partial M_1)$
obtained in Section \ref{subsec:acobordism}:

\begin{prop}
\label{prop:xinanhcob}
 $X$ is an $h$-cobordism.
\end{prop}

In order to prove Proposition \ref{prop:xinanhcob}
we consider the universal covering
$\pi\colon\thinspace\widetilde{M}\rightarrow M$ provided with the
contact form $\tilde{\alpha}:=\pi^*\alpha$,
which satisfies $\inf_0(\tilde{\alpha})>\pi$.
Observe that the covering $\pi$ is infinite
as $\pi_1M=\mathbb Z^d$, see
Section \ref{subsec:homologytypefundamantal}.
The induced covering
on the boundary
$\pi|_{\partial\widetilde{M}}\colon\thinspace\partial\widetilde{M}\rightarrow\partial M$
is given by the universal covering
$\mathbb R^d\times S^{2n-d}\rightarrow T^d\times S^{2n-d}$
caused by the $\pi_1$-isomorphicity
of the inclusion $\partial M\subset M$, see again
Section \ref{subsec:homologytypefundamantal}.
Furthermore,
the universal covering space of $\hat{Z}$
is made up by the analogue
of the gluing construction of $\hat{Z}$
involving this time
$\widetilde{M}$ and
\[
\widetilde{Z}=
\mathbb R\times T^*\mathbb R^d\times D^2\times\mathbb C^{n-1-d}=
D^2\times\mathbb R^{2n-1}
\]
with the gluing map being a lift of $\varphi\circ\pi$.
The universal covering map of $\hat{Z}$
restricts to $\pi$ on $\widetilde{M}$.

Similarly to Section \ref{sec:abdryvalprob}
and \cite[Section 6]{bgz}
one defines a moduli space $\mathcal W'$
of holomorphic discs
in the covering space of $\hat{Z}$
with respect to the lift
of the almost complex structure $J$.
This results into a covering
$\mathcal W'\rightarrow\mathcal W$ of moduli spaces.
As in \cite[Lemma 6.1]{bgz} one shows
that the evaluation map
\[
\ev\colon\thinspace\mathcal W'\times\mathbb D\longrightarrow\widetilde{\hat{Z}}
\]
is proper of degree $1$
because the projections of all holomorphic discs
in $\mathcal W'$ to the $\mathbb R$-factor of the symplectisation
are contained in a uniform compact interval.
Alternatively, one can compensate
the non-compactness caused by
$\widetilde{M}$
with the results in \cite{bwz}
applied to the {\it trivial} virtually contact structure
given by the universal covering of $\hat{Z}$.
Here, by a {\bf virtually contact structure} we mean
a Riemannian covering together with a contact form primitive
of the pull back of an odd-symplectic form on the base
that is uniformly bounded from below and above.
The virtually contact structure is {\bf trivial}
if the contact form is obtained from a primitive
of the odd-symplectic from on the base by pull back.

As in Section \ref{subsec:homologytypefundamantal}
one considers a diagram
\begin{diagram}
\mathcal W ' \times \mathbb D &
	\rTo^{\qquad\qquad\ev} &
	\widetilde{\hat{Z}} \\
\uTo_\subset &
	&
	\uTo_\subset \\
\mathcal W ' \times\{1\} &
	\rTo^{\quad\quad\ev\quad\quad} &
	\mathbb R\times T^*\mathbb R^d\times\{1\}\times\mathbb C^{n-1-d} \\
\end{diagram}
and concludes that $\widetilde{M}$ is contractible,
cf.\ \cite[Proposition 6.2]{bgz}
and its preceding remarks in \cite{bgz}.

\begin{proof}[{\bf Proof of Proposition \ref{prop:xinanhcob}}]
  With the above shown contractibility of $\widetilde{M}$
  the claim follows with the purely topological
  argumentation used in \cite[Theorem 9.1]{bgz}.
  Alternatively, one can follow the reasoning
  in \cite[Section 8]{bgz} or \cite[Section 2.5]{kbdiss}
  invoking simplicity of the topological space $\partial M$
  and the cobordism diagram from
  Section \ref{subsec:acobordism} similarly to \cite[Lemma 6.3]{bgz}.
\end{proof}

\begin{proof}[{\bf Proof of Theorem \ref{thm:mainthm}}]
 With Proposition \ref{prop:xinanhcob}
 the claim follows with standard arguments
 based on Whitehead's theorem, $\mathrm{Wh}(\mathbb Z^d)=0$,
 and the $s$-cobordism theorem
 as done in \cite{kbdiss,bgz}.
\end{proof}

\section{Appendix: Local contact inversion \label{app:loconinv}}

As in knot theory we call
the complement of the interior of a tubular neighbourhood
of a submanifold $Q$ the {\bf exterior} of $Q$.
We show that a collar extension of the exterior
of a subcritically isotropic torus in a contact manifold
admits a positive contact inversion
along the boundary of the exterior.
This allows surgerial constructions for contact manifolds
near subcritically isotropic tori
similar to the considerations in Section \ref{subsec:filling}.

\subsection{Hypersurfaces transverse to a Liouville flow\label{subsec:hyptrnliouvecfield}}

Let $(V,\lambda)$ be a Liouville manifold
with symplectic form $\omega=\mathrm d\lambda$
and Liouville vector field $Y$ defined by
$i_Y\omega=\lambda$.
Let $M_0$ and $M_1$ be hypersurfaces in $V$
that are transverse to $Y$
such that $\alpha_i:=\lambda|_{TM_i}$, $i=0,1$,
is a contact form.
We assume that
each flow line of $Y$
intersects each of $M_0$ and $M_1$ in a single point
so that the resulting correspondence
forms a bijection $M_0\rightarrow M_1$.

Under the stated assumptions
we find a domain $D$ in $\mathbb R\times M_0$
that contains $\{0\}\times M_0$
such that for all $p\in M_0$
the intersection of $D$ with each line $\mathbb R\times\{p\}$
is the maximal interval on which the flow line
$t\mapsto\varphi_t(p)$ of $Y$ is defined.
This defines an
embedding $\Phi\colon\thinspace D\rightarrow V$ of Liouville manifolds
(i.e.\ $\Phi^*\lambda=\mathrm e^t\alpha_0$)
via $\Phi(t,p)=\varphi_t(p)$ for all $(t,p)\in D$
such that $M_1\subset\Phi(D)$
is the $\Phi$-image of the graph of a smooth function $f\colon\thinspace M_0\rightarrow\mathbb R$
and $\alpha_1$ corresponds to the contact form
$\mathrm e^f\alpha_0$ on $\Phi^{-1}(M_1)$.
Setting $\psi(0,p)=\big(f(p),p\big)$ for all $p\in M_0$
we obtain a strict contactomorphism
$\Phi\circ\psi\colon\thinspace(M_0,\mathrm e^f\!\alpha_0)\rightarrow(M_1,\alpha_1)$.

\subsection{A model involution\label{subsec:amodleinvolution}}

We consider the Liouville manifold
\[
\Big(T^*T^d\times\mathbb C^{n+1-d},
\mathbf p\!\;\mathrm d\mathbf q
+\frac12\big(\mathbf x\mathrm d\mathbf y-\mathbf y\mathrm d\mathbf x\big)\Big)
\]
with symplectic form
\[
\mathrm d\mathbf p\wedge\mathrm d\mathbf q
+\mathrm d\mathbf x\wedge\mathrm d\mathbf y
\]
and Liouville vector field
\[
Y_0=\mathbf p\!\;\partial_{\mathbf p}
+\frac12\big(\mathbf x\partial_{\mathbf x}+\mathbf y\partial_{\mathbf y}\big)\,,
\]
which is transverse to
\[
M=\Big\{|\mathbf p|^2+|\mathbf x|^2+|\mathbf y|^2=1\Big\}
\]
defining a contact form $\alpha_0$ on $M$.
The involution
\[
\iota(\mathbf q,\mathbf p;\mathbf x,\mathbf y)=(\mathbf q,\mathbf p;-x_1,x_2,\ldots,x_{n+1-d},-y_1,y_2,\ldots,y_{n+1-d})
\]
preserves the Liouville form
and the $M$-defining distance function
inducing a strict contactomorphism of $(M,\alpha_0)$.
Observe that $\iota$ interchanges
\[
M\cap\big\{\!\pm\!x_1\geq0\big\}\cong T^d\times D^{2n+1-d}
\]
as well as the isotropic tori
\[
T_{\pm}=\big\{(\mathbf q,{\mathbf 0};\pm\!1,0,\ldots,0)\,\big|\,\mathbf q\in T^d\big\}\,.
\]

We remark
that the (by a Hamiltonian vector field)
shifted Liouville vector field
\[
Y_1=Y_0+\frac14\partial_{x_1}
\]
defines the standard (rotationally invariant) contact form
\[
\alpha_1=\mathrm d t+\mathbf p\!\;\mathrm d\mathbf q
+\frac12\big(\mathbf x\mathrm d\mathbf y-\mathbf y\mathrm d\mathbf x\big)
\]
on
\[
M_1=T^*T^d\times\left\{x_1=\frac23\right\}\times\mathbb R_{y_1}\times\mathbb C^{n-d}
\equiv\mathbb R_t\times T^*T^d\times\mathbb C^{n-d}\,,
\]
where $y_1$ is renamed in $t$ and
where now $\mathbf x$ and $\mathbf y$
stand for the corresponding tuples
with $x_1$ and $y_1$ deleted.
With \cite[Example 2.1.3]{gei08}
$\alpha_1$ can be brought to $\alpha_Z$
by a strict contactomorphism.

\subsection{Interpolating Liouville vector fields\label{subsec:intliouvec}}

We continue the discussion from Section
\ref{subsec:amodleinvolution}.
Let $\chi\equiv\chi(\mathbf q,\mathbf p;\mathbf x,\mathbf y)$ be the cut off function
$\tilde{\chi}\big(|\mathbf p|^2+|\mathbf x|^2+|\mathbf y|^2\big)$
induced by a smooth function $\tilde{\chi}\colon\thinspace[0,\infty)\rightarrow[0,1/4]$
that is equal to $0$ on $[0,1]$, strictly increasing on $(1,3/2)$,
and equal to $1/4$ on $[3/2,\infty)$.
With respect to the Hamiltonian function $H=-\chi y_1$
we define the Liouville vector field $Y=Y_0+X_H$,
where
\[
X_H=
2\tilde{\chi}'y_1
\big(
-\mathbf p\!\;\partial_{\mathbf q}+\mathbf y\partial_{\mathbf x}-\mathbf x\partial_{\mathbf y}
\big)
+\chi\partial_{x_1}
\]
denotes the Hamiltonian vector field of $H$.
The Liouville vector field $Y$ equals $Y_0$
on $\big\{|\mathbf p|^2+|\mathbf x|^2+|\mathbf y|^2\leq1\big\}$
and $Y_1$ on
$\big\{|\mathbf p|^2+|\mathbf x|^2+|\mathbf y|^2\geq3/2\big\}$;
each flow line of $Y$ connects $M$ with
$\big\{|\mathbf p|^2+|\mathbf x|^2+|\mathbf y|^2=3/2\big\}$
as
\[
\mathrm d\big(|\mathbf p|^2+|\mathbf x|^2+|\mathbf y|^2\big)(Y)
=|\mathbf p|^2+\frac12\big(|\mathbf x|^2+|\mathbf y|^2\big)+\chi x_1
\]
defining a bijection between the two hypersurfaces.
Denote by $M_0$ the set of intersection points
of $M$ with those flow lines of $Y$ that intersect
$\big\{|\mathbf p|^2+|\mathbf x|^2+|\mathbf y|^2=3/2\big\}$
along $\big\{x_1>-1/2\big\}$.
Observe that $M_0$ is an open neighbourhood of
$M\cap\{x_1=0\}$ in $M$
as
\[\mathrm d x_1(Y)=\frac12x_1+2\tilde{\chi}'y_1^2+\chi\]
is positive on $\big\{1<|\mathbf p|^2+|\mathbf x|^2+|\mathbf y|^2\big\}$
along $\{x_1=0\}$.
With the considerations from Section
\ref{subsec:amodleinvolution}
we obtain a strict contactomorphism
$\varphi\colon\thinspace(M_0,\mathrm e^f\!\alpha_0)\rightarrow(M_1,\alpha_1)$
defined in a neighbourhood of the invariant set $M\cap\{x_1=0\}$
of the involution $\iota$.

\subsection{Local inversion\label{subsec:locinv}}

Continuing the discussion from Section
\ref{subsec:intliouvec} we observe
that $\varphi\circ\iota\circ\varphi^{-1}$ defines
a strict contactomorphism of $\alpha_1$ on
$\varphi\big(M_0\cap\iota(M_0)\big)$
that leaves $\varphi\big(M_0\cap\{x_1=0\}\big)$
invariant changing (co-)orientations.
We call such a positive (not necessarily strict)
contactomorphism a {\bf contact inversion}
along $\varphi\big(M_0\cap\{x_1=0\}\big)$.

Observe that $\varphi\big(M_0\cap\{x_1=0\}\big)$
as boundary of $\varphi\big(M_0\cap\{x_1\geq0\}\big)$
can be brought into any neighbourhood
of the zero section
$\varphi\big(M_0\cap T_+\big)$ of
$T^*T^d\oplus\underline{\mathbb R}^{2n+1-2d}$
using the contact vector field
\[
X=t\partial_t+\mathbf p\!\;\partial_{\mathbf p}
+\frac12\big(\mathbf x\partial_{\mathbf x}+\mathbf y\partial_{\mathbf y}\big)
\]
on $\big(\mathbb R\times T^*T^d\times\mathbb C^{n-d},\alpha_1\big)$.
Conjugating the contact inversion
$\varphi\circ\iota\circ\varphi^{-1}$
with the flow of $X$ we obtain combined with
the isotropic neighbourhood theorem:

\begin{prop}
\label{prop:contact-inversion}
 Any isotropic submanifold
 of a given contact manifold,
 whose conformally symplectic normal bundle
 possesses a non-vanishing section,
 admits a tubular neighbourhood $U$ together with a
 contact inversion along the boundary $\partial U$.
\end{prop}

Indeed, this is because
the submanifold being a subcritically isotropic torus
with trivial conformally symplectic normal bundle
is not (really) used in the above construction.

\begin{ack}
  This work could not ever been finished without the
  immense support of
  Peter Albers and Hansj\"org Geiges.
  We would like to thank Joel Fish
  for drawing our attention to {\it symplectic dynamics}
  and Marc Kegel for showing us the beauty of knot complements.
\end{ack}

redpajama_set_name: RedPajamaArXiv


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Mourad Hamyd describes himself as a normal 18-year-old who lives with his parents but on the day France was gripped by the Charlie Hebdo massacre, he became known as "the third suspect" - even though he was in class at the time of the shooting.
The brother-in-law of one of the two gunmen turned himself into police on Wednesday, horrified and baffled to hear his name circulating in the news and on social media.
But the teenager was released without charge on Friday evening, relieved but badly shaken.
"I was stunned, completely overwhelmed by the events but the police officers were very correct with me," the visibly tired high school student told AFP in the presence of his lawyer Marie Calleghaer and several family members.
Numerous witnesses verified he was in school when brothers Cherif and Said Kouachi gunned down 12 people at the Paris office of the satirical magazine Charlie Hebdo, kicking off three days of violence that left a total of 17 people dead and traumatised a nation.
"I'm in shock, people said horrible and false things about me on social media even though I am a normal student who lives quietly with his parents," said the bespectacled teenager who one day hopes to study medicine.
"The attack was horrific and my thoughts are with the victims," he added.
After a massive manhunt, the Kouachi brothers were killed in a dramatic police standoff on Friday.
It remains unclear how Hamyd, who until this week was completely unknown to the police, became linked to the magazine attack.
But as he reflected on his name appearing in news reports and online posts across the world, he said he feared the wrongful accusation would haunt him for years to come.
"I only hope this won't taint my future," he said.
"I have nothing to do with this whole thing, Cherif is just my brother-in-law with whom we have a fairly distant relationship," he said, adding that Cherif lived in the Paris region and rarely visited his northern home town.
Hamyd's sister, who is married to Cherif, was taken into custody on Wednesday and freed on Saturday.
"I am sure she is also innocent," an unnamed relative told AFP.
As France prepared for a huge march in Paris in honour of the victims on Sunday, Hamyd's family told AFP they would attend a rally in their town of Charleville-Mezieres to pay their own tribute.
But Hamyd, hoping to fade back into anonymity as soon as possible, won't be joining them.


The Canadian statistical agency said Thursday the US government shutdown is having impacts beyond its borders, stemming data sharing and forcing Ottawa to suspend publication of trade figures. The United States is Canada's largest trading partner, with about 75 percent of Canadian exports sent to its southern neighbor. Statistics Canada and the US Census Bureau (USCB) share economic data collected by the two nations' respective customs agencies to produce trade reports.


text:
BHS DISTINGUISHED GRADS APPROVED STARTING WITH TEAM 2023
Team 2023, will be recognized at the end of their four years at Brownsburg High School under a new Latin honors system. The Distinguished Graduate recognition program was determined by a committee of parent volunteers, teachers, and administrators, and approved by the BCSC Board of School Trustees for Team 2023 and beyond. Beginning with graduation in 2023, students will be recognized:
Summa Cum Laude (greatest honors) for a achieving a cumulative GPA of 4.20 or higher
Magna Cum Laude (great honors) for achieving a cumulative GPA from 4.00 to 4.199
Cum Laude (with honors) for achieving a cumulative GPA of 3.80 to 3.999
Dale Sharpe
Rebekah Frazer
Sarah Mahan
Carlos Calero
Jamie Tewell
College / Career Visit Form

redpajama_set_name: RedPajamaCommonCrawl


text:
MATCH RATINGS : Manchester United 1-1 Chelsea – FA Youth Cup
Jonny Sutherland (7): Was the quieter of the two keepers. He dealt well with what was thrown at him and could do nothing about the goal. I was very impressed with his composure on the ball, he was always looking to play it to a United defenders feet.
Liam Grimshaw (7) A good game from Liam. He was defensively solid and attacked as much as he could and he provided a great outlet down the right flank. His passing was sloppy in patches but a good all round performance from him.
Luke McCullough (7) A very solid game from him. He won everything in the air and his passing was as impressive as always. Hesitated for a split second for Chelsea's goal but marshalled the defence magnificently.
Nic Ioannou (9) (MOTM) A brilliant performance from the 16 year old Cypriot. He won everything in the air,brought the ball out from the back and his fantastic pace meant that he cancelled out the Chelsea attacks greatest asset. Our most mature performance came from our youngest player on the night.
Tyler Blackett (5) It is fair to say that this was not one of Tyler's better games. He was often caught out of position which helped Chelsea counter attack. He was very wasteful with the ball, often playing a sloppy pass and almost always opting to cut inside on to his weaker right foot when, which mainly resulted in him losing the ball.
Jack Barmby (8) This lad continues to impress me. He was our only attacking threat, and at times we relied on him too much. Jack beat his man countless times and was always willing to take on players and inject place into our game. Every time he was on the ball, you felt that he would cause Chelsea a problem, and more often than not he did.
Joe Rothwell (8) A magnificent performance from Joe. Everything we did went through him. His composure and vision is very good, and he has a great level of footballing intelligence about him for one so young. He makes the game look so simple with his pass and move mentality.
Ben Pearson (6) A very good first half from Ben, he was everywhere, constantly snapping at Chelsea heels and putting them under pressure. As the game wore on, he tired, which resulted in him giving away a lot of sloppy balls.
Gyliano Van Velzen (4) A very poor game from Gyliano. It seemed like everything that could have gone wrong for him, did. He was very poor on the ball, losing it far too often. In the second half he drifted out of the game completely and it was like we were playing with ten men.
Mats Daehli (5) A very ineffective performance from Mats. He struggled to get involved in the game at all. The game just bypassed him completely and I could count on one hand the amount of meaningful touches on the ball he had. Was substituted with ten minutes left.
Sam Byrne (5) Was playing as lone a lone striker and found it very difficult to hold the ball up when challenged by Chelsea's centre backs. He was not helped by the fact he rarely had any support. He worked tirelessly as usual, but to no avail. Unlucky not to get on the score sheet.
One to watch: This weeks "one to watch" is a player who has impresses me every time I have seen him play this season. Jack Barmby, son of Nick, is a left footed right winger, who has just about everything required to make it at the highest level. He is the U18's top scorer this year and has scored in every round of the FA Youth Cup.
A very tricky winger with a great burst of acceleration, Jack is a huge threat to opposition defences. He is always willing to take on the full back, but his great intelligence allows him to know when to release the ball. He is at his best on the right wing, where he can cut inside on his left peg and deliver a long range shot, another great attribute of his.
Jack also has great upper body strength,he is not knocked off the ball easily. This is a great trait for him to have as he will no doubt be the subject of some vicious challenges in the future due to his confident nature.
Jack has no doubt been the stand out player for the U18's this term, and if he keeps impressing and performing like he is doing, a first team appearance may be on the cards within the next two years.
Follow @19Steve19
Follow @PeoplesPerson_


The speed of Twitter can be intimidating to even the most social media-savvy communicator. But Krisleigh Hoermann, community manager for the American Heart Association, believes the platform is the perfect place to host a conversation—albeit a fast-paced one—with influencers, community members and the media. Hoermann will discuss engagement tactics, how to build a following on Twitter and more at PR News' Digital PR Summit on Feb. 16 in San Francisco. We offer a preview of her presentation here.
PR News: What are the core elements of an effective Twitter PR strategy?
Krisleigh Hoermann: You should always begin by listening. Listen to what is being said about you, about your competitors and what your target audience and brand advocates are talking about. Target your audience just as you do in traditional PR methods. Who is in your sweet spot? Find them on Twitter—if they are on Twitter—and talk to them. Think about your personality—make your corporate culture clear to your followers through your own personality. Don't forget to measure your results. Find the stats that matter to your brand and track them.
PR News: How often do you post new content or engage with your followers on Twitter?
Hoermann: Think of Twitter as a chat. It's a faster pace than many other digital methods and therefore can handle more tweets within one day. The American Heart Association strives to send out four content-driven messages a day. We also try to respond to most, if not all, mentions of @American_Heart and be an active participant in relevant conversations around heart health and stroke. You have to scale it to your brand. Some brands only need three content-driven posts a week while others need more.
PR News: How many staffers are assigned to social media and much time are you spending a day on Twitter efforts?
Hoermann: My job is social media, therefore HootSuite is running all day. We have two people actively posting from our Twitter handle and have about 10 people who are subject matter experts for different questions and content areas. Organizationally, we have about 50 account holders from national down to our affiliate staff.
PR News: What's one tip you'll offer attendees at the Feb. 16 Digital PR Summit?
Hoermann: I'll show you how fun and easy it is to engage people on Twitter—whether through content-rich posts or Twitter parties. Your community is waiting.
Attend PR News' Digital PR Summit on Feb. 16 in San Francisco and learn more from social media experts like Krisleigh Hoermann.


How much does Viscira pay in San Francisco, CA?
Salaries at Viscira range from an average of $43,716 to $117,637 a year. A Project Management Director at Viscira makes the most with an average salary of $117,637 per year, while a Office Manager makes the least with an average salary of $43,716 per year.


text:
Q: RTF Datawindow and RTF Control Content how to find the text by color, Font name? i stored the RTF file format in my ASA database.
How to select the content by using the color or Font name, Alignment.
for ex:
i have a content it has 100 lines with formatted text in between it contains the text which is in blue color, i need to select dynamically by giving the color "blue"
is it possible?

A: In RTF, you have a colour table at the top which defines all the colours used.
With only blue defined, it will look something like this:
{\colortbl ;\red0\green0\blue255;}

Then in the actual text, you have \cf1 to change to the first defined colour, and \cf0 to change back to black, like this:
some black text\par
\cf1 this is blue text\cf0\par
more black text\par

(\par means  paragraph)

redpajama_set_name: RedPajamaStackExchange


text:
Indie Spotlight: Good Time (2017)
For this weeks Indie Spotlight, Tom Leach reviews Good Time starring Robert Pattinson.
Tom Leach
Image:IMDB
In recent years, New York natives Josh and Benny Safdie have compounded themselves as two of the most exciting young directors in the industry today. Their brand of stressful and character-driven crime dramas entered the mainstream with the release of the critically acclaimed Uncut Gems in 2019, but 2017's Good Time, starring Robert Pattinson, is unmissable.
Following a bank robbery, Connie Nikas (Pattinson) must try and make bail to keep his developmentally disabled brother Nick (Benny Safdie) from prison. Set in a gritty Queens, New York, the film reflects on criminality, family, and inner-city life. Good Time is no cheap thrill; the characters are flawed, people acting with ambiguous morals, provoking thought about how far one would go to protect the ones they love. Good Times does not provide a cut-and-dry 'good guy', not even the archetypal anti-hero of many crime films.
Director Benny Safdie as stars in the movie as Nick Nikas. Image:IMDB
Stand out performances include Pattinson and Safdie as brothers Connie and Nick Nikas. Good Time was revelatory for many viewers, showing Pattinson's skill post-Twilight. His performance is packed with strong, volatile emotion, creating great chemistry with other characters, driving the anxiety of the film. Benny Safdie, director by trade, and neurotypical, performs a respectful portrayal of a man with unnamed developmental disabilities. There is discussion to be had about boosting the presence of disabled actors in the film industry, though Safdie makes sure not to play a caricature. Other actors are exemplary of the Safdie Brother's brilliant casting of normal, first time actors, often natives of New York and individuals formerly/currently involved in the lives presented on screen. This method provides a layer of realism and uniqueness to the performances of characters that would otherwise be unfaithfully done by the legions of the upper-middle classes in Hollywood.
Alongside the acting, Good Time uses the cinematography to drive the turbulent energy. Most of the shots in the film are close-up, creating an uncomfortable and claustrophobic atmosphere. The close-ups also allow the audience to feel the emotions of the characters on screen. There are also visual motifs of vivid colours, such as Connie's signature red coat and bleach blonde hair, or the almost magical look of the green sprite bottle (an important plot device) that draw focus against the dark background – a metaphorical light in the darkness of the setting.
Underpinning it all is the frantic electronic soundtrack of Oneohtrix Point Never (Daniel Lopatin), who also worked with the Safdies on Uncut Gems. Reminiscent of the 1970s soundtracks of Tangerine Dream, the music keeps the energy moving throughout the film. A particular stand out track is the decidedly slower and more poignant The Pure and The Damned, featuring Iggy Pop, which scores the heart-wrenching ending of the film.
Good Time excels in intensity, spurred by its strong acting and stunning visuals. Its grimy and realistic characters do not lead to a particularly easy watch, but its energy makes for an extremely fresh experience in an age of slow and not-very-interesting thrillers.
AUTHOR: Tom Leach
Spanish and German student. Interested in cultural studies and left-wing politics globally. Twitter: @tleachleach
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Re-seen on the big screen


text:
Q: How can I get the Tweeter in Animal Crossing New Leaf? In Animal Crossing New Leaf, I searched and searched, but I found no tweeter! Am I supposed to invite friends to my town? Or am I supposed to earn it?

A: According to a wiki

It is obtained from visitors in the Happy Home Showcase you have streetpassed 9 or 10 times.
It can also be obtained one from Isabelle at the New Year's Eve countdown if the player speaks to her twice.

A: According to the Animal Crossing Wikia Tweeter page, you're supposed to Streetpass™ a single person ten times to get it.
From the AC3Wiki:

*

*First time: Nothing

*Second time: balloon

*Third time: balloon

*Fourth time: pinwheel

*Fifth time: bunny balloon

*Sixth time: bubble wand

*Seventh time: heart balloon

*Eighth time: ice cream

*Ninth time: soft-serve ice cream

*Tenth time: tweeter

*More than ten times: Random item listed above

You will have to go to the plaza and talk to the person you Streetpassed™.
Supposedly, they will be holding your prize and they will give it to you after speaking with them.
Alternatively, if you talk to Isabelle twice on New Year, she'll give you one.

A: You get the tweeter through streetpass in your happy homes showcase. The tenth time you get someone, a copy of them will be standing in the front with a tweeter, and when you talk to them, they will hand it to you.

A: In order to receive the tweeter,you have to invite the same person 10 times.Then,you have to talk to that certain person and they will give you the tweeter.


Kornelia Carpet, Upholstery & Air Duct Cleaners of Highland Park is a professional company that has been serving Highland Park and suburbs for years. With years of experience we are able to assist our customers in the best possible way. We have large truck-mounted cleaning units prepared for any size job, as well as a portable unit for jobs that are not accessible by our trucks. Our technicians are trained and ready to assist you.
Our technicians will carefully inspect your upholstery and determine the best cleaning technique. Our technicians understand that not all fibers are the same and that some upholstery needs special care. Let us pick the best cleaning method and you won't be disappointed.
Indoor Air Quality is important to all of us. Having a clean, properly functioning heating, ventilating and air conditioning system is vital to maintaining a clean healthy and comfortable living environment. And a newly cleaned system runs more efficiently than a dirty one, resulting in lower utility bills and a cleaner, less dusty home.
At Kornelia carpet cleaners of Highland Park each area rug is treated individuality with special care and proper cleaning. We understand that your area rugs are your precious possessions and will do everything to restore its value.
Need new carpet? Need a carpet repair? This is the place to call. We specialize in all sorts of carpet repairs from burn marks that need to be replacing to removal of the entire carpet/pad after water damage loss. We work with many carpet dealers; we can find you the carpet you are looking for, for the right price.
Our 24/7 service will help you in every situation with your water damage problem. Whether the damage has been caused by a broken pipe, flood, or sewage back up are team of experts will be there to assist you. We specialize in clean up and restoration services.


Tony Stark doesn't have an issue with confidence and you won't have to worry either with the Iron Man Stay Fly Kids Tank Top! Made for the littl'lins, the Iron Man Stay Fly Kids Tank Top is crafted from a soft heather red blend of 58% cotton 42% polyester. What's that mean? It means one will be looking great while displaying a slightly retro image of Marvel's Man in a Can!


Music star Wretch 32, global icon David Beckham and NBA star Derrick Rose will front three 30 second adverts, premiering April 14. There's just over a 100 days until the Olympics, and the iconic Adidas brand is bringing together stars from the worlds of sport, street and style to identify, nurture and celebrate talent. The new multi-channel campaign's first 60 second commercial debuted on April 7 at 20:12 on ITV – sending out the message that there is a stage for everyone in 2012. In the words of Wretch 32 'greatness is made in the city I'm from'.
This advert celebrates all things London but Adidas is giving people across Britain the opportunity to take the stage through its website. All challenges open on April 14, offering once in a lifetime opportunities. Enter and you're in with a chance to be the photographer on set with David Beckham, ball with Derrick Rose in Chicago or open for London's own Wretch 32 at his upcoming tour in May.
Adidas marketing Director, Nick Craggs, has said: 'our new take the stage campaign is designed to showcase London but also the talent that exists within Great Britain. Adidas is providing young people across Britain once in a lifetime opportunities that could change their lives. We feel the campaign is a great way to celebrate London and inspire the rest of Britain'.
Adidas commissioned Wretch 32, who'll tour with fellow Adidas star Example this month, to make the official soundtrack for the campaign. Produced by Wizzy Wow, it is based on insights from young people across the country and focuses on the various ways they are able to showcase their talents today. This homegrown, inspiring track is available for you to download for free.
Even with everything Adidas is offering through this great campaign, it's still giving you a little extra. Everyone will get an insight into the life of an olympian as comedian and TV presenter, Keith Lemon, uses his all access pass to get behind the scenes with some of the biggest names in sport. Keith will have access to everything the brand will be doing in 2012 and creating his own very special brand of content, that shows Team GB like they've never been seen before.
Team GB athletes Victoria Pendleton, Louis Smith and Tom Daley star alongside massive music and entertainment stars such Wretch 32, Example and Keith Lemon in this televised campaign, to showcase how you can go from the top deck of the bus to the top of the charts overnight.
For more information on the Adidas #TakeTheStage campaign visit the official website www.adidas.com/all2012. You can also keep up with all the latest brand news by following the official Twitter @AdidasUK.


Our Favorite 4th of July Recipes!
If you're anything like us, you've got some BBQ's, pool parties, and get togethers on the docket. But what to bring? We've got you covered with these vegan versions of classic Independence Day fare, all perfect for celebrating the independence all people and animals deserve!
Have a favorite 4th of July recipe? We'd love to hear your favorites!


text:
Nintendo Direct set for tomorrow, another source claims
UPDATE: The next Nintendo Direct digital event will take place this Tuesday, September 13
12th Sep 2022 / 10:58 am
Another credible source has pointed to a Nintendo Direct broadcast taking place tomorrow, Tuesday September 13.
GamesBeat journalist Mike Minotti and Giant Bomb's Jeff Grubb first claimed that a new Direct would take place during the week starting September 12, and then reported a Tuesday live stream date last weekend.
Now, respected Nintendo insider Emily Rogers has corroborated this claim, stating on Famiboards that a Direct will take place early on Tuesday morning, PT time.
Update 12th Sep 2022 / 3:08 pm
Nintendo has confirmed its next Nintendo Direct digital event will take place this Tuesday, September 13.
Wii U - The Legend of Zelda: The Wind Waker HD E3 Trailer
However, Rogers claimed she did not know the contents or format of the Direct and speculated that the early US kick-off time could even suggest a smaller "Mini" style event.
"According to my sources, the Direct is still set for Tuesday," she wrote. "However, I'm also hearing Tuesday's Direct will air in the early morning PDT (US time). Not Tuesday evening (US time). I was told the exact time/hour when it would air.
"Why is this notable? Because 'General' Directs are typically aired at the start of the evening for the US, around midnight in Europe and in the morning for Japan. Normally around 2pm-3pm PDT. So, that early morning time slot feels very much Mini-esque."
Rogers went on to note that the last time a general Nintendo Direct aired in the early morning (PT) was back in 2018, and stated that this was why she was tempering her expectations.
"To be fair, that doesn't guarantee that Tuesday's Direct is a 'mini'. Patterns and trends can be always broken. Nintendo's marketing department can break their own rules whenever they want. But I can at least confirm that this Direct will air in the early morning (US time PDT) on Tuesday."
A Wind Waker and Metroid Prime Nintendo Direct is reportedly still coming next week
UPDATE: The next Nintendo Direct digital event will take place on Tuesday, September 13
Jeff Grubb reported this weekend that the event airing on a Tuesday – which is unusual for Nintendo Directs – was likely down to this week's Tokyo Game Show, which kicks off on Thursday the 15th.
Minotti and Grubb previously claimed that Zelda and Metroid ports would be part of the next direct, and the latter said this weekend he understands that's still the case.
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text:
Op-Eds distributed in the 2002-03 academic year
Timothy Chambers
Op-Ed: What academia means to me
Sometimes dropping a tiny piece of information into position allows the larger picture to snap into sharper focus. The hunt for that tiny piece of information and for the deeper understanding it makes possible is a major satisfaction of the academic life. 02-154 (distributed June 18, 2003)
Elliott Colla
Op-Ed: Occupational hazards
If in the coming months Americans begin to notice cutbacks in schools, libraries and public transportation, remember this: Our leaders decided to fund military occupation in Iraq rather than vital services here. And what's worse: If we fail to provide services in Iraq – services that will be expensive – we should expect nothing but chaos and violence from the occupation. 02-153 (distributed June 18, 2003)
William O. Beeman
Op-Ed: Wisdom from Phil: Bush is in trouble
A San Diego barber and one of the world's most famous anthropologists both understand Americans' unspoken rules for the use of force and their strong preference for straight talk and a square deal. While they might follow different lines of reasoning, the barber and anthropologist would end up at the same conclusion: Bush is in trouble. 02-152 (distributed June 12, 2003)
Op-Ed: Meanwhile across the border, Iran's war apprehension is growing
What effect might the American military action in Iraq have next door in Iran? Iranians may be overly cautious, but the signs are everywhere that Iran is seriously concerned about a U.S. military threat. 02-111 (distributed April 10, 2003)
Brent Stuart Goodwin
Op-Ed: The Shock and Awe approach: Target the means and will to fight
War is an unfortunate feature of this world. As long as there are wars, the humane thing to do is to seek to make them as quick and decisive as possible, which goes a long way toward preventing them in the first place. 02-099 (distributed March 25, 2003)
Op-Ed: 'Coalition of the unwilling' falters as allied leaders battle populace
In America, the president stays in office for four years unless impeached. In a parliamentary system, the prime minister can be removed anytime on a vote of no confidence. If the United States is in the middle of an attack on Baghdad when an ally suddenly changes leadership, the war effort will revert, by default, to the unilateral action Americans fear. 02-078 (distributed February 28, 2003)
James Morone
Op-Ed: History lessons about God and war
America's past crusades offer three great lessons about God and arms. First, remember humility. Second, wartime anxieties spill over to domestic suspects. Finally, fighting for our ideals abroad can remind us to take them more seriously at home. 02-065 (distributed February 13, 2003)
Op-Ed: Clipping the Barbershop controversy
Critics charge that the film Barbershop is disrespectful to civil rights leaders Rosa Parks and Dr. Martin Luther King Jr. Defenders say the film carries an important message about freedom of speech. In either case, news that one of the film's stars will emcee the NAACP's Image Awards show suggests that the controversy might be at an end. 02-060 (distributed January 31, 2003)
David C. Lesis, M.D.
Op-Ed: Should you drink for your health?
A new study in The New England Journal of Medicine reports on the health benefits of moderate alcohol consumption. However, that study does not mean alcohol is a health food for everyone. 02-059 (distributed January 30, 2003)
Op-Ed: Who can stop White House warriors? Only the American public
With public opinion so crucial in shaping the actions of the Bush administration, it seems certain that Americans will get a war in Iraq unless they say emphatically that they don't want it. The militants in the White House are champing at the bit. Only their fear of the voters holds them in check. 02-053 (distributed January 22, 2003)
James G. Blight and janet M. Lang
Op-Ed: 40 years later, George Bush hijacks the Cuban Missile Crisis
If the historical John F. Kennedy had done in October 1962 what the Bush administration says he did – asserting a doctrine of "preemptive defense" – the likely outcome would have been a nuclear holocaust. The historical evidence is now unambiguous. 02-027 (distributed October 21, 2002)
Op-Ed: Specialists on Iran-Iraq region see U.S. action in Iraq as a mistake
U.S. interventions in Afghanistan and its contemplated interventions in Iraq will demand a depth of non-military historical, social and political expertise that the United States currently does not have. The author reports from an international conference in Tajikistan. 02-021 (distributed September 23, 2002)
Ross Cheit
Op-Ed: False-conviction chic in the Berkshires
Robert Halsey was convicted in 1993 of sexually abusing two boys on his school van route in Lanesboro, Mass. There was a mountain of evidence against him, and he was sentenced to two consecutive life terms. Now a growing movement is trying to suggest that Halsey was unjustly convicted. A country that cherishes the presumption of innocence still needs to learn something about the presumption of guilt. 02-009 (distributed September 20, 2002)
Herschel I. Grossman
Op-Ed: For peace in the Middle East: A two-state solution?
Palestinian Arabs often take the colonial experiences of Algeria and South Africa as models for their conflict with Zionism: As Europeans ultimately lost power in both Algeria and South Africa, perhaps Jewish settlers will not prove to be permanent. But suppose that analogy is wrong. What if the situation is more like the example of Ireland, where a two-state solution has worked? 02-006 (distributed August 14, 2002)
Lara Harb
Op-Ed: Is education the next security risk in the Occupied Territories?
The Israeli military raid on Al-Quds University in Jerusalem is ironic in light of the recent claim by the Israeli military that Palestinian textbooks incite students against the Israeli government. That claim, repeatedly debunked by U.S. academics, misses the point. The occupation – not to mention the ongoing attack on primary schools and universities – provides a far more provocative education than anything a textbook could ever convey. 02-004 (distributed July 15, 2002)


text:
Il collegio elettorale plurinominale Emilia-Romagna - 01 è stato un collegio elettorale plurinominale della Repubblica Italiana per l'elezione del Senato tra il 2017 ed il 2022.

Territorio
Come previsto dalla legge elettorale italiana del 2017, il collegio era stato definito tramite decreto legislativo all'interno della circoscrizione Emilia-Romagna.

Il collegio comprendeva la zona definita dai quattro collegi uninominali Emilia-Romagna - 01 (Rimini), Emilia-Romagna - 02 (Ravenna), Emilia-Romagna - 03 (Ferrara) e Emilia-Romagna - 04 (Bologna).

Dati elettorali

XVIII legislatura

Come previsto dalla legge elettorale, 193 senatori erano eletti in 33 collegi plurinominali con ripartizione proporzionale a livello regionale tra le coalizioni e le singole liste che avessero superato la soglia di sbarramento stabilita.

Nel collegio venivano eletti 12 senatori.

Eletti

Eletti maggioritario

Eletti proporzionale

Note

Voci correlate
 Circoscrizioni e collegi elettorali del Senato della Repubblica del 2017

Collegi elettorali del Senato della Repubblica del 2017 in Emilia-Romagna
Collegi elettorali plurinominali del Senato della Repubblica del 2017

redpajama_set_name: RedPajamaWikipedia


text:
Bahamaspress.com
Home Local Roberts reveals the mast of Corruption in the FNM Government
Roberts reveals the mast of Corruption in the FNM Government
PLP Chairman, Bradley Roberts
ROBERTS REMARKS:
One year ago today the people of Elisabeth went to the polls following the induced resignation of Malcolm Adderley. The full weight of the FNM Government was brought to bear. During the bye elections millions of tax payers' dollars was spent by the Government, voter intimidation was at the worst ever seen in the history of the Bahamas by FNM Ministers and with the forecast of victory by 300 votes by Hubert Ingraham, Dwayne Sands went down in flames.
Just like the straw market in 2001 and massive fire at Kelly's dock just a few days ago. The destructive torch of the FNM was put out by the water of the mighty PLP. The voters of Elizabeth sang bring the water out the torch, we don't want the FNM no more.
Despite the tremendous odds, Leo Ryan Pinder emerged as the winner and Dr. Dwayne Sands was forced to put his tail between his legs and return to the practice of medicine.
Your Progressive Liberal Party continues to give thanks and praise to Almighty God for the Bye-Election victory and with His continued support and the support of the Bahamian people we shall once again emerge victorious as the next Government of The Bahamas.
Elizabeth's new Member of Parliament Leo Ryan Pinder has in my opinion equipped himself extremely well. He's focused, fearless and he is aggressive in pursuing the interest of the good people of Elizabeth and indeed the entire Bahamas.
Ladies and Gentlemen, the record is there for all to see, that since the FNM's return to power in 2007 with the economy of the Bahamas at its highest peak in its history thanks to the PLP, Hubert Ingraham and the FNM with their Stop, Review and Cancel policy have brought the Bahamas to its knees. Crime and the fear of crime under the FNM have taken the Bahamas to a new low and the Government, despite its promise to have a plan to fight crime, is visionless and clueless. Many Bahamian Families have lost their homes, lost their cars, lost their way and many Families have broken up because of major differences over lack of income. The lives of large numbers of children have been adversely affected.
My Brothers and Sisters, whilst thousands of Bahamians were joining the unemployed lines under the FNM watch, Bahamians suffered the indignity of hearing Deputy Prime Minister Brent Symonette shamelessly boast about how much revenue the Government is receiving due to work permits being issued to foreigners by the immigration department throughout the Bahamas for many jobs that can be performed by Bahamians.
Foreigners and Bahamians no longer complain about not being able to get work permits under Hubert Ingraham's government. In-fact I just read where the Government is being complimented for approving work permits in "rapid time". As an example I was told that Sandals Hotel in Nassau and Exuma are employing non-Bahamians for jobs that Bahamians are fully qualified to perform. That a special official in the Department of Immigration has been assigned exclusively to handle the high volume of Sandals work permit applications. They also tell me that Hubert Ingraham is a regular visitor to Sandals for breakfast and appears not to pay for anything. If that is true what a cheap price to sell out your own people for – some food.
During the entire time that the good people of Grand Cay were exposed to and being provided with foul smelling water which Phenton Neymour said was safe for the people to drink, Hubert Ingraham never opened his mouth and the press appeared to be afraid to confront him. I am fully convinced that Hubert has lost interest in the people of Abaco.
My fellow Bahamians, in particular those who are unemployed, clearly the FNM Government is not concerned about your not being employed because they continue to give work permits to foreigners for jobs that Bahamians are fully capable of doing. The Chinese are no longer coming, they are here. Look around and see my brothers and sisters.
Ladies and Gentlemen, Hubert Ingraham and the FNM government have signed an agreement to sell the majority of shares in the Bahamas Telecommunications Company Limited. The most successful of all the Government's Corporations to a United Kingdom owned Cable and Wireless; the worst telecommunications company in the entire Caribbean who has been given the name by the people of the Caribbean that they have well-earned as Careless and Worthless.
I am both deeply and very troubled over the entire FNM privatization exercise as there are too many unanswered questions and the odour emanating from transactions from within BTC and its Chairman raise alarm bells and the failure of the BTC Chairman and Government Ministers to answer has led me to conclude that the privatization process was tainted and corrupt. Further, the chairman threatened to sue me on two occasions for statements issued in the name of the Party and after weeks, no writ has been forthcoming. It was all a sham to try an silence the PLP. As you are aware, Cable & Wireless never took part in the bidding process and there is strong evidence to suggest that discussions were going on with that Company before the bidding process was terminated.
In fact they brought in this man Sadat, a former CEO of Cable and Wireless, to head up URCA. The PLP brought this to the attention of the Press and they said nothing and the only newspaper that bothered to carry the story was The Bahama Journal. And when the Reporter asked Junior Minister Zharvigo Laing to respond to the concerns raised by the PLP, he refused to comment and referred them to URCA.
URCA is supposed to be an independent body free of influence by the government, but all indications strongly suggest that agreements were made in advance to accommodate a recent employee of Cable and Wireless.
Cable& Wireless, a company that Hubert Ingraham said he will never do business with, came in through the back door.
Who facilitated their entry???? And why should a so called transparent government allow Cable & Wireless to come in through the back door. This smells bad and it tastes bad just like the water in Grand Cay, Abaco.
Another interesting and disturbing fact is that when the PLP received the draft MOU between the Government of The Bahamas and Cable & Wireless and released to the press, our so called free press never published the document because Tommy Turnquest said it was bogus. When the original MOU was tabled in the House of Assembly and was found to be identical to the copy released by the PLP with the exception of two missing pages, we again brought this to the attention of the press.
We issued a Press Statement drawing to the public's attention that both documents were the same, essentially calling Tommy a liar, but the Press refused to carry it.
I am satisfied that the Government has extended its influence to what goes into the press.
I will have more to say on this at a different time. It is a dangerous matter.
My brothers and sisters, what is amazing about this pending sale is that Hubert Ingraham; a man who "says, what he says and means what he says" spoke the following words as published in the FNM's own newspaper: The Torch in March, 1997. I invite you to listen to me carefully as I quote Hubert Ingraham:
"Because I believe; that Bahamians ought to own the majority in Batelco, I shall never, never, ever sell the majority holding in Batelco to anyone other than Bahamians".
Hubert Ingraham has without explanation, without rationalization, without justification, unashamedly turned his back on the promise he made to "never, never, ever sell the majority holding in Batelco to anyone other than Bahamians".
Ladies and gentlemen allocated time does not allow me to continue with additional damming details but I should do so at the next opportunity. So please stay tune.
So I wish to speak on a matter that affects the little man and woman, the vendors on the streets throughout The Bahamas.
When I was Minister responsible for BTC, many commercial companies wanted to get into the whole sale prepaid business and I said NO.
This is a $23 Million industry in The Bahamas.
I wanted to ensure that that part of the business remains in the hands of the small business men. As a result many persons are on the streets in New Providences, Grand Bahama and other Islands selling Cards for their livelihood.
It has come to my attention that BTC/LIME will be implementing an electronic system that will make the present cards obsolete.
In essence what I am saying is that the hundreds of persons on the roads and in small business making their living through this means, their jobs will be obsolete when this system is implemented.
Ladies and Gentlemen I also want to inform you that Wholesale Vendors enjoy a 25% percent commission from BTC. That is, when Wholesalers buy $100,000 worth of phone cards they pay $75,000.00.
When Careless and Wuthless buys BTC, Careless and Wutless presently in the Caribbean gives Wholesalers a 10% commission.
That is when a Wholesaler buys $100,000 worth of cards he will pay $90,000 or an additional $15,000 out of pocket.
What does this means for the customers on the streets in Nassau?
Do you think that the Street vendors will still be able to sell you three twenty dollar cards for $50?
So I wish to conclude by lending my total support to the many voices that are crying out for a full and impartial hearing into the operations of BTC from 2007 to date and the corrupt privatization exercise of BTC.
Good night and see you soon.
BTC PRIVATIZATION TRANSACTION DOCUMENTS
http://www.mediafire.com/?90egp5q622wb3ww
SECTION 1a
http://www.mediafire.com/?2ignrhrmgkf0k66
SECTION 1b
http://www.mediafire.com/?nw9c8hhj97k48gx
http://www.mediafire.com/?1wt726hw026tld6
http://www.mediafire.com/?s4sv3zb7k434ngc
http://www.mediafire.com/?ocvph0rbo70f7ww
http://www.mediafire.com/?42xn1q3z682v9j1
http://www.mediafire.com/?r3ncm6z85w88ix9
http://www.mediafire.com/?t6edhptvq05xiqd
http://www.mediafire.com/?tc2hd4bbgxdkbpv
http://www.mediafire.com/?2hdc77i793oflul
http://www.mediafire.com/?3s1bgs0b5dxuzy3
http://www.mediafire.com/?02krunn7l4zu6ug
http://www.mediafire.com/?7dq4zc9o0bk3b04
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Darren Symonette
So……..Ummmmmmm, did Bradley Roberts address or reveal any of the allege Corruption that sitting members of Parliament, that belong to his party, were said to be involved in? I have to agree with the FNM here on this, because come on people, the Blue water fiasco was less than 5 years ago, not to mention the many alleged improprieties, members of the then Christie Government were accused of. Bottom line, all these people are just alike. "Peter is no better than Paul".
So Mr.Brent Symnotte, I think oh! sorry wrong Symonette, I think you have a dog in this race, why are you talking about Bluewater, that's the FNM's talking points, who is the government, let's deal with the state of our country at this time, and stop trying to dig up a baseless past, if the PLP were on trial here, I would be down their throat, but alas, it's actually the present government for whom there is adequate evidence for prosecution, your name alone lends bias opinion to this subject, good day my friend.
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text:
Q: Referencing cells from a different worksheet Im trying to reference data within a cell of first work sheet (active) on a second worksheet (trucks). When I use =active!$L$34 on my second worksheet (trucks) it bring the right number over. However if I go back to "active" worksheet and use the "cut" and "paste" on cell L34 to move it somewhere else and replace the data in that cell in the same manner with data from a different cell, when I go back to "trucks" and go to cell A1 which is where I used =ACTIVE!$L$34 the value is the same and the formula has changed to =ACTIVE!$B$26 (which is where I copy/pasted the original data to) is there any way to make it so that the formula doesn't change (=ACTIVE!$L$34 stays =ACTIVE!$L$34 regardless of where I put the data in that cell?)
So what I'm trying to do is this.
Worksheet active is manually populated with vehicle ID number(2234) and its assignment(2A26). The Assignment value 2A26 never changes, however the vehicle ID number can change frequently. The work sheet is broken up in to 3 categories: in service, out of service, special use. The vehicle numbers will move all around the worksheet typically using the keyboard shortcut (Ctrl+X) and then placed in its appropriate spot with (Ctrl+V)
Worksheet ACTIVE
UNIT    TIME    STN     EHS #
A1  0600/1600   TER     2925
A2  0500/1700   SPR     2740
A3  0600/1600   GNA     2727
A4  0600/1600   PYL
A5  0630/1830   BON     2103

Worksheet TRUCK
IN SERVICE      Out Of Service      Special service
EHS#    Asmnt       EHS#            EHS#    Asmnt
2727      A3        2234            2256     2P2
2740      A2
2925      A1

For Worksheet trucks is broken with the same 3 categories as active.
the difference with this worksheet is that I want to use the sort function to  rearrange the 3 categories in to numerical order based off the EHS#. However I need the EHS# and Asmnt info to come from EHS# and UNIT on worksheet active. Once this is done I want worksheet TRUCK to update and resort as changes are made to worksheet ACTIVE
Worksheet ACTIVE
UNIT    TIME    STN     EHS #
A1  0600/1600   TER     2924
A2  0500/1700   SPR     2918
A3  0600/1600   GNA     2782

Worksheet Truck
IN SERVICE      Out Of Service      Special service
EHS#    Asmnt       EHS#            EHS#    Asmnt
2918     2A2        2234            2256    2P2
2924     2A1        2925

Hope the above helps a little.
Currently I'm using Excel 2007 for the most part. Some computers this will be run on will be running Excel 2010.

A: Use the following:
=INDIRECT("ACTIVE!L34")

A: Thanks! It worked thanks!
Dont suppose you would be able to help me out with an auto sort code??
Now that I have the truck numbers coming across Im trying to sort them on to the new worksheet.
Sample of what Im trying to do is below.
ROAD TRUCKS
EHS #   Assignment
2740    A1
0   A2
2727    A3
0   A4
2103    A5
0   A6
2924    A7
2918    A8
2782    A9
2919    A10
0   A11
2793    A12
2386    A13
2929    A14
2894    A15
2250    A16
0   A17
As the EHS# changes in the "ACTIVE" worksheet it now updates it properly in the "truck" worksheet. Now Im trying to figure out how to update the sort function when there are changes made to the "truck" worksheet when the changes are made in the "ACTIVE" worksheet.
Thanks again!