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\pagestyle{fancy} \lhead{{\sf Homework 12 $|$ Due November 21 (Thursday)
}} \rhead{\thepage} \cfoot{{\sf MthSc 4560/6560 $|$ Topology $|$ Fall
    2013 $|$ M.~Macauley}}



\begin{document}

$\;$

\begin{enumerate}

%m53.3
\item Let $p\colon E\to B$ be a convering map of a connected space.
  \begin{enumerate}
    \item Show that if $p^{-1}(b_0)$ has $k$ elements for some $b_0\in
      B$, then $p^{-1}(b)$ has $k$ elements for every $b\in B$. Such a
      covering is called a \emph{$k$-fold covering} of $B$. 
    \item Show by example how this can fail if $B$ is disconnected. \\
  \end{enumerate}


  
  %h1.3.10
\item Up to homeomorphism, there are three connected $2$-fold covers
  of $S^1\vee S^1$ and seven connected $3$-fold covers. Find them. \\



% h1.3.3
\item Let $p\colon E\to B$ be a covering space with $p^{-1}(b)$ finite
  and nonempty for all $b\in B$. Show that $E$ is compact and
  Hausdorff iff $X$ is compact and Hausdorff. \\


% s03.8
\item The \emph{Hawaiian earring} is the subspace $H$ of $\C$ defined
  as follows:
  \[
  H=\bigcup_{n=1}^\infty C_n\,,\qquad C_n=\left\{\tfrac{1}{n}(1+e^{2\pi i
    z}):z\in\C\right\}\hspace{25mm}
  \begin{tikzpicture}[baseline=5.5ex]
    \draw (0,1) circle(1);
    \draw (0,1/2) circle(1/2);
    \draw (0,1/3) circle(1/3);
    \draw (0,1/4) circle(1/4);
    \draw (0,1/5) circle(1/5);
    \draw (0,1/6) circle(1/6);
    \draw (0,.15) node{$\cdot$};
    \draw (0,.24) node{$\cdot$};
    \draw (0,.05) node{$\cdot$};
  \end{tikzpicture}
  \]
  Let $z_0=0\in\C$ be the basepoint. This space retracts onto the
  $n^{\rm th}$ circle $C_n$ via the map
  \[
  r_n\colon H\longto C_n\,,\qquad\quad r_n(z)=\begin{cases}z & z\in C_n
  \\ 0 & z\not\in C_n. \end{cases}
  \]
  \begin{enumerate}
  \item Prove that the induced homomorphism
    $(r_n)_*\colon\pi_1(H,z_0)\to\pi_1(C_n,z_0)$ is onto.
  \item Assuming that $\pi(C_n,z_0)\cong\Z$, prove that $\pi_1(H,z_0)$
    is uncountable. [\emph{Hint}: For each binary sequence $(b_n)$,
      define an element of $\pi_1(H,z_0)$ which goes $b_n$ times
      around $C_n$.]
  \end{enumerate}
  
  


\end{enumerate}

\end{document}
