% October 30, 2007
% Math 589

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 \begin{document}
 
\begin{frame}[allowframebreaks]{Groupoid}
\begin{exam}
 Let $X$ be a manifold and$$PH(X)=\{(x,[\sigma],y)\mid x,y\in X\}$$  where $[\sigma]$ is
the homotopy class of paths such that $\sigma(0)=x$, $\sigma(1)=y
$. Then $PH(X)$ is a groupoid on $X$ with the rules:
$\alpha(x,[\sigma],y)=x$, $\beta(x,[\sigma],y)=y$,
$\mu((x,[\sigma],y),(y' ,[\tau],z) )=(x,[\sigma\circ\tau],z)$ if
and only if $y = y'$, where $\sigma\circ\tau$ is the concatenation
of paths $\sigma$ and $\tau$, $\epsilon(x) = (x,[constant ],x)$
and $i(x,[\sigma],y)=(y,[\sigma^{-1} ],x)$, where $\sigma^{-1}
(t)=\sigma(1-t)$, $t\in[0,1]$.
$(PH(X),\alpha,\beta,\epsilon,i,\mu;X)$ is a groupoid.
\end{exam}
\begin{exam}[Cont.]
 If $PH(X)$ is equipped
with the quotient topology of the compact open topology on the
space of paths of $X$, then $\alpha\times\beta:PH(X)\rightarrow
X\times X$ is a covering map. It follows that $PH(X)$ is a Lie
groupoid on $X$, called it the \textit{Poincar\'{e} groupoid
associated to} $X$. Then $(PH(X),\alpha,\beta,\epsilon,i,\mu;X)$
is a Lie groupoid.
\end{exam}
 \end{frame} 

\end{document}