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\usepackage{fancyhdr} \usepackage{enumerate} \theoremstyle{definition}
\newtheorem{exercise}[]{Exercise} \newtheorem{bonus}{Bonus}[exercise]

% Common mathematical symbols
\newcommand{\RR}{\mathbb{R}} \newcommand{\CC}{\mathbb{C}}
\newcommand{\ZZ}{\mathbb{Z}} \newcommand{\del}{\partial}
\newcommand{\Cc}{\mathcal{C}}\newcommand{\Dd}{\mathcal{D}}
\newcommand{\IR}{\mathbf{I}}\newcommand{\Id}{\mathrm{id}}

\begin{document}
\pagestyle{fancy} \lhead{1301 Topology II, Assignment 3} \rhead{Due
  date: March 14, 2013} \cfoot{}
\noindent {\bf Reading:} Hatcher \S 2.1 and 2.2.


\begin{exercise}
Let $X$ be the complement of two disjoint discs in a larger disc, as shown below, and let $\gamma$ be an immersed circle in $X$ as shown below.
\begin{center}
\includegraphics[scale=0.5]{drawing.pdf}
\end{center}
\begin{enumerate}
\item Choose generators for the fundamental group of $X$ and express $[\gamma]$ in terms of these generators.  Show explicitly the homotopy which justifies your claim. 
\item Choose generators for the first homology of $X$, and express $[\gamma]$ in terms of these generators.  Show explicitly why $\gamma$ is cohomologous to the combination of generators you claim it is.
\end{enumerate}

\end{exercise}

 \begin{exercise}
 Let $X$ be the closed unit disc in $\CC$ and let
 \[
 A_k = \{z\ :\ z^k = 1\}, 
 \]
 for a fixed $k\in\mathbb{N}$.  Compute the relative homology groups $H^i(X,A_k;\ZZ)$ for all $i,k$, and provide representatives of generators for these groups.   
\end{exercise}

\begin{exercise}
Hatcher, Section 2.1, exercise 7, 12, 15.
\end{exercise}

\begin{exercise}
Section 2.2, exercise 2, 3, 4, 10.
\end{exercise}


\end{document}
