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\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due April 17 (Tuesday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 481 $|$ Topics in Geometry and Topology
    $|$ Spring 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\begin{enumerate}

\item Let $f,g\in\Isom(\H^2)$ be orientation-preserving, and suppose
  that $f$ is parabolic. Prove that either there exists $x\in\hat{\R}$
  such that $f$ and $g$ both fix $x$, or $f^ng$ is an isometry of
  hyperbolic type for some value of $n$. [\emph{Hint}: Use
    conjugation and ``standard position.'']


\item For $n\in\N$, a group element $f$ has \emph{order $n$} if $n$ is
  the smallest positive integer such that $f^n=1$. If such an $n$
  exists, then $f$ has \emph{finite order}, otherwise it has
  \emph{infinite order}.
  \begin{enumerate}
    \item Prove that conjugate elements in any group have the same order. 
    \item Let $f\in\Isom(\H^2)$ be orientation-preserving. Prove that
      if $f$ has finite order, then $f$ is elliptic. 
    \item Suppose that $f\in\Isom(\H^2)$ is elliptic of order $5$. Find
      all possible value of $\tau(f)$. [\emph{Hint}: Use conjugation
        and basic properties of $2\times 2$ rotation matrices.]
  \end{enumerate}


\item Show that the connected sum of two projective planes is
  homeomorphic to the Klein bottle. You may use ``cut-and-paste''
  arguments.


\item Recall that a side-paring takes a polygon and pairs its sides in
  either an orientation-preserving or an orientation-reversing manner.
  \begin{enumerate}
    \item Find all possible ways to side-pair a square, up to
      rotational symmetry of the square.
    \item For each side-pairing you found in Part~(a), compute the
      Euler characteristic of the resulting surface $S$, and identify
      $S$ has either a sphere, torus, Klein bottle, or projective
      plane.
  \end{enumerate}


\item Let $S$ be the surface obtained by identifying opposite sides of
  an $18$-gon in an orientation-preserving manner, that is, with the
  defining relation $a_1a_2\cdots a_9a_1^{-1}a_2^{-1}\cdots
  a_9^{-1}$. Find with proof either a connected sum of tori $T^2$
  and/or projective planes $P^2$ that is homeomorphic to $S$.


\item Prove that $P^2\#T^2$ is homeomorphic to $P^2\#P^2\#P^2$. 

\end{enumerate}

\end{document}
