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% Common mathematical symbols
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
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\begin{document}
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\lhead{1300F Geometry and Topology, Assignment 1}
\rhead{Due date: October 6, 2016}
\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=1ex{\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 $h\in \Gamma$, that is, for all $h\neq 1$.

\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{enumerate}
\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=1ex{\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}\mbox{}
  Let the group of order two, $C_2=\{1,-1\}$, act on $S^n$ via $x\mapsto
  -x$.  Show that $S^n/C_2$ is homeomorphic to the projective space
  $\RR P^n$, as it was defined in class.
\end{exercise}

\begin{exercise}
Prove that $\CC P^1$ is homeomorphic to $S^2$.  Try proving this using the given coordinate charts. 
\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 the notes for $S^3$ and $\CC P^1$,
  compute $p$ in coordinates (one chart on each of the domain and codomain will suffice).
\end{exercise}

\end{document}
