text:
\section{Introduction}

Central Lie group extensions can be obtained
by pullback of the prequantization central extension.
The ingredients are a connected Lie group $G$ with Lie algebra $\g$,
a connected prequantizable symplectic manifold
$(\M,\Om)$, and
a Hamiltonian action of $G$ on $\M$.

As $\M$ is prequantizable, it has a prequantum
$S^1$-bundle $\cP \rightarrow \M$ with connection 1-form $\Theta$,
giving rise to the \emph{quantomorphism group}
$\Quant(\cP,\Theta)$ of connection-preserving
automorphisms of this bundle.
Since its identity component $\Quant(\cP,\Theta)_0$
is a central $S^1$-extension of the Hamiltonian diffeomorphism group
$\Ham(\M,\Omega)$,
its pullback by the Hamiltonian action $G \rightarrow \Ham(\M,\Omega)$
yields a central $S^1$-extension $\widehat{G}$
of $G$,
\begin{equation}\label{pbdiag}
\xymatrix
{
\widehat{G} \ar[r] \ar[d]&
\Quant(\cP,\Theta)_{0}\ar[d]
\\
G \ar[r] &
\Ham(\M,\Omega)\,.
}
\end{equation}
If the manifold $\M$ and the Lie group $G$ are infinite dimensional,
then this construction remains valid; the pullback $\widehat{G}$ is still a  Lie
group, even though this may not be the case for $\Ham(\M,\Omega)$ and
$\Quant(\cP,\Theta)$
(cf.\ \cite{NV03}).

We apply this construction
in the following setting.
Suppose that $\pi:P \rightarrow M$ is a prequantum
$S^1$-bundle over a compact, symplectic manifold $(M,\omega)$ of dimension $2n$, and that $\theta$ is a connection 1-form with curvature $\omega$.
The identity component
$G =\Quant(P,\th)_0$
of the {quantomorphism group} is then a Fr\'echet Lie group,
with Lie algebra $\g$ isomorphic to the Poisson Lie algebra $C^{\infty}(M)$.
The infinite dimensional symplectic manifold $\M$ on which $G$ acts will be
a connected component of the
\emph{nonlinear Grassmannian} $\mathrm{Gr}_{2n-1}(P)$
of codimension two closed, oriented,
embedded submanifolds of $P$.
This Fr\'echet manifold
is prequantizable by \cite{I96,HV04}, and the natural action of $G$ on $\M$ is Hamiltonian.
In this way, we obtain central $S^1$-extensions $\widehat{G} \rightarrow G$
of the identity component $G = \Quant(P,\th)_0$ of the quantomorphism group.

For any (Fr\'echet) Lie group $G$, the central extensions of
$G$ by $S^1$ play
a pivotal role in the theory of \emph{projective unitary $G$-representations}.
Every such
representation
gives rise to a central $S^1$-extension $\widehat{G}$, together with a \emph{linear}
unitary $\widehat{G}$-representation
\cite{PS86, TL99, JN15}.
Passing to the infinitesimal level, one obtains information on the
projective $G$-representations from the
(often more accessible) linear representation theory
of the corresponding central Lie algebra extensions $\widehat{\g}$.

In the passage to the infinitesimal level, however, one important piece of information is lost:
not every Lie algebra extension $\widehat{\g} \rightarrow \g$
integrates to a group extension $\widehat{G} \rightarrow G$.
The ones that do, determine a lattice $\Lambda \subseteq H^2(\g,\R)$
in the continuous second Lie algebra cohomology of $\g$,
called the \emph{lattice of integrable classes}.

In the context of quantomorphism groups,
the continuous second Lie algebra cohomology
of the Poisson Lie algebra $C^{\infty}(M)$
has been explicitly determined: in \cite{JV}, we proved that
\[
	H^2(C^\infty(M),\R) \simeq H^1(M,\R)\,.
\]
To the best of our knowledge, it remains an open problem to determine the
full lattice $\Lambda \subseteq H^1(M,\R)$ of integrable classes; it appears
that
the period homomorphism
governing integrability (cf.\ \cite[Thm.\ 7.9]{N02}) is not easy to calculate
in the setting of quantomorphism groups.

In the present paper, we contribute towards a solution by
explicitly determining
the sublattice $\Lambda_0 \subseteq \Lambda$
corresponding to the group extensions $\widehat{G} \rightarrow G$
described above.
We find that
\[
	\Lambda_0 = \frac{n+1}{2\pi \mathrm{vol}(M)}\pi_! (H^2(P,\R)_{\Z})\,,
\]
where $H^2(P,\R)_{\Z}$ is the lattice of integral classes in de Rham cohomology,
and $\pi_{!}$ denotes  fiber integration.
This formula is easily evaluated in concrete situations.
If $M$ is a compact surface,  then
$\Lambda_0 \subseteq H^1(M,\R)$ is of full rank.
On the other extreme, we find that $\Lambda_0 = \{0\}$
if $M$ is a compact K\"ahler manifold of dimension $2n \geq 4$.
Intermediate behavior is displayed by \emph{nilmanifolds}.
Thurston's nilmanifold $M^4$, for example, affords a lattice $\Lambda_0$
that is of rank $1$ in the 3-dimensional vector space
$H^1(M^4,\R)$.

We expect that in the representation theory of the Poisson Lie algebra $C^{\infty}(M)$,
the lattice $\Lambda$
of integrable classes will play the same role as the
\emph{integral level condition} in the
representation theory of loop algebras and affine Kac-Moody
algebras \cite[\S 12]{Ka90}. From a differential geometric point of view,
integrality of the level
for a loop algebra $\g$ (possibly twisted,
over a simple Lie algebra) is precisely the condition
that the induced class in $H^2(\g,\R) \simeq H^1(S^1,\R)$
corresponds to a Lie group extension,
cf.\ \cite[\S 4]{PS86}.

\section{Prequantization central extension}

Let $G$ be a connected Lie group with Lie algebra $\g$,  and let
$(\M, \Omega)$ be a connected
prequantizable symplectic manifold with a Hamiltonian $G$-action.
Both $G$ and $\M$ are allowed to be
infinite dimensional manifolds, modeled on locally convex spaces.
Let $\pi\colon \cP\to \M$ be a {\it prequantum bundle},
\ie a principal $S^1$-bundle with principal connection
1-form $\Th$ and curvature $\Om$. In particular,
the identity $\pi^*\Om=d\Th$ holds.

\subsection{The Kostant-Souriau extension}

The prequantum bundle $\cP \rightarrow \M$ gives rise to the
{\it prequantization central extension} \cite{Ko70, S70}
\begin{equation}\label{kostant}
S^1\to\Quant(\cP,\Th)_0\to\Ham(\M,\Om),
\end{equation}
where $\Ham(\M,\Om)$ is the group of Hamiltonian diffeomorphisms, and
$\Quant (\cP,\Th)_0$
is the identity component of the \emph{quantomorphism group}
\[
	\Quant(\cP,\Th)=\{\ph\in\Diff(\cP):\ph^*\Th=\Th\}\,.
\]
Note that $\ph^*\Th = \Th$ implies $\ph_*E = E$, where $E \in \X(P)$ is the infinitesimal generator
of the $S^1$-action. In particular, every quantomorphism is a bundle automorphism.

The infinitesimal counterpart of \eqref{kostant} is the
central
extension
\begin{equation}\label{extquant}
\R \rightarrow \X(\cP,\Th) \rightarrow \X_{\ham}(\M,\Omega)
\end{equation}
of the Lie algebra of \emph{Hamiltonian} vector fields
\[
	\X_{\ham}(\M,\Omega) := \{ X_{f} \in \X(\M) : i_{X_{f}}\Omega = -df \},
\]
namely the \emph{quantomorphism Lie algebra}
\[
\X(\cP,\Th) := \{X\in\X(\cP) \colon L_X\Th=0\}\,.
\]
It is isomorphic to the Poisson Lie algebra $C^{\infty}(\M)$
via the Lie algebra isomorphism
\begin{equation}\label{iso}
\zeta \colon C^{\infty}(\M) \rightarrow \X(\cP,\Th),
\quad\zeta_{f}:=X_{f}^{\rm hor}+(\pi^*f)E,
\end{equation}
where $Y^{\rm hor}$ denotes the horizontal lift of the vector field $Y\in\X(\M)$.
The central extension \eqref{extquant} can thus be identified with the \emph{Kostant-Souriau} extension
\begin{equation}\label{poissonrijtje}
\R \rightarrow C^{\infty}(\M) \rightarrow \X_{\ham}(\M,\Omega)\,,
\end{equation}
induced by the map $f \mapsto X_{f}$.

The central extensions of a locally convex Lie algebra $\g$ are classified by
its \emph{continuous second Lie algebra cohomology} $H^2(\g,\R)$, cf.\ e.g.\ \cite[\S 2.3]{JV}.
This is the cohomology of the cochain complex $C^{n}(\g,\R)$
of continuous, alternating, $n$-linear maps $\g^n \rightarrow \R$, with differential
$\delta \colon C^{n}(\g,\R) \rightarrow C^{n+1}(\g,\R)$ defined by
\begin{equation}\label{CE}
\delta \psi(x_0,\ldots,x_{n}):= \sum_{0\leq i<j\leq n}
(-1)^{i+j} \psi([x_i,x_j],x_0,\ldots,\widehat{x}_i, \ldots,
\widehat{x}_j, \ldots, x_n)\,.
\end{equation}

A continuous linear splitting $s \colon \g \rightarrow \widehat{\g}$ of the central extension
$\R \rightarrow \widehat{\g} \rightarrow \g$
gives rise to the
2-cocycle $\psi(X,Y) := [s(X),s(Y)] - s([X,Y])$.
Given a point $x_0 \in \M$, we split \eqref{poissonrijtje} by mapping
$X_{f} \in \X_{\ham}(\M,\Omega)$ to the unique Hamiltonian function $f^{x_0}$
vanishing on $x_0$.
The splitting
\begin{equation}\label{splitting}
s_{x_0} \colon X_{f} \mapsto f^{x_0}
\end{equation}
yields the \emph{Kostant-Souriau cocycle} $\psi_{KS}$ on $ \X_{\ham}(\M)$
that is given by
\begin{equation}\label{KS}
\ps_{KS}(X_f,X_g)=\{f,g\}(x_0) = \Om(X_f,X_g)(x_0)\,.
\end{equation}

\subsection{Group extensions from Hamiltonian actions}

If $\M$ is a compact (hence finite dimensional) manifold,
then $\Ham(\M,\Om)$
and $\Quant(\cP,\Theta)$ are both Fr\'echet Lie groups, with Lie algebras
$\X_{\ham}(\M,\Omega)$ and $\X(\cP,\Th)$, respectively, see
 \cite[\S 3]{RS},
{\cite[\S VIII.4]{Om74}}.
Even though this need no longer be the case if $\M$ is infinite dimensional,
it is still true that the pullback $\widehat{G}$ of \eqref{kostant} under the
Hamiltonian action of a Lie group $G$ on $\M$ has a smooth Lie group structure.
\begin{Theorem}{\rm \cite[Thm 3.4]{NV03}}\label{pullb}
Let $(\M,\Om)$ be
a prequantizable, symplectic manifold
with a Hamiltonian action of
a connected Lie group $G$.
Then
the pullback of the prequantization central extension \eqref{kostant}
by the action $G\to\Ham(\M,\Om)$
provides a  central Lie group extension
\begin{equation}
S^1 \rightarrow \widehat{G} \rightarrow G\,.
\end{equation}
\end{Theorem}
The derived Lie algebra extension
$\R \rightarrow \widehat{\g} \rightarrow \g$
is given by the pullback
of the Kostant-Souriau extension \eqref{poissonrijtje}
along the infinitesimal action $\g \rightarrow \X_{\ham}(\M,\Omega)$.
The linear splitting \eqref{splitting}
therefore induces a linear
splitting of $\widehat{\g} \rightarrow \g$, and
the corresponding 2-cocycle $\si$ on $\g$
is the pullback by $\g \rightarrow \X_{\ham}(\M,\Omega)$ of the Kostant-Souriau cocycle.
It is given explicitly by
\begin{equation}\label{splittingCE}
\si(\xi,\et)= \Om(\xi_\M,\et_\M)(x_0)\,,
\end{equation}
where  $\xi_\M$ denotes the fundamental vector field on $\M$ for $\xi\in\g$.

\section{Exact volume preserving diffeomorphisms}

In order to obtain central extensions of the Lie group $G = \Diff_{\ex}(M,\nu)$ of exact volume
preserving diffeomorphisms of a compact manifold $M$
endowed with volume form $\nu$, we consider its Hamiltonian action
on the non-linear Grassmannian of codimension 2
embedded submanifolds of $M$.

\subsection{Non-linear Grassmannians}

Let $M$ be a closed, connected manifold of  dimension $m$.
The {\it non-linear Grassmannian} $\Gr_{k}(M)$ consists of $k$-dimensional, closed, oriented, embedded
submanifolds $N \subseteq M$.  It is a Fr\'echet manifold in a natural way, cf.\
\cite{KM97, GV14}.
The tangent space of $\OGr_k(M)$ at $N$ can
naturally be identified with the space of smooth sections of the normal
bundle $TN^\perp:=(TM|_N)/TN$.

For every $r\geq 0$, the transgression map $\tau \colon \Omega^{k+r}(M) \rightarrow \Omega^{r}(\Gr_{k}(M))$ is defined by
$$
(\tau{\alpha})_{N}([Y_1],\ldots,[Y_r]) := \int_{N}i_{Y_r}\ldots i_{Y_{1}}(\alpha|_{N}).
$$
Here all $[Y_j]$ are tangent vectors at $N\in\OGr_k(M)$, \ie sections
of $TN^\perp$. The expression above is independent of the vector fields $Y_j$ on $M$ along $N$ chosen to represent $[Y_j]$.

The natural group action of $\mathrm{Diff}(M)$ on $\Gr_{k}(M)$, defined by
$(\ph , N) \mapsto \ph(N)$, is smooth, since it
descends from the action of $\Diff(M)$ on the manifold
of embeddings into $M$ defined by composition
$(\ph , f) \mapsto \ph \o f$. It differentiates to
the Lie algebra action  $\X(M) \rightarrow \X(\Gr_{k}(M))$
given by $X \mapsto \ta_X$ with $\ta_X(N) = [X|_{N}]$.
The transgression enjoys the following functorial properties:
\begin{align}\label{fun}
&d\circ \tau = \tau \circ d,\quad\quad\quad \ph^* \circ \tau = \tau \circ \ph^*, \\
&i_{\ta_{X}}\circ \tau = \tau\circ i_{X},\quad L_{\ta_X} \circ \tau = \tau\circ L_{X}.\nonumber
\end{align}

\begin{Theorem}{\rm \cite[\S 25.3]{I96} \cite[Thm.\ 1]{HV04}}\label{taual}
Let $\al\in\Om^{k+2}(M)$ be a closed differential form with integral cohomology class. Then the non-linear Grassmannian $\OGr_{k}(M)$ endowed with the closed 2-form $\Om=\tau\al$ is prequantizable,
\ie there exist an $S^1$-bundle
$\mathcal P\to\OGr_{k}(M)$ with
connection form $\Th\in\Omega^1(\mathcal P)$ and curvature~$\Om$.
\end{Theorem}

\subsection{Lichnerowicz central extensions}

Let $\nu \in \Omega^{m}(M)$ be a volume form on $M$, normalized so that $\mathrm{vol}_{\nu}(M) = 1$.
It induces a symplectic form $\Om=\tau\nu$
on the codimension two non-linear Grassmannian $\OGr_{m-2}(M)$ \cite{I96}.
This is the higher dimensional version of the natural symplectic form on the space of knots in $\R^3$ \cite{MW83}.
The natural action of the group $\Diff(M,\nu)$ of volume preserving diffeomorphisms on $\OGr_{m-2}(M)$ is symplectic, as
$\ph^*\ta\nu=\ta\ph^*\nu=\ta\nu$ for all $\ph\in\Diff(M,\nu)$
by \eqref{fun}. To get a Hamiltonian action, we have to restrict to the subgroup $\Diff_{\ex}(M,\nu)$ of \emph{exact volume preserving diffeomorphisms}.

Its Lie algebra ${\X}_{\ex}(M,\nu)$ of \emph{exact divergence free vector fields}
is the kernel of the infinitesimal flux homomorphism, defined on the Lie algebra
$\X(M,\nu)$ of divergence free vector fields by
\begin{equation}\label{flu}
\X(M,\nu)\to H^{m-1}(M,\R),\quad X\mapsto[i_X\nu].
\end{equation}
We denote by $X_\al$ the exact divergence free vector field with potential
$\al\in\Om^{m-2}(M)$, \ie $i_{X_\al}\nu=d\al$.

The Lie algebra homomorphism \eqref{flu} is integrated by
\emph{Thurston's flux homomorphism}.
On the universal cover of the identity component of the group
of volume preserving diffeomorphisms, we define
\begin{equation}\label{split1}
\widetilde{\Flux}:\widetilde{\Diff}(M,\nu)_0\to H^{m-1}(M,\R) \quad \text{by} \quad
\widetilde{\Flux}([\ph_{t}])=\int_0^1[i_{X_t}\nu]dt\,,
\end{equation}
where $\ph_t$ is a volume preserving isotopy from the identity to $\ph$,
and $X_t$ is the time dependent vector field such that $\frac{d}{dt}\ph_t=X_t\o\ph_t$.
By \cite[Thm.\ 3.1.1]{B}, this is a well defined group homomorphism.

For any codimension one submanifold $N \subset M$,
the integral $\int_{N} \widetilde{\Flux}([\ph_{t}])$
is the volume swept out by  $N$ under $\ph_{t}$.
Therefore, the monodromy
subgroup $\Gamma := \widetilde{\Flux}(\pi_1(\Diff(M,\nu)_0))$  of $H^{m-1}(M,\R)$
is discrete. It follows that
equation \eqref{split1} factors through a Lie group homomorphism
\begin{equation}\label{split}
\Flux:\Diff(M,\nu)_0\to H^{m-1}(M,\R)/\Ga,\quad
\Flux(\ph)=\int_0^1[i_{X_t}\nu]dt\mod\Ga\,.
\end{equation}
The group of exact volume
preserving diffeomorphisms $\Diff_{\ex}(M,\nu)$ is now defined as the kernel of the
Flux homomorphism;
it is a Lie group with Lie algebra ${\X}_{\ex}(M,\nu)$ \cite{B, KM97}.

Since $\mathrm{vol}_{\nu}(M)$ is normalized to $1$, the cohomology class $[k\nu]\in H^m(M,\R)$
is integral for every $k\in \Z$.
By Theorem \ref{taual}, this implies that
the manifold $\OGr_{m-2}(M)$ with symplectic form $\Om=k\, \tau\nu$ is prequantizable.
The natural action of $\Diff_{\ex}(M,\nu)$
on $\OGr_{m-2}(M)$ is Hamiltonian,
as $i_{\ta_{X_\al}}\ta\nu=\ta i_{X_\al}\nu=\ta d\al=d\ta\al$
for all $X_\al\in\X_{\ex}(M,\nu)$ by \eqref{fun}.
Now we can apply Theorem \ref{pullb} to this Hamiltonian action
on a connected component $\mathcal M$ of $\OGr_{m-2}(M)$.
This yields the central Lie group extension
\begin{equation}\label{extdiffexgroup}
S^1 \rightarrow \widehat{\Diff}_{\ex}(M,\nu) \rightarrow \Diff_{\ex}(M,\nu)
\end{equation}
of the group of exact
volume preserving diffeomorphisms.

To obtain the corresponding Lie algebra 2-cocycle, we fix a point $Q \in \M$, that is, a codimension two submanifold $Q \subset M$ in the connected component $\M$
of the nonlinear Grassmannian.
By \eqref{splittingCE}, the Lie algebra extension of $\X_{\ex}(M,\nu)$ corresponding to
\eqref{extdiffexgroup} for $k = 1$, is described by the
Lie algebra 2-cocycle
\begin{equation}\label{lich}
\la^{\nu}_{Q}(X,Y)=(\tau \nu)_{Q}(\ta_X,\ta_Y)
= \int_{Q}i_Yi_X\nu
\end{equation}
on $\X_{\ex}(M,\nu)$, which we call the \emph{singular Lichnerowicz cocycle}.
If the class $[k\nu]$ is used to construct the extension, then the corresponding
2-cocycle is $k\la_{Q}^{\nu}$.

\begin{Theorem}\label{lint}{\rm\cite[\S 25.5]{I96}\cite[Thm.\ 2]{HV04}}
Let $\nu$ be a volume form on $M$ with $\mathrm{vol}_{\nu}(M)=1$
and $Q$ a codimension two embedded submanifold of $M$. Then the
Lie algebra extensions defined by integral multiples of the
cocycle $\la^{\nu}_{Q}$ of equation \eqref{lich}
integrate to central Lie group
extensions of the group of exact volume preserving diffeomorphisms $\Diff_{\ex}(M,\nu)$.
\end{Theorem}

Recall that two classes $[Q] \in H_{m-k}(M,\R)$ and
$[\alpha] \in H^{k}(M,\R)$ are called \emph{Poincar\'e dual} if
$\int_{Q} \gamma = \int_{M} \eta \wedge \gamma$ for all closed
$\gamma \in \Omega^{m-k}(M)$.
If $[\et]\in H^2(M,\R)$ is Poincar\'e dual to
$[Q]\in H_{m-2}(M,\R)$, then by \cite[Prop.\ 2]{Vi}
the cocycle $\la^{\nu}_{Q}$ is cohomologous to the {\it Lichnerowicz cocycle} \cite{Li74}
\begin{equation}\label{lichcoc}
\la^{\nu}_\et(X,Y)=\int_M\et(X, Y)\nu=\int_M\et\wedge i_Yi_X\nu.
\end{equation}
{If $\dim M\ge 3$,} the map  $[\et]\mapsto[\la^\nu_\et]$ is believed to be an isomorphism between
$H^2(M,\R)$ and the second Lie algebra cohomology
group $H^2(\X_{\ex}(M,\nu),\R)$, see \cite[\S 10]{R95} for
the outline of a proof.

\begin{Remark}\label{pd}
If $[\eta]$ is Poincar\'e dual to $[Q]$ with $Q\in\Gr_{m-2}(M)$, then in particular, it is
an integral cohomology class. Conversely,
every integral cohomology class $[\et]\in H^2(M,\R)$ is the Poincar\'e dual
of a closed submanifold of codimension two in $M$;
it can be obtained (cf.\ \cite[Prop.\ 12.8]{BT82}) as the zero set of a section transversal to the zero section
in a rank two vector bundle with Euler class $[\et]$.
\end{Remark}

We infer that the Lichnerowicz cocycle \eqref{lichcoc} gives rise to an
\emph{integrable}
Lie algebra extension if $[\eta] \in H^2(M,\R)$ is an \emph{integral} class
in de Rham cohomology
(in the sense that on integral singular 2-cycles, it evaluates to an integer).
We denote by  $H^2(M,\R)_{\Z}$
the space of integral de Rham classes.
It follows that the image of $H^2(M,\R)_{\Z}$
by the map $[\et]\mapsto [\la_\et^{\nu}]$,
which coincides with the image of $H_{n-2}(M,\Z)$ by the map $[Q]\mapsto[\la_Q^{\nu}]$,
lies in the lattice of integrable classes in $H^2(\X_{\ex}(M,\nu),\R)$.

\section{Strict contactomorphisms}
Let $P$ be a compact manifold of dimension $2n+1$,
equipped with a contact 1-form $\th$.
The group $\Diff(P,\th)$ of strict contactomorphisms
is a subgroup of the volume preserving diffeomorphism group $\Diff(P,\mu)$,
where the volume form $\mu$ is a constant multiple of $\th\wedge(d\th)^n$.
The group of {\it exact strict contactomorphisms} is defined as
\begin{equation}
\Diff_{\ex}(P,\th) := \Diff(P,\th)_{0} \cap \Diff_{\ex}(P,\mu)\,.
\end{equation}
We use Theorem \ref{lint} to investigate central extensions of
locally convex Lie groups $G$
that act on $P$ by exact strict contactomorphisms.

The motivating example is the case where $P \rightarrow M$ is a
prequantum bundle over a compact symplectic manifold
$(M,\omega)$; in this case $\Diff(P,\th)$ is the quantomorphism group, while
$\Diff_{\ex}(P,\th)$
coincides with the identity component $\Quant(P,\th)_0$ of the quantomorphism group.
Since the latter is a locally convex Lie group (it is even an ILH-Lie group by
\cite[VIII.4]{Om74}), we thus obtain central
Lie group extensions of $G = \Quant(P,\th)_0$ by $S^1$.

\subsection{Strict contactomorphisms}
The contact form $\th$ gives rise to the volume form $\th \wedge (d\th)^n$. We will use
the following two normalizations of this form:
\begin{equation}\label{normalisation}
\mu := \frac{1}{(n+1)!}\,\th \wedge (d\th)^{n}\,, \quad \text{and}
\quad
\nu :=  \frac{1}{\mathrm{vol}_{\mu}(P)} \,\mu\,.
\end{equation}
The \emph{Reeb vector field} $E \in \X(P)$ is uniquely determined by $i_{E}\th = 1$ and ${i_{E}d\th = 0}$.
We define the \emph{strict contactomorphism group}
\[
\mathrm{Diff}(P,\th) := \{\ph \in \mathrm{Diff}(P)\,;\, \ph^* \th = \th\}\subset\Diff(P,\mu)
\]
and the \emph{Lie algebra of strict contact vector fields}
\[
\X(P,\th) = \{X \in \X(P)\,;\, L_{X} \th = 0\}\subset\X(P,\mu)\,.
\]
For every isotopy $\ph_t\in\Diff(P)$, starting at the identity and
corresponding to the time dependent vector field $X_t\in\X(P)$,
we have $\ph_t\in\Diff(P,\th)$ for all $t$ if and only if
the vector field $X_t\in\X(P,\th)$ for all $t$.

A Hamiltonian function $f \in C^{\infty}(P)^{E} =\{f\in C^\infty(P):L_Ef=0\}$
defines a unique  strict contact vector field
$\zeta_{f} \in \X(P,\th)$ by
\[i_{\zeta_{f}} \th = f, \quad i_{\zeta_{f}}d\th = -df\,.\]
The corresponding map
\begin{equation}\label{zet}
\ze:C^{\infty}(P)^{E} \rightarrow \X(P,\th)
\end{equation}
is an isomorphism of Fr\'echet Lie algebras, if $C^\infty(P)^E$ is equipped with the
Lie bracket
\begin{equation}\label{lb}
\{f,g\} = d\th(\zeta_{f},\zeta_{g})=L_{\ze_f}g\,.
\end{equation}

\begin{Proposition}\label{trekterug}
Let $Q$ be a codimension two submanifold of the contact manifold $(P,\th)$.
Then the pullback to $C^\infty(P)^E$
of the singular Lichnerowicz cocycle $\la^{\mu}_{Q}$
on $\X(P,\mu)$ by the map $\ze$ in \eqref{zet}
is given by
\[
(\zeta^* \la^{\mu}_{Q})(f,g) = \si_{Q}(f,g) + \frac{1}{n+1} \delta \rho_{Q}(f,g)\,,
\]
where
the 2-cocycle $\si_{Q}$ and the 1-cochain $\rho_{Q}$ on $C^{\infty}(P)^{E}$ are given by

\begin{eqnarray}\label{sg2}
\si_Q(f,g) & :=  & \phantom{-}\! \int_Q gdf\wedge(d\th)^{n-1}/(n-1)!\,,\\
\rho_{Q}(h) & := & -\int_{Q}h\th\wedge(d\th)^{n-1}/(n-1)!\,.
\end{eqnarray}
\end{Proposition}
\begin{proof}
We calculate $\zeta^*\la_{Q}^{\mu}(f,g) = \la^{\mu}_{Q}(\zeta_{f}, \zeta_{g}) =
\int_{Q} i_{\zeta_{g}}i_{\zeta_{f}}(\th\wedge (d\th)^{n})/(n+1)!$.
First, note that
\begin{eqnarray}
i_{\zeta_{g}}i_{\zeta_{f}}(\th\wedge (d\th)^{n}/n!) &=&
i_{\zeta_{g}}(f(d\th)^n/n! + \th\wedge df \wedge (d\th)^{n-1}/(n-1)!) \nonumber\\
&=& -(fdg-gdf)\wedge(d\th)^{n-1}/(n-1)!  \label{pullLichnerowicz}\\
&& + \{f,g\} \,\th \wedge (d\th)^{n-1}/(n-1)!\nonumber \\
& &-\th\wedge df \wedge dg \wedge (d\th)^{n-2}/(n-2)!\,.\nonumber
\end{eqnarray}
Expanding $d\left(\th\wedge(fdg-gdf) \wedge (d\th)^{n-2}\right)$, we obtain
\begin{eqnarray*}
\th\wedge df \wedge dg \wedge (d\th)^{n-2}/(n-2)!
&=&
\phantom{+}{\textstyle \frac{1}{2}}(n-1) (fdg-gdf)\wedge(d\th)^{n-1}/(n-1)!\\
& & - {\textstyle \frac{1}{2}} d\left(\th\wedge(fdg-gdf) \wedge (d\th)^{n-2}/(n-2)!\right).
\end{eqnarray*}
Inserting this into \eqref{pullLichnerowicz} yields
\begin{eqnarray}
i_{\zeta_{g}}i_{\zeta_{f}}(\th\wedge (d\th)^{n}/n!)
&=& - (n+1){\textstyle \frac{1}{2}} (fdg-gdf)\wedge(d\th)^{n-1}/(n-1)!  \label{pullLichnerowicz2}\\
&& +  \{f,g\}\th\wedge (d\th)^{n-1}/(n-1)!\nonumber \\
& &+ {\textstyle \frac{1}{2}} d\left(\th\wedge(fdg-gdf) \wedge (d\th)^{n-2}/(n-2)!\right)\,.\nonumber
\end{eqnarray}
Since the value of $\la_{Q}^{\mu}(\zeta_{f}, \zeta_{g})$ is obtained by
integrating the above expression over $Q$ and dividing by $n+1$,
the last term vanishes ($Q$ is closed),
the middle term yields a multiple of the coboundary $\delta \rho_{Q}(f,g) = \rho_{Q}(-\{f,g\})$, and
the first term yields the cocycle $\si_{Q}(f,g)$.
\end{proof}
In particular the classes $\ze^*[\la_Q^\mu]$ and $[\si_Q]$ in
$H^2(C^\infty(P)^E,\R)$ coincide.

\begin{Remark}[\bf Regular contact manifolds]\label{BWremark}
For us, the motivating example is the total space $(P,\th)$ of a prequantum $S^1$-bundle
$\pi \colon P \rightarrow M$
 over a compact, symplectic manifold $(M,\omega)$.
 These are called \emph{regular} or
 \emph{Boothby-Wang} contact manifolds \cite{BW}.
The top form
\[\mu = \frac{1}{(n+1)!}\th \wedge (d\th)^{n} = \frac{1}{(n+1)!}\th \wedge \pi^*\omega^n\]
is a volume form, and the Reeb vector field
$E\in\X(P)$ coincides with the infinitesimal generator
 of the principal $S^1$-action.
The group  of strict contactomorphisms coincides with
the quantomorphism group, hence it is a
Fr\'echet Lie group \cite{Om74,RS}.

The pullback by $\pi $ is an isomorphism
between the Poisson Lie algebra $C^\infty(M)$ and
the Lie algebra $C^\infty(P)^E$ with Lie bracket \eqref{lb}.
Under the identification $C^{\infty}(M) \simeq C^{\infty}(P)^{E}$,
the isomorphisms \eqref{iso} and \eqref{zet}, both denoted by $\ze$, coincide.
Moreover, the cocycle $\si_Q$ in \eqref{sg2} can be identified with the cocycle $\psi_{\pi_*Q}$
on the Poisson Lie algebra, determined by
the singular $2n-1$ cycle $C=\pi_*Q$ in $M$ by the formula \cite{JV}:
\[
\ps_C(f,g)=\int_Cgdf\wedge\om^{n-1}/(n-1)!,\quad f,g\in  C^\infty(M)\,.
\]
More details will be given in Section 5.
\end{Remark}

\subsection{Exact strict contactomorphisms}

Suppose that $G$ is a locally convex Lie group that acts
smoothly {and effectively} on $P$
by exact strict contact transformations.
{Its Lie algebra $\g$ is then a subalgebra of
$\X(P,\th)\simeq C^\infty(P)^E$.
We investigate the integrability of
the pullback $\iota^*[\si_Q] \in H^2(\g,\R)$
along the inclusion $\iota \colon \g \hookrightarrow C^{\infty}(P)^{E}$, where
$\si_Q$ is the cocycle
$\si_Q(f,g) =  \int_Q gdf\wedge(d\th)^{n-1}/(n-1)!$ of  \eqref{sg2}.}
For general contact manifolds, we have to impose the condition
of exactness so that we can make use of Theorem \ref{lint}.
However,
in the important special case of regular contact manifolds,
we will show that
the exactness condition is automatically satisfied.

Analogous to the group $\Diff_{\ex}(P,\th) := \Diff(P,\th)_{0} \cap \Diff_{\ex}(P,\mu)$
of exact strict contact transformations, we define the Lie algebra of
{\it exact strict contact vector fields} by
$$\X_{\ex}(P,\th):=\X(P,\th) \cap \X_{\ex}(P,\mu)\,.$$

For every isotopy $\ph_t\in\Diff(P)$, starting at the identity
and determined by the time dependent vector field $X_t\in\X(P)$,
we have $\ph_t\in \Diff_{\ex}(P,\th)$ for all $t$ if and only if
$X_t\in\X_{\ex}(P,\th)$ for all $t$.

\begin{Lemma}
The function space
\[
C^{\infty}_{0}(P)^{E} := \{f \in C^{\infty}(P)^{E}\,;\,  f(d \th)^n \in d\Omega^{2n-1}(P)\}
\]
is a Lie subalgebra of $C^\infty(P)^E$ of finite codimension
$\le\dim H^{2n}(P,\R)$,
isomorphic under $f\mapsto\ze_f$  to
the Lie algebra $\X_{\ex}(P,\th)$
of  exact strict contact vector fields.
\end{Lemma}

\begin{proof}
As $f(d\th)^n$ is closed for all $f \in C^{\infty}(P)^{E}$, we can define the
linear map
\begin{equation}\label{liho}
C^{\infty}(P)^{E} \rightarrow H^{2n}(P,\R),
\quad f \mapsto \frac{1}{n!}[f (d\th)^n]
\end{equation}
with kernel  $C^{\infty}_{0}(P)^{E}$.
This is a Lie algebra homomorphism, as
$\{f,g\}(d\th)^n =ndf\wedge dg\wedge (d\th)^{n-1}$ is exact.
It coincides, under the identification \eqref{zet}, with the flux homomorphism \eqref{flu}
for the volume form $\mu$
restricted to  $\X(P,\th)\simeq C^{\infty}(P)^{E}$, as
\begin{eqnarray*}
(n+1)! [i_{\ze_f}\mu]&= &[i_{\ze_f}(\th\wedge (d\th)^n)]\\
&=& [f(d\th)^n- n (df)\wedge\th\wedge(d\th)^{n-1}]\\
&=& [f (d\th)^n- n d(f\th\wedge(d\th)^{n-1}) + n f (d\th)^n]\\
&=& (n+1)[f (d\th)^n]\,.
\end{eqnarray*}
It follows that the kernel $C_0^\infty(P)^E$ of \eqref{liho}
is identified under $\ze$ with the exact strict contact vector fields
$\X(P,\th)\cap\X_{\ex}(P,\mu)=\X_{\ex}(P,\th)$.
\end{proof}

\begin{Proposition}\label{LAsame}
If the contact manifold $(P,\th)$ is regular, i.e.\ the total space of a prequantum bundle $\pi:P\to M$,
then the Lie algebras of strict contact and exact strict contact vector fields coincide:
$\X(P,\th)=\X_{\ex}(P,\th)$.
Moreover,  the group of exact strict contact diffeomorphisms
is precisely the connected component of the
quantomorphism group:
\[ \Diff_{\ex}(P,\th)=\Diff(P,\th)_{0} \,.\]
\end{Proposition}

\begin{proof}
Any $f\in C^\infty(P)^E$
is of the form $\pi^*\bar f$ for a smooth function $\bar f$ on the compact symplectic manifold $M$.
If we write $\bar f =\bar f_0 + c$ with $\int_{M} \bar{f}_{0} \omega^n = 0$, then
$\bar f_0 \omega^n = d\gamma$ is exact, so that also
 $f (d\th)^n=c(d\th)^n+\pi^*(\bar f_0\om^n) = c d(\th \wedge (d\th)^{n-1}) + d\pi^*\gamma$ is exact.
Hence $C_0^\infty(P)^E=C^\infty(P)^E\simeq C^\infty(M)$ and the conclusion follows.
\end{proof}

The following example shows that for contact manifolds that are not regular,
the Lie algebra
$C_{0}^{\infty}(P)^{E}$ can be strictly smaller than $C^{\infty}(P)^{E}$.

\begin{Example}
An example of a non-regular contact form on the 3-torus $P=\T^3$
is $\th=\cos z dx+\sin z dy$.
The orbits of the Reeb vector field $E=\cos z \partial_x+\sin z \partial_y$
determine constant slope foliations
on each 2-torus of constant $z$.
We show that $C_0^\infty(P)^E \simeq \X_{\ex}(P,\th)$
has codimension two in $C^\infty(P)^E\simeq\X(P,\th)$.

We use the inclusion $\X(P,\th)\subset\X(P,\mu)$.
Any divergence free vector field $X\in\X(P,\mu)$ is the sum $X = X_0 + X_{\alpha}$
of an exact divergence free vector field
$X_\al$
with potential 1-form $\al = Adx + Bdy + Cdz$ and
a constant vector field $X_0 = a \partial_x + b \partial_y + c \partial_z$.
With volume form $\mu ={\textstyle \frac{1}{2}}\th\wedge d\th= -{\textstyle \frac{1}{2}} dx\wedge dy\wedge dz$, we have
$X_\al=2(B_z-C_y)\partial_x+2(C_x-A_z)\partial_y+2(A_y-B_x)\partial_z$.
The vector field $X$ is  strict contact if $L_{X}\th = 0$,
which amounts to
\begin{eqnarray*}
\sin z(C_{xx}-A_{xz}-A_y+B_x-c)+\cos z(B_{xz}-C_{xy})&=&0\\
\cos z(B_{yz}-C_{yy}+A_y-B_x+c)+\sin z(C_{xy}-A_{yz})&=&0\\
\cos z (B_{zz} - C_{yz}) + \sin z (C_{xz} - A_{zz}) &=& 0.
\end{eqnarray*}
Thus $a, b \in \R$ are arbitrary, $c=0$ (as can be seen by integrating the above equations
over $x$ and $y$), and $X_\al$ is {a strict contact vector field.}
We find {that} $\X(P,\th)$ is isomorphic to the semidirect product
$\mathrm{Span}\{\partial_x, \partial_y\} \ltimes \X_{\rm ex}(P,\th)$, and {that}
the flux homomorphism \eqref{flu} restricted to $\X(P,\th)$
has 2-dimensional  image generated by $[dy\wedge dz], [dx\wedge dz]\in H^2(P,\R)$.
\end{Example}

We apply Theorem \ref{lint}
to the contact manifold $P$ with integral volume form $k\nu=\frac{k}{\mathrm{vol}_{\mu}(P)}\mu$ for $k \in \Z$
and we obtain the following central result.

\begin{Theorem}\label{centralresult}
Let $G$ be a Lie group acting {smoothly} on $(P,\th)$ by exact strict contact transformations.
Then the restriction to $\g$ of the class
$\frac{k}{\mathrm{vol}_{\mu}(P)} [\si_{Q}]$
is integrable to a central Lie group extension of $G$.
\end{Theorem}

\begin{proof}
Since $G \subseteq \Diff_{\ex}(P,\th)$ and $\Diff_{\ex}(P,\th) \subseteq \Diff_{\ex}(P,k\nu)$,
the action of $G$ on the connected component $\M$ of $\mathrm{Gr}_{2n-1}(P)$
is Hamiltonian.
Theorem~\ref{pullb} then yields a
Lie group extension
\[S^1 \rightarrow \widehat{G} \rightarrow G\,.\]
By Theorem~\ref{lint}, the corresponding class in $H^2(\g,\R)$
is the pullback along the inclusion $\iota \colon \g \hookrightarrow \X_{\mathrm{ex}}(M,\nu)$
of
$[\lambda^{k\nu}_{Q}] = \frac{k}{\mathrm{vol}_{\mu}(P)}[\lambda^{\mu}_{Q}]$.
By Proposition~\ref{trekterug}, this is the restriction to $\g$ of
the class
\begin{equation*}\label{integrclass}
	\frac{k}{\mathrm{vol}_{\mu}(P)} [\si_{Q}]\in H^2(C_0^{\infty}(P)^{E}, \R)\,,
\end{equation*}
where $\sigma_{Q}$ is the the cocycle
$\si_Q(f,g) =  \int_Q gdf\wedge(d\th)^{n-1}/(n-1)!$ of  \eqref{sg2}.
\end{proof}

\section{The quantomorphism group}

Let $P \rightarrow M$ be a prequantum bundle over a
compact symplectic manifold $(M,\omega)$, and let $\th \in \Omega^1(P)$
be a connection 1-form with curvature $\omega$.
We apply Theorem~\ref{centralresult} to the
identity component $G = \Quant(P,\th)_0$
of the quantomorphism group.
As its Lie algebra is isomorphic to the Poisson Lie algebra $\g = C^\infty(M)$,
an explicit description of the second Lie algebra cohomology
is available \cite{JV}.
With the above construction,
we obtain a lattice of integrable classes in
$H^2(C^{\infty}(M),\R)$.

\subsection{Cohomology of the Poisson Lie algebra}

We describe the second Lie algebra cohomology $H^2(C^{\infty}(M), \R)$
in two different ways:
using \emph{Roger cocycles} related to $H^1(M,\R)$,
and
using \emph{singular cocycles} related to $H_{2n-1}(M,\R)$.
The two pictures are linked by Poincar\'e duality.

\begin{Definition}
The \emph{Roger cocycle} \cite[\S 9]{R95} associated to a closed 1-form $\alpha$
on the $2n$-dimensional symplectic manifold $(M,\om)$
is defined by
\begin{equation}
\label{rog}
\psi_\al(f,g):=\int_M f\al(X_g)\om^n/n!
=-\int_M\al\wedge fdg\wedge \om^{n-1}/(n-1)!.
\end{equation}
It was first defined for surfaces in \cite{Ki90}.
\end{Definition}

The Roger cocycles link the first de Rham cohomology
of $M$ to the second Lie algebra cohomology of the Poisson Lie algebra $C^{\infty}(M)$.

\begin{Theorem}\label{LAcohomology}{\rm\cite[\S 9]{R95}\cite[\S4]{JV}}
The Roger cocycles $\psi_{\alpha}$ and $\psi_{\alpha'}$ are cohomologous
if and only if $\alpha - \alpha'$ is exact, and
the corresponding map $[\alpha] \mapsto [\psi_{\alpha}]$ is an isomorphism
\[
	H^1(M,\R) \stackrel{\!\sim\,}{\rightarrow} H^2(C^{\infty}(M), \R)\,.
\]
\end{Theorem}

This shows (\cf \cite[\S2]{JV}) that every (locally convex) central extension
$\R \rightarrow \widehat{\g} \rightarrow C^{\infty}(M)$
corresponds to a Roger cocycle \eqref{rog} with respect to some linear splitting
$C^{\infty}(M) \rightarrow \widehat{\g}$.
However, the cocycles that come from the Hamiltonian action
of $\Quant(P,\th)$ on $\Gr_{2n-1}(P)$, using splittings of type \eqref{splitting},  are
more closely related to \emph{singular} homology.

\begin{Definition}\label{defpsiC}
The \emph{singular cocycle} $\psi_{C}$ on $C^{\infty}(M)$, associated to a singular $(2n-1)$-cycle $C$ on $M$, is defined by
\begin{equation}\label{sg22}
\psi_C(f,g):=
  \int_C gdf\wedge\om^{n-1}/(n-1)!\,.
\end{equation}
\end{Definition}
The Lie algebra 2-cocycles $\psi_{C}$ and $\psi_{C' }$ are cohomologous
if and only if $C - C'$ is a boundary.
\begin{Proposition}\label{cohthm}
The singular cocycle $\psi_{C}$ is cohomologous to the Roger cocycle $\psi_{\alpha}$
if and only if $[C] \in H_{2n-1}(M,\R)$ is Poincar\'e dual to $ [\alpha]\in H^1(M,\R)$.
In particular, the map
$[C] \mapsto [\psi_{C}]$ is an isomorphism
\begin{equation}\label{FullLACohomology}
H_{2n-1}(M,\R) \stackrel{\!\sim\,}{\rightarrow} H^2(C^{\infty}(M),\R)\,.
\end{equation}
\end{Proposition}
\begin{proof}
In view of the fact that $C^{\infty}(M) \simeq \R \oplus \X_{\ham}(M,\om)$ for compact $M$,
this follows from the discussion at the end of \cite[\S 5]{JV}.
\end{proof}

\subsection{An integrable lattice in $H^2(C^{\infty}(M),\R)$}

By applying Theorem \ref{centralresult} to the regular contact manifold $(P,\th)$
we obtain a lattice of integrable classes in
the Lie algebra cohomology $H^2(C^{\infty}(M),\R)$.
In the following, we denote the lattice of integral classes by
$H_{*}(M,\R)_{\Z}$ (homology) or $H^*(M,\R)_{\Z}$ (cohomology).

\begin{Corollary}[{\bf Singular version}]\label{singversion}
Let $[C] \in H_{2n-1}(M,\R)_{\Z}$ be in the image under $\pi_{*}$
of $H_{2n-1}(P,\R)_{\Z}$. Then
the Lie algebra extension corresponding to the class
\[
	\frac{n+1}{2\pi\, \mathrm{vol}(M)} [\psi_{C}]
\]
integrates to a central
extension
of $\Quant(P,\th)_0$ by~$S^1$.
In the above expression,  $\mathrm{vol}(M) = \int_{M}\omega^n/n!$ is the Liouville volume of $M$,
and $[\psi_{C}] \in H^2(C^{\infty}(M), \R)$ is the singular class \eqref{sg22}.
\end{Corollary}
\begin{proof}
We apply Theorem~\ref{centralresult} to the unit component
$G = \Diff(P,\th)_0$ of the quantomorphism group.
By \cite[VIII.4]{Om74}, this
is a Fr\'echet Lie group, with Lie algebra $\g$ isomorphic to the
Poisson Lie algebra $C^{\infty}(M)$.
By Proposition~\ref{LAsame}, the group $\Diff(P,\th)_0$
coincides with the group $\Diff_{\ex}(P,\th)$ of exact
strict contactomorphisms.

Recall that $(P,\th)$ is a regular contact manifold,
for which
$d\th = \pi^*\omega$ and $C^{\infty}(P)^{E} = \{\pi^*f \,;\, f \in C^{\infty}(M)\}$.
If $Q \subseteq P$ is an embedded, closed, oriented submanifold, then
the Hamiltonian action of $\Quant(P,\th)_{0}$ on the connected component $\M \subseteq \Gr_{2n-1}(P)$
of $Q$ gives rise
to a central Lie group extension with Lie algebra cocycle on $C^\infty(M)$
\[
\psi(f,g) = \frac{1}{\mathrm{vol}_{\mu}(P)}\si_Q(\pi^*f,\pi^*g)= \frac{1}{\mathrm{vol}_{\mu}(P)} \int_{Q} \pi^* \left(gdf\wedge\omega^{n-1}\right)/(n-1)!\,.
\]
Expressing the volume of $P$ as
$\mathrm{vol}_{\mu}(P)=\frac{2\pi}{n+1}\mathrm{vol}(M)$, we find
\[
\psi(f,g) = \frac{n+1}{2\pi \,\mathrm{vol}(M)} \,\psi_{\pi_*Q}(f,g)\,,\\
\]
where  $\pi_* Q$
is the pushforward along $\pi \colon P \rightarrow M$ of the singular $(2n-1)$-cycle represented by
the embedded closed submanifold $Q \subseteq P$, and $\psi_{\pi_*Q}$
is the singular cocycle of \eqref{sg22}.
By Remark \ref{pd}, every class in $H^2(P,\R)_{\Z}$ can be represented
by an oriented, embedded submanifold $Q$, so with $[C] = \pi_* [Q]$ in
$H_{2n-1}(M,\R)_{\Z}$, the result follows.
\end{proof}

\begin{Remark}[\bf Triviality of Lie algebra extensions]\label{Rk:Triviality}
Note that from the above proof,
it follows that the Lie algebra extension corresponding to the Hamiltonian action of
$\Quant(P,\th)_{0}$ on the connected component $\M$ of $Q$ in $ \Gr_{2n-1}(P)$ is trivial if and only if
$[\psi_{\pi_{*}Q}] \in H^2(C^{\infty}(M),\R)$ is zero. By Proposition \ref{cohthm}, this is the case
if and only if
$\pi_*[Q] \in H_{2n-1}(M,\R)$ vanishes.
\end{Remark}

Using Poincar\'e duality, we translate this to Roger cocycles and de Rham cohomology.
For a smooth map $f \colon M \rightarrow N$, we denote by $f_{!} \colon H^*(M,\Z) \rightarrow H^*(N,\Z)$
the map that corresponds to $f_* \colon H_*(M,\Z) \rightarrow H_*(N,\Z)$ under Poincar\'e duality.
For the prequantum bundle $\pi \colon P \rightarrow M$, the induced map
\[\pi_{!} \colon H^{k}(P,\R)_{\Z} \rightarrow H^{k-1}(M,\R)_{\Z}\] on
integral classes in de Rham cohomology is fiber integration.

\begin{Corollary}[{\bf de Rham version}]\label{derhamversion}
For every class $[\alpha]$ in the sublattice $\pi_{!} (H^{2}(P,\R)_{\Z})$
of $H^{1}(M,\R)$,
the Lie algebra extension corresponding to the class
\[
	\frac{n+1}{2\pi\, \mathrm{vol}(M)} [\psi_{\alpha}]
\]
in $H^2(C^{\infty}(M), \R)$ integrates to a central
extension
of $\Quant(P,\th)_0$ by~$S^1$.
\end{Corollary}
\begin{proof}
The Roger cocycle
$\psi_{\alpha}$ is cohomologous to the singular cocycle $\psi_{C}$
if $[C] \in H_{2n-1}(M,\R)_{\Z}$ is Poincar\'e dual to
$[\alpha] \in H^1(M,\R)_{\Z}$.
If $[C] = \pi_* [Q]$ for $[Q] \in H_{2n-1}(Q,\R)_{\Z}$, then
$[\alpha] = \pi_{!} [\eta_{Q}]$ for the Poincar\'e dual $[\eta_{Q}]$ of $Q$, as
\[\int_{M} \pi_{!} \eta_{Q} \wedge \gamma = \int_{P}\eta_{Q} \wedge \pi^*\gamma = \int_{Q} \pi^*\gamma = \int_{C}\gamma\,.\]
The result now follows from Corollary \ref{singversion}.
\end{proof}

The lattice $\pi_{!}(H^2(P,\R)_{\Z})$,
which yields the integrable classes in Lie algebra cohomology,
is contained in the lattice $(\pi_{!}H^2(P,\R))_{\Z}$ of integral classes
in $\pi_{!}H^2(P,\R)$.
Note however that it can be strictly smaller, cf.\  \S \ref{sec:thurston}.
The following proposition
is helpful in determining this lattice.

\begin{Proposition}\label{Gysin}
The image of $\pi_{!} \colon H^{2}(P,\Z) \rightarrow H^{2}(M,\Z)$
is the kernel of taking the cup product with the Euler class $[P]\in H^2(M,\Z)$
of the bundle $P$,
\begin{equation}\label{imageofpr}
\pi_{!}H^{2}(P,\Z) = \{[\alpha] \in H^1(M,\Z)\,;\, [P] \smallsmile [\alpha] = 0\}.
\end{equation}
\end{Proposition}
\noindent Note that the image of $[P]$ in $H^2(M,\R)$
is the class $[\omega]$ of the symplectic form.
\begin{proof}
This follows immediately from
the Gysin long exact sequence
in integral cohomology, associated to the principal $S^1$-bundle $P\to M$,
\[
\dots \to
H^2(P,\Z)\stackrel{\pi_!}{\to}
H^1(M,\Z)\stackrel{[P]\smallsmile \,\cdot\,}{\to}H^3(M,\Z)\stackrel{\pi^*}{\to}H^3(P,\Z)\to\dots,
\]
see e.g.\ \cite[\S VI.13]{Br93}.
\end{proof}

Similarly, it follows from Poincar\'e duality (or the Gysin sequence in homology, \cite[\S 9.3]{Sp66}), that
\begin{equation}\label{GysinHomology}
\pi_{*} (H_{2n-1}(P,\Z)) = \{[C] \in H_{2n-1}(M,\Z)\,;\, [C_{P}] \smallfrown [C] = 0\}\,,
\end{equation}
where $[C_{P}]\in H_{2n-2}(M,\Z)$, Poincar\'e dual to $[P]$, is the zero set of a transversal
section of the prequantum line bundle
$P \times_{S^1}\mathbb{C} \rightarrow M$.

\subsection{Examples}
We calculate the integrable classes in $H^2(C^{\infty}(M),\R)$ that correspond to our
group extensions for a number of explicit examples.
For compact surfaces, they span the second Lie algebra cohomology,
whereas for compact K\"ahler manifolds of $\mathrm{dim}_{\R}\geq 4$,
they are all trivial.
For non-K\"ahler symplectic manifolds, our method yields non-trivial integrable
classes, but they do not necessarily span $H^2(C^\infty(M),\R)$.
We illustrate this at the hand of Thurston's nilmanifold,
which was historically the first example of a non-K\"ahler symplectic manifold.

\subsubsection{Compact surfaces}
Let $M$ be a compact orientable 2-dimensional manifold of genus $g$, with generators $[a_{1}], \ldots, [a_{g}]$ and
$[b_{1}], \ldots, [b_{g}]$ of
$H_{1}(M,\Z)$.
A symplectic form $\omega \in \Omega^2(M)$ is prequantizable
if and only if $\mathrm{vol}(M) \in \Z$.
As $H_3(M,\Z) = \{0\}$, equation \eqref{GysinHomology} shows that
$\pi_* H_1(P,\Z) = H_{1}(M,\Z)$.
From Corollary \ref{singversion}, we thus obtain:
\begin{Corollary}
For $k_i, l_i \in \Z$, the Lie algebra cocycles
on the Poisson Lie algebra $C^\infty(M)$
\begin{equation} \label{2dcase}
\psi(f,g) =\frac{1}{\pi \mathrm{vol}(M)} \left(\sum_{i=1}^{g} k_i \int_{a_{i}}  gdf +
\sum_{i=1}^{g} l_i \int_{b_{i}} gdf\right)
\end{equation}
integrate to central $S^1$-extensions of the group $\Quant(P,\th)_{0}$
of quantomorphisms.
\end{Corollary}
By Theorem~\ref{LAcohomology}, the $\R$-span of this integrable lattice
is the full second Lie algebra cohomology of the Poisson Lie algebra $C^{\infty}(M)$.
{The above result on integrable cocycles appears to be new.}

\subsubsection{K\"ahler manifolds}

If $M$ is a prequantizable compact K\"ahler manifold
of dimension $2n$, $n\ge 2$, then the map
\[H^1(M,\R) \rightarrow H^{2n-1}(M,\R) \,;\quad [\alpha] \mapsto [\omega]^{n-1}\wedge [\alpha]\]
is an isomorphism by the Hard Lefschetz Theorem.
Since $n\ge 2$, the map $[\alpha] \mapsto [\omega] \wedge [\alpha]$ is injective,
so Proposition \ref{Gysin} implies that $\pi_{!}H^{2}(P,\R)_{\Z} = \{0\}$.
From Remark \ref{Rk:Triviality}, we then obtain the following result.
\begin{Corollary}
If $M$ is a compact K\"ahler manifold of real dimension $\geq 4$,
then the central Lie group extension derived from the Hamiltonian action of
$\Quant(P,\th)_{0}$ on
the connected component $\M$ of  $\mathrm{Gr}_{2n-1}(P)$ splits at the Lie algebra level.
\end{Corollary}

In particular, the Hamiltonian action of $\Quant(P,\th)_{0}$ on $\M$
lifts to
an action of the universal cover $\widetilde{\Quant}(P,\th)_{0}$ on
the prequantum bundle $\cP \rightarrow \M$.
For compact K\"ahler manifolds of $\mathrm{dim}_{\R} \geq 4$, we thus
obtain a \emph{linear} representation of
$\widetilde{\Quant}(P,\th)_{0}$ on the space
of sections of the prequantum line bundle $\mathcal{L} \rightarrow \M$
associated to $\cP$. This marks a qualitative difference with
the case $\mathrm{dim}_{\R} = 2$, where
the central Lie algebra deformation \eqref{2dcase} occurs.

\subsubsection{Thurston's nilmanifold}\label{sec:thurston}
A \emph{nilmanifold} $M = \Gamma \backslash N$ is a compact homogeneous space for a
connected nilpotent Lie group $N$.
Without loss of generality, one may assume that $N$ is 1-connected, and
$\Gamma \subseteq N$ discrete and co-compact \cite{Ma49}.
If $\n$ is the Lie algebra of $N$, then
by \cite{No54},
the inclusion
$\bigwedge \n^* \hookrightarrow \Omega(M)$ as left invariant forms
yields an isomorphism
between the Lie algebra cohomology $H^*(\n,\R)$ of $\n$ and the de Rham cohomology
$H^{*}(M,\R)$ of $M$.
This remains true over rings of integers localized at small primes \cite{LP82}.

To illustrate that nontrivial lattices of integrable cocycles for $C^{\infty}(M)$ exist
in dimension $\geq 4$, we consider
the quotient $M_{r} = \Gamma \backslash N$ with $N = \mathrm{Heis}(\R,r) \times \R$
and $\Gamma = \mathrm{Heis}(\Z,r) \times \Z$, where $\mathrm{Heis}(R,r)$ is
the \emph{Heisenberg group} over the ring $R$ at level $r\in \N$,
\[
\mathrm{Heis}(R,r) :=
\left \{
\begin{pmatrix}
1 & u & h/r\\
0 & 1 & v\\
0 & 0 & 1
\end{pmatrix}
 \,;\, u, v, h \in R
\right\}\,.
\]
For $r=1$, this is Thurston's symplectic manifold $M^4$ \cite{Th76}. We
include the case $r\neq 1$ to illustrate the role that the torsion of $H^2(M,\Z)$
plays in determining the lattice of integrable cocycles.

The Lie algebra $\n$ is generated by
$x,p,h$ and $z$, with $h,z$ central and $[x, p] = r h$.
The left invariant forms corresponding to the dual basis are
$x^* = du$, $p^* = dv$, $z^* = dz$ and $h^* = dh + r \,udv$.
The differential $\delta \colon \bigwedge\n^* \rightarrow \bigwedge \n^*$
in  \eqref{CE}
is determined by its action on generators:  $\delta h^* = r\, x^* \wedge p^*$ and
$\delta x^* = \delta p^* = \delta z^* = 0$.

The cohomology of the Eilenberg-MacLane space $M_{r} \simeq K(\Gamma, 1)$
is readily calculated from $H^*(B\Gamma,\Z)$.
From \cite[\S 6.1]{LP82} and the K\"unneth formula, one finds
\[
H^*(M_{r},\Z) \simeq H^*(\n,\Z) = \bigwedge \left [x^*,p^*, z^*, x^*\wedge h^*, p^*\wedge h^* \right ] /(r x^*\wedge p^*)\,.
\]
For $a,b \in \Z - \{0\}$, we define the (integral) symplectic form $\omega_{ab} \in \Omega^2(M_{r})$ by
\[\omega_{ab} = a h^* \wedge x^* +  b z^* \wedge p^* = a (dh\wedge du - r u du \wedge dv) + b\, dz \wedge dv\,.\]
It determines the Euler class $[P_{abc}] \in H^2(M_{r},\Z)$ of the prequantum line bundle only up to torsion;
\[ [P_{abc}] = a h^* \wedge x^* +  b z^* \wedge p^* + c x^* \wedge p^*\,, \]
where $c \in \{ 0, \ldots, r-1\}$ labels the different prequantum line bundles with
the same curvature class $[\omega_{ab}] \in H^2(M_{r}, \R)$.
The kernel of the cup product with the Euler class is given by
\[
\mathrm{Ker}
\Big(
[P_{abc}] \smallsmile \,\cdot\, \colon H^1(M_{r},\Z) \rightarrow H^3(M_{r},\Z)
\Big) = \pi_{!} H^2(P_{abc},\Z) =
\{tx^* \,;\, r|tb\}\,,
\]
as $[P_{abc}] \smallsmile t x^* = tb z^* \wedge p^* \wedge x^*$ is a multiple of
$\delta (z^* \wedge h^*)$ if and only if $r | tb$.

From Corollary \ref{derhamversion} with $n=2$ and $\mathrm{vol}(M_{r}) = ab$, we then obtain the following lattice of integrable
classes in second
Lie algebra cohomology:
\begin{Corollary}
For $(M_{r}, \omega_{ab})$ and $k\in \Z$, the 2-cocycles
\[
\psi(f,g) = \frac{3 r  k}{2\pi a \,\,\mathrm{gcd}(r,b)} \int_{M} f dg \wedge du \wedge dv \wedge dz
\]
for the Poisson Lie algebra $C^{\infty}(M_{r})$
are integrable to the identity component of the quantomorphism group
$\Quant(P_{abc},\th)_{0}$.
\end{Corollary}
Since $H^2(C^{\infty}(M_r),\R) \simeq H^1(M_r,\R)$
by Theorem \ref{LAcohomology}, we find a single ray spanned by
integrable classes in this 3-dimensional cohomology space.

\section{The Hamiltonian group}

In this final section, we briefly describe how
the central extensions of the quantomorphism group
$\Quant(P,\th)_{0}$, obtained in Corollary \ref{derhamversion},
can be pulled back by a homomorphism
$\widetilde{\Ham}(M, \om) \rightarrow \Quant(P,\th)_0$. This
yields central $S^1$-extensions
of the universal covering group $\widetilde{\Ham}(M,\om)$.

The homomorphism $\widetilde{\Ham}(M, \om) \rightarrow \Quant(P,\th)_0$
is obtained as follows.
Since $M$ is compact, the Kostant-Souriau extension
\eqref{poissonrijtje}
is split (\cite[Corollary 3.5]{JV})
by the Lie algebra homomorphisms
\[
\R \stackrel{\rho}{\leftrightarrows} C^{\infty}(M) \stackrel{\kappa}{\leftrightarrows} \X_{\ham}(M),
\]
defined by
\[
\rho(f) := \frac{1}{\mathrm{vol}(M)}\int_{M} f \omega^n/n!\,,
\quad
\kappa(X_{f}) := f - \rho(f)\,.
\]
By Lie's Second Theorem for regular Lie groups
\cite[Thm.\ 40.3]{KM97},
these Lie algebra homomorphisms integrate to group homomorphisms
\[
\R \stackrel{R}{\leftrightarrows} \widetilde{\Quant}(P,\th)_{0} \stackrel{K}{\leftrightarrows} \widetilde{\Ham}(M,\om),
\]
on the universal covering groups.
This yields the following commutative diagram:
\begin{center}
\begin{tikzpicture}
  \matrix (m) [matrix of math nodes,row sep=1.5em,column sep=2em,minimum width=2em]
  {
      & & \widehat{\Quant}(P,\th)_{0}\\[-4pt]
     \R & \widetilde{\Quant}(P,\th)_{0} & \Quant(P,\th)_{0} \\
     & \widetilde{\Ham}(M,\om) & \Ham(M,\om). \\};
  \path[-stealth]
    (m-1-3) edge (m-2-3)
    (m-2-3) edge (m-3-3)
    (m-2-1) edge (m-2-2)
    (m-2-2) edge [bend right, dashed] node[above]{\scriptsize $R$} (m-2-1)
    (m-2-2) edge node [above] {{\scriptsize Pr}} (m-2-3)
    (m-3-2) edge (m-3-3)
    (m-3-2) edge [dashed] (m-2-3)
    (m-2-2) edge (m-3-2)
    (m-3-2) edge [dashed, bend right] node[right]{{\scriptsize $K$}} (m-2-2);
\end{tikzpicture}
\end{center}
If we pull back the central $S^1$-extension
$\widehat{\Quant}(P,\th)_{0} \rightarrow \Quant(P,\th)_{0}$
along the homomorphism
$\mathrm{Pr}\o K \colon \widetilde{\Ham}(M, \om) \rightarrow \Quant(P,\th)_0$,
we obtain a central Lie group extension
$\widehat H \rightarrow \widetilde{\Ham}(M, \om)$
by $S^1$,
\begin{center}
\begin{tikzpicture}
  \matrix (m) [matrix of math nodes,row sep=1.5em,column sep=3em,minimum width=2em]
  {
      \widehat H& \widehat{\Quant}(P,\th)_{0}\\
     \widetilde{\Ham}(M,\om)_{0} & \Quant(P,\th)_{0}. \\};
  \path[-stealth]
    (m-1-1) edge [dashed] (m-1-2)
    (m-1-1) edge [dashed](m-2-1)
    (m-1-2) edge (m-2-2)
    (m-2-1) edge node [above] {{\scriptsize $\mathrm{Pr}\circ K$}} (m-2-2);
\end{tikzpicture}
\end{center}
If $\psi$ is the cocycle of $C^{\infty}(M)$ corresponding to the central extension
$\widehat{\Quant}(P,\th)_0$ of $\Quant(P,\th)_{0}$, then
$\kappa^*\psi$ is
the corresponding Lie algebra cocycle on $\X_{\ham}(M)$.
The pullback by $\ka$ of the Roger cocycle \eqref{rog}
is $$(\ka^*\psi_\al)(X_f,X_g)=\int_M f\al(X_g)\om^n/n!\,,$$
and {the pullback} of the singular cocycle \eqref{sg22} is
$$(\ka^*\psi_C)(X_f,X_g)=
\int_C gdf \wedge\omega^{n-1}/(n-1)!\,.
$$
Both expressions are independent of
the choice of Hamiltonian functions.

\begin{Proposition}\label{intham}
If the  2-cocycle $\psi$ on $C^\infty(M)$ can be integrated to a central extension of the group of quantomorphisms $\Quant(P,\th)_0$,
then $\kappa^*\ps$
can be integrated to a central extension of the universal covering group
$\widetilde\Ham(M,\om)$ of the group of Hamiltonian diffeomorphisms.
\end{Proposition}

\section*{Acknowledgements}
We thank Stefan Haller for valuable comments and remarks,
which improved the final version of the paper.
C.V.\ was supported by the grant PN-II-ID-PCE-2011-3-0921
of the Romanian National Authority for Scientific Research.
B.J.\ was supported by the NWO grant
613.001.214 ``Generalised Lie algebra sheaves".

redpajama_set_name: RedPajamaArXiv


The economy and tuition costs may have led to an unexpected shift at the University of Northern Iowa — more part-time students and fewer full-time students.
The shift to part-time enrollment among undergraduate students — nearly 240 more part-time and 220 fewer full-time — contributed to a $1.4 million tuition shortfall this year, UNI officials said. That lost revenue contributed to a $5 million deficit at UNI this year, also due to state budget cuts and other factors.
While UNI leaders can't say for sure without more study what caused the enrollment shift, the economy and students' need to work more hours while taking classes are likely culprits, they said.
In fall 2010, students enrolled part time made up 8.9 percent of UNI's undergraduate population, while full-timers were 91.1 percent. That part-time percentage had been on a slow decline since 2006, when it was 10.2 percent. This fall, the number of part-time students jumped to 11 percent of undergrad enrollment.
"It's to that point where either full-time tuition has become unaffordable to students, so they have to drop down to part-time at a reduced tuition rate, or that in order to afford to attend college they need to keep a job or get a job," UNI student government President Spencer Walrath said.
Several other Iowa colleges and universities said they have heard anecdotally from students who dropped to part-time status to work more hours or because they couldn't afford the larger tuition bill, but most schools did not see the noticeable change that UNI had this fall.
Iowa State University, in fact, has seen its percentage of part-time enrollment decrease over the past five years, from 13 percent to 11.5 percent of total enrollment this fall — ISU's lowest part-time enrollment in the past 20 years, Registrar Kathy Jones said. Undergraduate part-time enrollment was 6.3 percent in 2007 and is 5.1 percent this fall at ISU.
Kirkwood Community College student Shawn Vorba is taking 25 credits this fall but will drop to part-time status in the spring, so he can start a job as a sales associate at a Cedar Rapids car dealership. The 28-year-old is studying business management and marketing, and will finish his necessary classes online so he can work full time.
Kirkwood this fall has 48.2 percent of its students enrolled part time, compared with 45.7 percent last fall. It was in the 44 percent to 45 percent range for several years before that. Breaking those part-time numbers down, enrollment of half-time students was up this fall, while students who are less than half time declined, said Kristie Fisher, vice president for enrollment.
Kirkwood has increased weekend class offerings at the request of students who want to work more hours while working toward a degree, Fisher said.
Mount Mercy University in Cedar Rapids has seen enrollment growth in all areas in the past several years, though the percentage of part time and full time hasn't changed much, officials there said. Part-time students were about 39 percent of enrollment this fall.
The University of Iowa heard more frequently from students struggling financially to stay full time in 2008 and 2009, when the economic crisis first hit, Registrar Larry Lockwood said, bit it's not as common to hear about it now. Part-time enrollment among UI undergraduates has bounced between 9.9 percent and 10.7 percent for the past five years.
UI officials have noticed an increase in the average number of credits undergraduates are taking, up to 13.82 this semester from about 13 credits six years ago. Students are trying to graduate sooner, he said.


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text:
Arcadis signs alliance with Indian based Chola MS
29 October 2014 Consultancy.uk 1 min. read
Global engineering and consulting firm Arcadis has signed a strategic alliance with Cholamandalam MS Risk Services, one of India's leading risk engineering consulting companies. As part of the alliance, the firms collaborate on SRM and environmental projects primarily in the oil / gas and transportation industries across India and South East Asia.
Established in 1994, Cholamandalam MS Risk Services (Chola MS) is a Chennai based risk consulting firm with a focus on the construction sector. The firm, a 50:50 joint venture between the multinationals Murugappa Group (India) and Mitsui Sumitomo Insurance Group (Japan), operates mainly in the Asian market. Chola MS has a team of roughly 130 experts, working from offices in Chennai, Mumbai, New Delhi, Gurgaon, Ahmedabad and Pune.
With the strategic alliance, Arcadis significantly boosts its service portfolio in the area of safety and risk management (SRM), a fast-growing segment within the technical consulting industry. "The ability to identify and minimise risk exposure is an ever increasing concern for our clients as they enter more difficult environments such as offshore production and transfer facilities," explains Mark Fenner, Global Director for Environmental Solutions at Arcadis. With the partnership in place, Arcadis will be able to leverage the specialist skills of Chola MS and hence provide clients with a "completer and better service". In the coming months Arcadis will gradually integrate Chola MS's service offerings into its existing portfolio and introduce the firm to its existing clients in the region.
"Arcadis is a global leader in its field and we are very excited about the opportunities this will bring for our clients and our people. We have worked on a number of projects within and outside India over the years and this will allow us to work with more clients," says MM Murugappan, Chairman of Chola MS.
Partnerships news
More Partnerships
Accenture and AWS partner to support UK and US entrepreneurs Accenture and Amazon Web Services have announced an extension of their partnership to support entrepreneurs win investment.
FinTech marketplace TISAtech names NextWave a consulting partner NextWave Consulting has partnered with TISAtech, to supply the FinTech marketplace with expertise to drive the growth of its services.
Arcadis news
Limiting global warming to 1.5 degrees to cost at least €6 trillion According to a new report, if the global energy sector fails to rapidly decarbonize and increase its generation capacity, the world will lose the race to net zero.
Arcadis launches new City of London office Design consultancy Arcadis has announced the opening of its new office in the City of London next week. National interior design consultancy, Office Principles, has delivered the new offices.
Arcadis appoints Debbie Francis as North-West City Executive The North-Western wing of Arcadis' UK operations has appointed Debbie Francis as its new leader.

redpajama_set_name: RedPajamaCommonCrawl


text:
Another Look At Panic Shot! Rockman
A little while back, we learned of the existence of an official Rockman arcade machine by the name of Panic Shot! Rockman. Details were pretty scarce then, but recently uncovered fliers and advertisements for the machine offer a better idea of what the game was all about, and even reveals an interesting series milestone.
"Everyone's favorite hero, Rockman, has taken a big role in prize machines!" screams the flier(s). "The super popular characters from the Famicon and GameBoy, Rockman and the army of Dr. Wily have arrive in a flipper-type ball game. If you manage to hit through the card in the moving Wily figure, you win. When you hit the ball, the figure will perform a surprising action! The background and the vehicle lamps will also flash brightly! The first great achievement of the Rock Man series: Hear Dr. Wily's voice!"
The machine appeared in arcades in 1992, and was successful enough to warrant a followup version the following year -- the one we covered previously. This particular version of the game tasks players with knocking over Robot Master cards while avoiding a mechanical Wily figure that moves back and forth on a fixed path, blocking your shots. Version 2 swaps out Robot Masters for series trademark Mettools (right). Successfully knock them all down within the set time limit and you win prizes and such.
As mentioned in the fliers, the machine marks the first time in series history where Dr. Wily was voiced; predating his vocalization in 1993's Rockman: Wishing Upon A Star animated series. Wily here likely he taunts the player in you typical mad scientist mannerisms throughout gameplay and maybe even explains how to play the game. The identity of the voice actor remains unknown, but considering the game came out a year or so before Wishing Upon A Star, Ken'ichi Ogata likely assumed the role.
Regardless of its simplicity, Panic! Shot seems like a pretty fun game and definitely quite the collectors item if you're into obscure Rockman products. Scoping out Japanese auctions is probably your best bet at securing a machine, though they're extremely difficult to come by. If you do happen to spot one, it's best to make sure everything is functioning properly or else it really isn't worth the time and money to import. If you've got Panic Shot! at home, we'd love to see more of it, so pictures and videos would be greatly appreciated!
Credits: Images (The Arcade Flyer Archive), translation (Belie)
Update On Megamix, MMZX Volume 2
I bet you're wondering what happened to the second volumes in the Mega Man Megamix and Mega Man ZX manga series. It's the end of July and neither book made their target release dates -- May and early July, respectively. Where are they? Well, I've got a couple updates on just what's going down.
According to Udon's managing editor Matt Moylan, both books were recently approved by Capcom as of July 13, and are off to the printer for production. Matt wasn't able to provide a specific release date for either book; however, just recently Udon's own release schedule was updated showing both books with a mid-August release window.
Moylan will be revealing firmer release dates shortly. Until then, it's a matter of patience. The good news is these books are still on, so it's not like either series was canceled, if that's what you were worried about. In fact, Megamix volume 1 is one of Udon's better selling mangas to date!
More updates as they come in.
Rockman Online Headed To Japan?
Last month, Japanese site Nikkei Trendy posted an interview with Capcom of Japan's COO, Haruhiro Tsujimoto, where he discussed the release strategy for Rockman Online, the upcoming online RPG from Capcom and NeoWiz. In a nutshell, Tsujimoto revealed the game would remain exclusive to South Korea, passing up a Japanese (and Western) release in order to focus on branching the game's appeal to Asian markets. However, despite this statement, it would appear that Capcom is indeed looking at releasing Rockman Online in Japan, too.
According to My Game News Flash and NeoGaf, Capcom Japan recently registered the Rockman Online title at the Japanese Patent & Trademark Offices -- a surefire sign the game is making the jump to the Blue Bomber's native country. Taking into consideration the game wasn't initially set to release in Japan, I'd say this is a good chance we'll be seeing the game registered abroad, further fueling the chances of RMO hitting other regions.
A Western release is still up in the air, mind you. Capcom of America showed some interest in the title last month, when they asked fans via Capcom Unity, if this is something they'd like to see released stateside. The news was met with mostly positive reception; but there's been no indication if it'll sway the possibilities. Until we hear otherwise, keep rallying support.
Auction - EXE 3 Black Tokyo Game Show Edition
Now this is definitely one of those items for completists or the super hardcore collector. Currently up for grabs on eBay is the uncommon Tokyo Game Show edition of Rockman EXE 3 Black, a pre-release variant of the game that was limited to only 1000 copies, distributed a month ahead of Black's retail release on March 28, 2003 in Japan.
Three things separate the TGS edition from the standard March retail version: one, the cartridge came preloaded with special data; secret Battle Chips already unlocked. Two, a holographic Shadow Style Rockman trading card was bundled with the game, and three, to distinguish the TGS version from the retail version, the packaging lacks the gold CESA Game Award emblem present on all copies of the retail version.
While it is a rather interesting item, I don't recommend it to the casual collector. Nevertheless, you can purchase it here for $169 USD. The seller is willing to make an offer, so perhaps you can get him to knock it a bit.
Rockman Invades Wonder Festival
Over the weekend, garage kit convention Wonder Festival kicked off in Japan. Aside from the Kotobukiya figures, the show also played host to an impressive assortment of fan-made Rockman models and figures. From Classic and X to ZX and DASH, there were models of characters from practically every series on display.
Most if not all the models were for sale in limited quantities, and for those of us who couldn't attend, Tomopop has assembled a gallery's worth of photos of each individual figure from all sorts of angles.
Click the above link to check them out or view a few select photos after the break. Like what you see? Obtaining these custom figures is by no means an easy feat, yet they do pop up on eBay occasional. I suggest snooping around there, or better yet, check out Tokyo Hunter's import services.
Mega Man, Servbot Cameo In Street Fighter X Tekken
Capcom's big surprise yesterday was the unveiling of Street Fighter X Tekken (and Tekken X Street Fighter), a brand new collaborative project between Capcom and Namco Bandai that pits characters from both the Street Fighter and Tekken universes against each other. What's noteworthy about this announcement are a couple Mega Man nods featured in both the game's announcment trailer and subsequent gameplay footage...
Blues Getting Kotobukiya Treatment Too
At this weekend's Wonder Festival in Japan, Kotobukiya exhibited a poster teasing a new, third figure in their Rockman line: Blues (aka Proto Man).
Blues will be joining the ranks of the Rockman and Roll Kotobukiya figures some time later this year; Rockman in September, Roll in December.
Thanks, Steve [via Moeyeo and GA Graphic]
Mega Man Universe Has A Release Window
While Keiji Inafune wasn't too forthcoming about a release date for Mega Man Universe at last night's panel, the actual Mega Man Universe kiosk on display at Comic-Con says the game is dropping some time in "Winter." Capcom's jgonzo confirmed the release window during a video tour of Capcom's booth, which you can view below. Jump to the 5:00 point for the reveal.
No indication if this refers to a Winter 2010 release, or early 2011. I'm currently seeking out an official comment, so hang tight.
Capcom Opens Official Mega Man Universe Website
Hot on the heels of tonight's Mega Man Universe press conference, Capcom Japan has opened up an official Mega Man Universe website.
Nothing too spectacular going on right now, the site is limited to a mere two pages: the Unity page contains links to Capcom Unity's various posts and Tweets, and the movie page where you can watch the trailer for the game. Yup, that's about it.
Curiously, the site makes constant reference to the "Mega Man Universe" name and not "Rockman Universe." Perhaps the game will simply be called Mega man Universe in Japan? Makes sense, especially when you recall that the MMU title was even registered in the Japanese Trademark Offices.
Hopefully we'll be seeing some significant updates in the future.
Credit: 2ch
Minor Mega Man Universe Details From SDCC; "Oft-Requested" Mega Man Game Teased
Capcom's San Diego Comic-Con panel literally just came to an end, and as expected, Keiji Inafune was there to talk a little about the mysterious Mega Man Universe. Unfortunately, details were very, very slim as Capcom looks to continue keeping the game shrouded in mystery for the time being, but nevertheless, here's what we got:
According to Inafune, Mega Man Universe will be a game where players can play as their favorite Mega Man characters (including the macho, middle-aged Mega Man from the original game's NES boxart) or other Capcom characters. A primary goal for the title is to offer an experience where Mega Man fans can "unite", a possible implication that online multiplayer will play a key aspect.
Inafune then went on to say more details on the game are expected to arrive in the future. There was no gameplay shown, but Inafune assured the crowd he's determined to show us something soon. Concerning a release date, the game "won't be out for a while."
In closing, Inafune let loose that many more Mega Man announcements are coming this year, and confirmed that an "oft-requested thing that we all want" is coming to fruition. The crowd immediately began to shout for "Mega Man Legends 3", to which Inafune replied with a swift "Thank You."
So, tonight's event kind of turned out to be a bust for big MMU news. However, Inafune's tease about forthcoming announcements has certainly piqued the interests of many, especially that "oft-requested" game tease. Could it finally be the much demanded Mega Man Legends 3? I certainly hope so!
Capcom will be holding another panel Saturday where another game is expected to be announced; perhaps we'll learn more about MMU there.
Capcom's Comic-Con Panel Is Live (Ended)
Unfortunately, I'm not at Comic-Con this year, but my buddies from Eye Lazor Beams and the folks over at 1Up are. The 1Up crew are currently live blogging Capcom's press conference, which you can follow right here.
Mega Man Universe details will likely be revealed at the show, so expect to hear news about the title throughout the night. I'll be posting the relevant details from both 1Up's live blog and ELB after the break for your convenience. Should screen shots, videos, etc. come out, I'll be posting those in a separate post. Check back often!
Kotobukiya Roll Figure On The Way
Looks as if Roll is getting the Kotobukiya model treatment. The Mega Man Network reports the manufacturer plans on releasing a Roll model later this Fall, likely after Rockman's late September release.
The 1/10 scale figure is slated to come with swappable parts that can be arranged to resemble Roll's classic look (right) or her Mega Man 8 look (left). Three different face plates and three different types of hands will also be included.
Pricing has yet to be announced, but considering the fact the model will contain more pieces than its predecessor, it'll probably be a bit pricey. More details as they come!
Inafune Talks Mega Man Universe
I'll admit I'm a tad behind on this story, but nevertheless, it should keep your appetites whet for the impending flood of Mega Man Universe info coming within the next couple days, tomorrow the earliest.
Coinciding with the game's announcement trailer the other day, Keiji Inafune sat down with GameSpot to talk a little about the game. Mum on specifics, Inafune's words offer some subtle hints at just what fans can expect from MMU. Select portions of the interview after the break...
Mega Man Swag At Comic Con
Heading out to Comic Con later this week? Don't forget to stop by Capcom's booth for all sorts of free Mega Man swag. In addition to those cute Servbot Bobble Buds, here's some other Mega Man goods to expect at the show:
Mega Man Universe foamies: 50 of these will be given out every hour. 2000 pieces total.
Mega Man 10 Trucker Hats: Cosplayers who visit the Mega Man Universe kiosk will receive one of these nifty hats.
Lithographs: lithographs of sorts will be given out during producer signings.
Servbot Bobble Buds: 1000 of these will be on-hand for booth attendees.
Mega Man Kubricks: Capcom will be selling the Mega Man and Proto Man Kubrick figures at their booth. Limited to 500 pieces.
Grab Bags: Returning this year are the famous grab gabs featuring games, collectibles and misc. promo goods. You'll never know what kind of Mega Man stuff will be inside!
You can also score some merchandise on Saturday, July 24 at 1PM during a Mega Man cosplay contest. The first 50 cosplayers to show up will get both the trucker hat and foam busters, while the winner of the contest (as judged by Keiji Inafune and MMU producer Ito) will receive a sketch signed by Inafune!
Needless to say, I'm very jealous of everyone on the West coast! (Better bring me something back...) Regardless, please have fun! You can check out the complete list of other Capcom goods at the show and booth map here. It all kicks of July 22!
Kotobukiya Rockman In Greater Detail
Moeyo has released some nice, up close and personal glamor shots of the forthcoming 1/10 scale Kotobukiya Rockman model, currently on display at Kotobukiya's own store in Akihabara, Japan. Moeyo is a bit on the NSFW side, so I took the liberty of dumping the pics here. Check them out below and more after the break!
Mega Man X, Private Investigator
Remember that old Japanese commercial I featured last year? You know, the one featuring a guy in a Mega Man X costume riding a bicycle for no inexplicable reason? Turns out the costumed characters starred in a second commercial, one just as if not more bizarre than the first. This time, X and crew are attempting to solve a murder. The victim? The PlayStation 2 console itself.
Apparently, these zany commercials were part of a cross-promotion with Mega Man X7, Ape Escape, Monster Rancher 4, Bomberman, and Jikkyō Powerful Pro Baseball 10 that aired during the summer of 2003. There's supposed to be a couple more ads featuring the costume-clad heroes, but they've yet to surface online. I can't even begin to imagine just how bizarre the must be. Hopefully we'll see them one day!
UPDATE: According to a reader, these commercials are actually a spoof on a famous Japanese detective show whose opening theme is used in these commercials. Click here to see the original!
Mega Man Universe Revealed
Capcom official revealed Mega Man Universe today over at Capcom Unity. Slated to release on Xbox Live and PlayStation Network, the game is said to revolutionize the classic formula in new and creative ways. Keiji Inafune, directly involved in the project, had this to say:
"We are thrilled to finally be able to announce Mega Man Universe. This game will break the mold and challenge the conventional wisdom of what a Mega Man game can be. When fans finally get to see it, I believe they'll envision their most far-out Mega Man dreams coming true."
Click here for HD trailer
Not much else is know about the game at this point, but the above trailer does drops some cryptic hints that MMU might act as a sort of crossover with other Capcom franchises, namely Street Fighter and Ghosts 'n Goblins. The tagline, "Your world. Megafied", also seems to suggest that the game will support user created content and customization.
More details for the game are said to come from Comic-Con next week. The show will also play host to some cool MMU merchandise such as a foam buster, and a cosplay contest. More details on that here. Naturally, I'll be keeping you up to date with the news as it comes in.
Capcom's "Charging Up" For Something
Capcom's teasing something Mega Man related over at Capcom Unity. The site is featuring a single, cryptic image depicting Classic Mega Man and (possibly) X's Busters with the phrase "We're Charging Up" attached.
One week from today, Keiji Inafune is set to reveal a new game at Comic-Con. Perhaps this a teaser of what's to come?
A penny for your thoughts.
[via Capcom Unity]
More Homemade Crocheted Plushies
Craft savvy Mega Man fan Juri brings us another batch of photos of her crocheted Mega Man plushies. New to the crew this time around are Bubble Man, Top Man, Elec Man, and Crash Man. Check them out below!
...and yes, Top Man's wheels are buttons (creative!). Once again, Juri assures us she's planning on selling them via Etsy someday, but it's a ways off as she plans on making even more.
You can see the original plushies here. Thanks, Juri!
Want An Operate Shooting Star Translation Patch?
To the dismay of some, Capcom has chosen not bring to America Rockman EXE: Operate Shooting Star, the "enhanced" port of the original Mega Man Battle Network. However, that's not enough to keep one fan down.
A buddy of mine is toying with the notion of translating OSS into English. However, before work can begin on a patch, he'd like to know if this is something the community would want, otherwise it's not really worth the effort. So, question for you is, would you like an English patch for OSS?
The process of creating a patch would take at least three to four months, he notes. Most of the work involves lifting the script from the original GBA game, making the necessary changes, and translating the new material introduced in OSS that wasn't present in the original game (new dialog, new chips, etc.). Again, this will only happen if he sees quite a bit of positive reception to the idea. What do you think? Sound off in the comments!
Posted by Protodude at 10:04 AM 158 comments:
Servbot Bobble Buds Coming To Comic-Con
If you're making the journey to Comic-Con next week, you might want to zip ovet to Capcom's booth ASAP. The company announced they'll be selling an exclusive Servbot "Bobble Bud" figure at the show, limited to only a 1000 units.
"Designed by our good friends at SOTA toys, these servbots come in a little plastic ball (picture #1) which eventually becomes their head! Tucked away inside of its own head is the rest of the servbot's body (pic #2). Picture #3 shows you the figure fully assembled. Picture #4 shows you that you can either have the head in bobble mode (left) or in a fixed state (right)."
These are guaranteed to go fast, so grab em' while you can! If you're not going to the show, well, there's always Ebay.
via Capcom Unity
Hitoshi Ariga, Mega Man VAs To Attend FanExpo Canada
Anime News Network reports that Mega Man Megamix author and illustrator Hitoshi Ariga will be appearing at this year's FanExpo, held from August 27-29 at the Metro Toronto Convention Center. Ariga is slated to attend the "CNAnime" portion of the show and will likely be on-hand for autographs and the like.
Other guests of interest include Megumi Ogata, the voice of Sage Harpuia from the Mega Man Zero series and Brad Swaile who provided the voice of Lan Hikari in Mega Man NT Warrior. Needless to say, it might be a wise idea to bring some stuff to sign!
More details about the show, location and how to secure tickets here!
Credit: Anime News Network
Possible Release Date For Kotobukiya Rockman
Originally slated to releasing "sometime in September", it appears Kotobukiya's 1/10 scale Rockman model has been given a more specific release date, if Amazon Japan is to be believed. According to the site, the figure will ship out on September 25th, 2010 -- a date which was recently confirmed by the model's manufacturer through a pre-order confirmation message sent to PRC reader RollChan yesterday.
While Capcom themselves have yet to announce an official release, the Sept. 25 release is likely within the ballpark. We'll be keeping you posted!
Looking to purchase the model? A list of import shops selling the figure can be found here. Thanks for the tip, RollChan!
A Closer Look At The Interactive Rockman Storybook
A little while back we caught a small glimpse of a neat interactive Rockman storybook from Japan, which had readers taking on the eight Rockman 6 Robot Masters through solving puzzles and brain teasers. Furthermore, "boss battles" were carried out through a unique, built-in slot machine; line up the appropriate numbers and defeat the Robot Master!
Today we're taking another look at the book, but this time in its entirety. Thanks to the efforts of an anonymous reader, we've obtained photos of every page and activity the book has to offer. Yes, it's mostly in Japanese, but still pretty fun to look at. Images after the break!
Buy DQIX, Get Zero Collection Half Off At K-Mart
If you're looking to snag a copy of Mega Man Zero Collection for cheap, K-Mart has a pretty tantalizing deal going on. According to reports, the retailer is offering 50% off of Zero Collection and a number of other games if you buy Dragon Quest IX between today and July 17. Additionally, you'll also get a free $10 gaming coupon to put towards a future purchase.
Not a bad deal if you ask me, especially if you've yet to buy MMZC and also intend to purchase DQIX. Save yourself a trip and call the store ahead of time to see if the deal is going on in your area! Full details and more deals at Examiner.
Credit: GoNintendo
Hitoshi Ariga Drawing Rockman and Co. Live
Rockman Megamix author and illustrator Hitoshi Ariga has opened up a Ustream channel where fans can tune in and watch the artist draw Rockman characters and the like live. He even takes requests! The stream is currently off-air as of this writing, but Ariga is known to pop-in every now and then to sketch some stuff. Check back often!
The stream also sports a chat feature, too, so you can even communicate with Ariga and other viewers. Considering both Ariga and a majority of the viewers are Japanese, it's probably a good idea to drop comments in their native language if you're looking for a reply. A computer based translator like Babel Fish or Google Translate should suffice, however. Check out the channel here!
Credit: Rockman Perfect Memories
Udon Announces Art Of Capcom 2
Udon Entertainment has announced a follow up to their previous Art of Capcom art book. Entitled Art of Capcom 2, the new book contains more of Udon's Capcom art from the more recent games, comics, etc.
Why should you, the Mega Man fan care? Well according to our pal Matt Moylan, the book does contain some Mega Man-related art from Tatsunoko Vs. Capcom and Marvel Vs. Capcom 2. This includes promo art, character endings, sketches, etc.
The book will make its debut later this month at Comic Con where you can buy an exclusive hardcover version at Udon's booth. A soft cover version will follow suit, retailing everywhere in August.
Head on over to Udon's official blog for some scans and more details. Thanks, Matt!
Inafune To Announce New Game At Comic-Con
Mark your calenders for July 22nd. According to a blurb from Comic-Con's Thursday schedule, Keiji Inafune is expected to reveal a new game:
"4:45-5:45 Capcom 2010 and Beyond!— See the latest trailers and get the inside scoop on all of Capcom's most highly anticipated games for 2010 and beyond: Marvel Vs. Capcom 3, Sengoku Basara, Okamiden, Bionic Commando Rearmed 2, Ghost Trick, Dead Rising 2, and more. On top of all that, hear details on an exciting new game announcement by Capcom legend Keiji Inafune, the father of Mega Man! Room 6BCF "
No word if it's Mega Man-related or not, but it couldn't hurt to be a little hopeful. Naturally, there's all sorts of speculation as to what it could be, but I'm willing to bet that this is the event where Mega Man Universe will make its debut. Regardless, I'll be keeping you posted!
Servbot Makes Dead Rising 2 Cover
The official box art for Dead Rising 2 was revealed today. The illustration focuses in on protagonist Chuck bracing for an impending zombie horde; among their numbers shines a familiar face...
Mega Man 8-Bit Deathmatch Demo Released
At long last, the much anticipated demo for Mega Man: 8-Bit Deathmatch has finally hit the web! The fully playable demo consists of four maps (single and multiplayer), five characters, and over ten different weapons at your disposal. Download it here!
For the uninformed, 8-bit Deathmatch is a custom Doom mod developed by CutmanMike that "aims to put Doom and Mega Man into a blender and spew out a fun new way to blast your buddies online." The game is rendered entirely in 3D, with textures based upon classic 8-bit Mega Man sprites. You can play it solo in single player or duke it out online with other players.
For more information of 8-bit Deatmatch, check out the game's official blog here. Now, go out there and kick some ass!
Capcom Unity Reveals Mega Man 10 Package Design Winners
Following up on from the winning Japanese designs, Capcom Unity has announced their respective winners for the Mega Man 10 box design contest, chosen by community manager jgonzo and MM10 assistant producer Ito-san.
Winning designs and complementary judge commentary after the jump!
Looking Back At Axed Wave Two Retro Rotos
Now here's a rather neat blast from the past: a rare look at the canceled wave two line of Jazware's Retro Roto figures, coming from the archives of Figures.com and Toy News International.
As the story goes, wave two would have released back in 2006 shortly after wave one. The line was planned to consist of Heat Man, Wood Man, Ice Man, Snake Man and Dr. Wily figures, as seen above. Unfortunately, the market for the toys waned and many Jazewares Mega Man products were consequently canceled, wave two included. Till this day these figures have never resurfaced, not even in the form of prototypes.
Hope for the release of wave two is bright, however. The company recently expressed an interest in perusing the production of the figures according to an interview with Jazeware's Scott Levine in addition to new 3.5'' figures. Perhaps the re-release of wave one is a precursor of what's to come? We can hope. Actually, scratch that, you can do more than hope -- you can tell Jazware's you want them via Twitter of FaceBook. You never know, fan support could go a long way.
GameSide Magazine Features Rockman 10 Concept Artwork (Updated)
The latest issue of GameSide magazine is out in Japan, and with it comes a nice little feature on Rockman 10's development. The special spans six pages and even touts some conceptual art of RM10's Robot Masters and their respective stages. Scans below!
UPDATE (7/5): The rest of the scans are out. Check them over at The Mega Man Network or after the break. There's some pretty fascinating stuff, well worth a look!
Mega Man 10 XBLA Discounted On July 5
For one day only, July 5th, the Xbox 360 version of Mega Man 10 will be marked down from 800 MS points to 400 MS points (5 dollars) as part of the latest "Deal of the Week." That's half off its original price! Of course, there is a bit of a catch: you must have an Xbox Live Gold membership in order to purchase.
Again, it's a one day only deal so jump at the opportunity while you can. More information on Deal of the Week available here.
Thanks for the tip, Hardcore Hecxz and Mr. Flint!
Amazon Lists Mega Man Star Force: Official Complete Works
It appears Udon Entertainment is bringing over yet another Complete Works art book. According to a listing at Amazon, Mega Man Star Force: Official Complete Works is coming stateside on December 8, 2010, retailing at $39.99 ($26.57 via Amazon).
At 176 pages, the book "collects the stunning artwork of every Star Force game, and features character designs, promotional art, rarely seen pieces, and plenty of creator commentary." In other words, its chock full of art. Pre-order your copy here!
And don't forget, MMBN: Official Complete Works is on the way, too. The book is expected to arrive November 23, 2010. Of course, as with most Udon releases, take these release dates as tentative until something a bit more official pops up.
Auction - Promo MM Hats From CES
Currently up for bidding on Ebay are two promotional Mega Man hats from an old Consumer Electronics Show (CES), distributed only to attendees. Based on the hats' logo, I assume they might have come from CES 1989 where Mega Man 3 was featured. Yes, that CES.
Two colors are available, orange and pink, both never worn. I imagine they'd make for neat collector's item if you're into rare promo goods. Click the above respective links to bid!
Thanks, Skaarg.
Battle-Damaged Zero Figure Is Simply Awesome
Hot on the heels of the Rockman DASH models comes another custom creation slated to appear at garage-kit convention Wonder Festival. Dubbed "battle-damaged Zero", the figure depicts a battered and weathered Zero in the heat of battle, a display that fans of the red crusader will surely dig.
Created by a chap going by the name of Rockman@Leopard, the kit will be sold exclusively through the Wonder Festival convention, so don't go counting on getting a hold of one. Nevertheless, that shouldn't stop you from admiring some high quality photos of it over at FG. Click the link to chem 'em out!
Retro Roto Re-Releases Now Available
Largely out of stock for a couple years now, Jazeware's line of six-inch Retro Roto Mega Man figures are now returning to retail. These aren't new figures, mind you, but re-releases of the original line. However, unlike the original releases, the Build-a-Bot pieces used to create the "Full Charge" Mega Man figure are no longer included. Aside from slightly different packaging design, these are generally the same figures.
Where to buy? According to a few readers, the re-released versions have been sighted in-store at Toys R Us, Walgreens and CVS. Online retailers carrying the re-releases include GMKWorld, CmdStore and Big Bad ToyStore. There might be other stores, so shop around.
If you missed out on the figures the first time around, these are definitely for you. Thanks to XHunter, John, RollChan and Brad for the tip(s).
Amazon Lists Mega Man Star Force: Official Complet...
GameSide Magazine Features Rockman 10 Concept Artw...
Capcom Unity Reveals Mega Man 10 Package Design Wi...
Minor Mega Man Universe Details From SDCC; "Oft-Re...


The Amazon Kindle was launched with one real aim: to bring people books. The Kindle was never designed to compete with tablets and sticking to this principle has allowed the Kindle to evolve through to the present day and excel at its task.
The Kindle has remained one of our favourite devices. Where tablets, phones, PCs and TVs have seen radical changes, the technology and the functions of the Kindle have remained true to task, but evolving in pace with technology.
We've been looking over the archives of Kindle history, from Amazon's original ebook reader, through to its brand new, premium, Oasis, to pick out the important steps in the Kindle's journey.
Launched in November 2007, the Amazon Kindle was showcased in NewsWeek magazine. It went on sale on Amazon.com on 19 November and sold out within hours and was immediately dubbed "the iPod of reading".
The original Kindle launched with a 6-inch E Ink display, offered a free wireless connection over Sprint's EV-DO network, on the new Whispernet announced by Amazon.
There was no touch control, however, so the Kindle offered a full keyboard, navigation buttons and a quirky wedge-shaped design aiming to make it better to hold. It also offered a speaker and headphone socket, and expandable SD card storage.
Because page navigation was difficult, it featured a second display with a scroll wheel to help make selections on the display easier. That was needed because the E Ink display was too slow to refresh to give a natural navigation experience.
It cost $399 and was only available within the US, offering access to 90,000 books at launch.
On 9 February 2009 Amazon updated the Kindle with the launch of the Kindle 2. The crazy design of the 2007 Kindle was swapped for a more conventional, flatter, design, with a button lay-out that was flatter and less dominating.
It retained the 6-inch E Ink display, however, improving the technology for faster page turns and better refreshing, while moving from the original 4 shades of grey to 16. There was a storage boost too, moving up wit 2GB of internal storage for 1000s of books.
A new navigation controller was added to make it easier to select text and options on the screen.
The Kindle Store by this time had expanded to around 230,000 titles and the Kindle 2 launched with a Steven King exclusive, called Ur.
It cost $359 originally, discounted to $299 and then $259. The Kindle 2 was then dropped for the Kindle 2 International edition that was announced on 7 October and shifted to GSM for global wireless connections.
The Kindle also fractured off into a line of larger devices called the Kindle DX aimed at magazine reading, but only survived two generations before they stopped being offered.
The Kindle Keyboard was the first Kindle to sell natively in the UK, announced on 29 July 2010. It was launched originally as Kindle 3, an obvious evolution of the Kindle 2, but then changed its name to the Kindle Keyboard.
The Kindle Keyboard again made the page turn controls more compact in the edges and slotted the navigation controller alongside the keyboard. It still didn't offer a touchscreen, so that keyboard was used for browsing and buying from the Kindle store.
The other big departure was a Wi-Fi only version, meaning a lower pricing point. It was $139/£109 for Wi-Fi and the 3G version costing $189/£149. The new UK Kindle Store opened on 27 August 2010, with access to 400,000 books.
The display is still a 6-inch E Ink display, by now offering 600 x 800 pixels.
On 28 September 2011, Amazon had a huge day, announcing not only a fourth-generation Kindle, but also the Kindle Fire as it branched into tablets. However, for the Kindle family, the most important device was the Kindle Touch. The Touch was the first implementation of touchscreen, dropping the navigation keys and the keyboard.
The new device retained the 6-inch E Ink display, but now touch was added because Amazon felt the refresh rate was fast enough and the experience clean enough to make this move. There was 4GB of storage and a battery life of weeks.
The Kindle Touch was again available in Wi-Fi and 3G versions, it introduced Amazon's X-Ray feature. It was originally launched in the US, but became international in March 2012. The Kindle Touch cost $99 for Wi-Fi and $149 for 3G at launch in the US.
The rumours of a front-lit Kindle appeared not long after the launch of the Touch, but it was 6 September 2012 when Amazon announced the Kindle Paperwhite. The first generation device added illumination to the display which was a major breakthrough, meaning you could now read in the dark, with manual brightness adjustment.
It was originally launched with a 6-inch 212ppi display and in 3G and Wi-Fi editions, again relying solely on touchscreen navigation.
The second edition (Paperwhite 2) was announced on 3 September 2013 with an upgraded E Ink display offering better contrast and faster page turns thanks to a more powerful processor. The illumination was also improved for more a more even front light.
The third edition Kindle Paperwhite was announced on 30 June 2015 and stepped the E Ink display up to 300ppi, with twice the number of pixels of the 2012 model. This model also saw the debut of the Bookerly font, Amazon's own font designed for reading.
The Kindle Paperwhite (2015) costs £109.99 (Wi-Fi)/$119.99 (Wi-Fi) and £169.99 (3G)/$189.99 (3G) with special offers. The Kindle Paperwhite is Amazon's best selling Kindle.
While most of the attention has been taken by the advancements firstly to touch and then to the front lighting of the Paperwhite, the humble Kindle continued.
Announced in 2014, this version of the Amazon Kindle was heralded mostly for its price, launched alongside the advanced Kindle Voyage. This Kindle is perhaps basic by comparison, but with a price of only £59/$79, it's the cheapest Kindle, but still fully featured, offering a great entry point into the world of ebook readers.
The Kindle offers full touch control and a 6-inch E Ink display, but doesn't offer a 3G connection, sticking to Wi-Fi only. But there's 4GB of storage and a battery that will last you through weeks of reading.
The Kindle Voyage takes much of what Amazon has done elsewhere and refines it further. It was announced on 18 September 2014 alongside the £59 basic Kindle, offering quite a contrast in the Kindle family.
The Kindle Voyage looked to enhance the Kindle experience, removing the bezel and placing touch controllers next to the display to make page turning easier without having to swipe the display. At the same time the display steps-up offering adaptive front lighting, a feature unique to this model.
The display is still a 6-inch E Ink display, offering a 300ppi resolution. The Kindle Voyage aimed to give users a premium reading experience and it did so, but comes with hefty £169/$199 price for the Wi-Fi model, or £229.99/$289 for the 3G version.
The Amazon Kindle Oasis was announced on 13 April 2016 and is a radical departure for the Kindle design, breaking the device down and starting again.
It offers a device that's much thinner and lighter than any previous Kindle, pulling the hardware into the grip on one side and offering two top page turning buttons. The aim is to make it a superlative one-handed reading device.
It offers enhanced front lighting, but lacks the adaptive lighting of the Voyage, and again sticks to a 6-inch E Ink display with 300ppi.
It comes with a battery cover that will extend the life up to about 9 weeks, making this the longest-lasting Kindle so far. But it's also the most expensive, with a £269.99 price tag.
There's a new Oasis in town, and this time Amazon is making a big change we've not seen so far on this list: it's bumping the screen size up to 7 inches. While the resolution stays the same 300dpi, the new Oasis is a lot more ambitious than the 2016 version.
Announced on 11 October, the new Oasis offers IPX8 waterproofing, another first for Kindle, as well as offering support for Audible books – although that's also coming to some of the older models.
It also brings a welcome price drop, offering this skinny reading experience for £229.99, a £40 price decrease from the previous model. You also get 8GB storage for that (with 32GB an option), meaning more space for all those audiobooks.
It also now has adaptive front lighting too.


Here at LICKALIX we are obsessed with healthy breakfast smoothies, so we thought we would browse our Instagram and give you the low down on some of our favourites.
These homemade breakfast smoothies are a great replacement for a full english and they're jam-packed full of vitamins, minerals, antioxidants and yes, even fibre (thank you NutriBullet). So, for all you busy bees, it's the perfect quick and easy way to get in your daily dose of fruit and veg.
They not only make you look good but they also keep you fuller for longer, boost your energy, reduce cravings and they taste pretty darn good as well!


Cian O'Mahony is Chief Science Officer at Creme Global, a data science company specialising in predictive modelling and software. His background is in pure and applied mathematics, holding a first class honors degree from University College Cork followed by postgraduate studies in applied mathematics and pharmacy. He currently leads a team of analysts developing exposure, intake and risk assessment models in a number of areas including pesticides, food safety, predictive microbiology, nutrition, personal care products and cosmetics. Many of the models and analyses developed by his team at Creme Global are now built into a range of applications used by regulators, industry and academia worldwide.