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%\theoremstyle{remark}
\newtheorem{normlemma}[thm]{Normality Lemma}
\newtheorem{HBthm}[thm]{Heine-Borel Theorem}
\newtheorem{ASthm}[thm]{Alexander Sub-basis Theorem}
\newtheorem{urysohn}[thm]{Urysohn Lemma}
\newtheorem{Tietze}[thm]{Tietze Extension Theorem}
\newtheorem{Tych}[thm]{Tychonoff Theorem}
\newtheorem{mthm}[thm]{Metatheorem}

\begin{document}
\noindent Name:\\
\noindent Date:\\
\noindent Due: 11/2/07

\noindent Homework 7 (Chapter 3 3.67-3.105)\\
\noindent Do all of the problems.

\setcounter{chapter}{3}

\setcounter{section}{8}

\section{Covering spaces}

\setcounter{thm}{67}

\begin{thm}
Let $(\wt X,p)$ be a covering space of $X$. If $x$, $y\in X$, then
$|p^{-1} (x)| = |p^{-1} (y)|$.
\end{thm}


\begin{exercise}
{\rule{0cm}{0.1cm}}\newline\vspace*{-0.5cm}
\be

3. Describe all non-homeomorphic 3-fold covers of the wedge of two circles.
\ee
\end{exercise}

\setcounter{thm}{71}

\begin{exercise}
{\rule{0cm}{0.1cm}}\newline\vspace*{-0.5cm}
\be
\item Describe all non-homeomorphic 3-fold covers of the Klein bottle.
\item Describe all non-homeomorphic 2-fold covers of $\T^2\num\T^2$.
\item Describe all non-homeomorphic 3-fold covers of $\T^2\num\T^2\num\T^2$.
\item Describe all non-homeomorphic 3-fold covers of $\PP^2$.
\ee
\end{exercise}

\setcounter{thm}{73}

\begin{thm}
If $(\wt X,p)$ is a cover of $X$, $Y$ is connected, and $f$, $g:Y\to \wt X$
are continuous functions such that $p\circ f= p\circ g$, then $\{ y\mid
f(y) = g(y)\}$ is empty or all of $Y$.
\end{thm}

\begin{thm}
Let $(\wt X,p)$ be a cover of $X$ and let $f$ be a path in $X$. Then for
each $x_0\in \wt X$ such that $p(x_0) = f(0)$, there exists a unique
lift $\wt f$ of $f$ satisfying $\wt f(0) = x_0$.
\end{thm}

\setcounter{thm}{77}

\begin{thm}
If $(\wt X,p)$ is a cover of $X$, then $p_*$ is a monomorphism ({\em i.e.},
1--1 or injective) from $\pi_1(\wt X)$ into $\pi_1(X)$.
\end{thm}


\begin{thm}
Let $(\wt X,p)$ be a cover of $X$, $\alpha$ a loop in $X$, and $\wt x_0
\in\wt X$ such that $p(\wt x_0) =\alpha (0)$. Then $\alpha$ lifts to a
loop based at $\wt x_0$ if and only if $[\alpha]\in p_* (\pi_1(\wt X,\wt {x}_0))$.
\end{thm}

\setcounter{thm}{81}

\begin{thm}
Let $(\wt X,p)$ be a covering space of $X$.
Choose $x\in X$, then $|p^{-1}(x)| = [ \pi_1(X): p_* (\pi_1(\wt{X}))]$.
\end{thm}

\setcounter{thm}{92}

\begin{exercise}
{\rule{0cm}{0.1cm}}\newline\vspace*{-0.5cm}
\be
\item Describe all regular 3-fold covering spaces of a figure eight.
\item Describe all irregular 3-fold covering spaces of a figure eight.
\item Describe all regular 4-fold covering spaces of a figure eight.
\item Describe all irregular 4-fold covering spaces of a figure eight.
\item Describe all regular 3-fold covering spaces of a wedge of 3 circles.
\item Describe all regular 4-fold covering spaces of a wedge of 3 circles.
\ee
\end{exercise}


\begin{thm}
Let $(\wt X,p)$ be a regular covering space of $X$. Then ${\mathcal{C}}(\wt X,p)
\cong \pi_1 (X)/p_* (\pi_1(\wt X))$. In particular, ${\mathcal{C}}(\wt X,p)
\cong \pi_1 (X)$ if $\wt{X}$ is simply connected.
\end{thm}


\section{Theorems about groups}
\setcounter{thm}{100}
\begin{cor}
A subgroup $H$ of a free group $F_n$ is always a free group.
\end{cor}

\end{document}