\documentclass[11pt]{book}

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%\theoremstyle{remark}
\newtheorem{thm}{Theorem}
\newtheorem{THM*}{Theorem}
\newtheorem{thm*}[thm]{*Theorem}
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%\theoremstyle{remark}
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\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\D}{\mathbb{D}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\PP}{\mathbb{R}\mathrm{P}}
\newcommand{\Sph}{\mathbb{S}}
\newcommand{\T}{\mathbb{T}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\K}{\mathbb{K}}
\newcommand{\B}{\mathbb{B}}
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\def\dim{\mathop{\rm dim}\nolimits}
\def\image{\mathop{\rm Im}\nolimits}  
\def\interior{\mathop{\rm Int}\nolimits}
\def\star{\mathop{\rm St}\nolimits}
\def\kernel{\mathop{\rm Ker}\nolimits}
%\def\bd{\mathop{\rm Bd}\nolimits}
\def\deg{\mathop{\rm deg}\nolimits}
\def\num{\mathop{\#}\limits}
\def\cl{\mathop{\rm Cl}\nolimits}
\def\lub{\mathop{\rm lub}\nolimits}

\newcommand{\bd}{\mbox{Bd}}
\newcommand{\relation}{\mathcal R}
\newcommand{\cantor}{X_{C}}
\newcommand{\varep}{\varepsilon}
\newcommand{\np}{\vfill\eject} 
\newcommand{\vup}{\vspace{-.08in}} 


%\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: 10/22/07

\noindent Homework 6 (Chapter 3 3.52-3.66)\\
\noindent Do all of the problems.
\setcounter{chapter}{3}

\setcounter{section}{5}

\section{3-manifolds}

\subsection{Lens spaces}

\setcounter{figure}{7}

\begin{figure}[h]
\begin{center}
\includegraphics[height = 1.5 in]{lensspace}
\caption{\label{lensspace} Lens space as a quotient of a lens}
\end{center}
\end{figure}

\begin{exercise}
Show that isotopies form an equivalence relation 
on the set of all embeddings of
$X$ into $Y$.
\end{exercise}

\begin{figure}[h]
\begin{center}
\includegraphics[height = 1.5 in]{solidtorus}
\caption{\label{solidtorus} Solid torus with meridian}
\end{center}
\end{figure}

\setcounter{thm}{54}

\begin{lem}
Let $\{[ \mu ],[ \lambda ] \}$ be
a basis for $\pi_1(\bd(\D^2\times\Sph^1))\cong \Z\times\Z$.
Then $p[\mu]+q[\lambda]$ has a simple closed curve
representative if and only if $p$ and $q$ are relatively prime.
\end{lem}


\begin{exercise}
Use Van Kampen's Theorem to explicitly calculate a group presentation of 
$\pi_1(L(p,q))$.
\end{exercise}

\subsection{Knots in $\Sph^3$}

\setcounter{figure}{10}

\begin{figure}
\begin{center}
\includegraphics[height =2 in]{trefoilwitharrows}
\caption{\label{trefoilwitharrows} The arrows for the arcs of a trefoil knot}
\end{center}
\end{figure}

\begin{lem}
Every loop in $M_K$ is homotopic in $M_K$ to a product of $a_i$'s.
In other words, the loops
$\{a_i\}$ generate $\pi_1(M_K)$.
\end{lem}

\setcounter{thm}{59}

\begin{exercise}
Find the fundamental group of the complement of the unknot (See Figure \ref{unknot}).

\setcounter{figure}{12}

\begin{figure}
\begin{center}
\includegraphics{unknot}
\caption{\label{unknot} The unknot}
\end{center}
\end{figure}
\end{exercise}

\begin{exercise}
Find the fundamental group of the  complement of the trefoil knot.
\end{exercise}

\begin{exercise}
Find the fundamental group of the  complement of the figure-$8$ 
knot (See Figure \ref{figeight}).
\end{exercise}

\begin{figure}
\begin{center}
\includegraphics{fig8}
\caption{\label{figeight} The figure-$8$ knot}
\end{center}
\end{figure}



\section{Homotopy equivalence of spaces}

\setcounter{thm}{63}

\begin{thm}
If $X\sim Y$ then $\pi_1(X)\cong\pi_1(Y)$.
\end{thm}


\end{document}