\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig,amsmath,latexsym,enumerate}
\usepackage[all]{xy}
\usepackage{tikz}
\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in

\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Int}{Int}
\def\longto{\longrightarrow}
\def\into{\hookrightarrow}

\renewcommand{\footrulewidth}{1pt}
\newenvironment{proof}[1][Proof.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{fminipage}%
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}%
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
\newcommand{\qed}{\hfill \ensuremath{\Box}}
\newcommand{\disp}{\displaystyle}
\newcommand{\ds}{\displaystyle}


\pagestyle{fancy} \lhead{{\sf Homework 13 $|$ Due December 6 (Friday)
}} \rhead{\thepage} \cfoot{{\sf MthSc 4560/6560 $|$ Topology $|$ Fall
    2013 $|$ M.~Macauley}}



\begin{document}

$\;$

\begin{enumerate}


\item Show that there is no retraction from:
\begin{enumerate}
\item the solid torus $S^1\times D^2$ to its boundary torus $S^1\times
  S^1$;
\item the annulus $S^1\times I$ to its boundary; 
\item the M\"obius band to its boundary. \\
\end{enumerate}



\item Recall that a space $X$ is contractible if the identity map on
  $X$ is nullhomotopic. Show that $X$ is contractible if and only if
  $X$ has the homotopy type of a one-point space. \\



\item Let $S$ be a set. A free group on $S$ is a group $F$ together with
  a map $i\colon S\to F$ such that for any other map $j\colon S\to G$
  to a group $G$, there is a unique homomorphism $f\colon F\to G$ such
  that $j=f\circ i$:
  \[
  \xymatrix{S\ar[r]^i\ar[dr]_j & F\ar@{-->}[d]^f \\ & G}
  \]
  \begin{enumerate}[(i)]
    \item Show that $i$ must be injective.
    \item Show that if $F$ and $F'$ are free groups on $S$, then
      $F\cong F'$. \\
  \end{enumerate}


\end{enumerate}

\end{document}
