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% Common mathematical symbols
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\begin{document}
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\lhead{1300F Geometry and Topology, Assignment 1}
\rhead{Due date: October 3, 2012}
\cfoot{}
\begin{exercise}\label{firstone}
Let $\Gamma$ be a group, and give it the discrete topology. Suppose
$\Gamma$ acts continuously on the topological $n$-manifold $M$, meaning
that the action map
\begin{equation*}
\xymatrix@R=0{\Gamma\times M\ar[r]^-\rho & M\\
(h,x)\ar@{|->}[r]& h\cdot x}
\end{equation*}
is continuous. Suppose also that the action is \emph{free},
i.e. the stabilizer of each point is trivial. Finally, suppose the
action is \emph{properly discontinuous}, meaning that each $x\in M$
has a neighbourhood $U$ such that $h\cdot U$ is disjoint from
$U$ for all nontrivial $h\in \Gamma$.
\begin{enumerate}[i)]
\item Show that the quotient map $\pi:M\to M/\Gamma$ is a local
homeomorphism, where $M/\Gamma$ is given the quotient
topology. Conclude that $M/\Gamma$ is locally homeomorphic to
$\RR^n$.
\item Show that the quotient topology on $M/\Gamma$ is uniquely
determined by the requirement that $\pi$ is a local homeomorphism.
\item Show that $\pi$ is an open map.
\item Give an example where $M/\Gamma$ is not Hausdorff.
\end{exercise}
\begin{exercise}
Let $(\Gamma, M, \rho)$ be as in Exercise~\ref{firstone}, and let
$f:M\to N$ be a continuous map such that
\begin{equation*}
f(h\cdot x) = f(x)
\end{equation*}
for all $x\in M$ and $h\in\Gamma$. Show that there is a unique map
$\bar f:M/\Gamma\to N$ such that $\bar f(\pi(x)) = f(x)$ for all
$x\in M$, and show that it is continuous.
\end{exercise}
\begin{exercise}
Let $(\Gamma, M, \rho)$ be as in Exercise~\ref{firstone}. Prove
that $M/\Gamma$ is Hausdorff if and only if the image of the map
\begin{equation*}
\xymatrix@R=0{\Gamma\times M\ar[r]& M\times M\\
(g, x)\ar@{|->}[r]&(gx, x)}
\end{equation*}
is closed in $M\times M$.
\end{exercise}
\begin{exercise}
Let $ M = \CC^n\setminus\{0\}$ and let the generator of $\Gamma=\ZZ$
act via $x\mapsto 2x$, for $x\in M$. Show that the quotient
$M/\Gamma$ is a manifold homeomorphic to $S^{2n-1}\times S^1$.
\end{exercise}
\begin{exercise}\mbox{}
Let $M = S^{n}$ and let $\Gamma = \ZZ_2$ act on $M$ via $x\mapsto
-x$. Show that $M/\Gamma$ is homeomorphic to the projective space
$\RR P^n$, as it was defined in class.
\end{exercise}
\begin{exercise}
Consider the 3-sphere $S^3\subset \RR^4$. Using the isomorphism
$\RR^4 \cong \CC^2$, we obtain the inclusion $\iota: S^3\to
\CC^2\setminus\{0\}$. Composing with the projection map $\pi:\CC^2\setminus\{0\}\to
\CC P^1$, we obtain
\begin{equation*}
p = \pi\circ\iota: S^3\to \CC P^1,
\end{equation*}
known as the ``Hopf fibration''.
Using the coordinate charts given in class for $S^3$ and $\CC P^1$,
compute $p$ in coordinates (one chart on each of the domain and codomain should suffice).
\end{exercise}
\end{document}