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% 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 2} \rhead{Due
  date: February 15, 2013} \cfoot{}
\noindent {\bf Reading:} Hatcher \S 1.2 and 1.3.

\begin{exercise}
Hatcher \S 1.2, Exercises 6, 7. Do, but do not hand in 14 and 22.  
\end{exercise}

 \begin{exercise}
 If $G$ is a non-abelian group, then its abelianization is $G^{ab}:=G/[G,G]$, where $[G,G]$ is the subgroup generated by all commutators $[g,h] = ghg^{-1}h^{-1},\ g,h\in G$.

Prove that the fundamental group of any knot complement must have abelianization isomorphic to $\ZZ$. Recall that a knot is an embedded circle in $\RR^3$.
\end{exercise}

\begin{exercise}
	Compute the fundamental group of $\CC^2\setminus\{\ell_1\cup \ell_2\cup \ell_3\}$, where $\ell_i$ are three different lines through the origin.  
\end{exercise}

\begin{exercise}
Let $X = S^2 \times S^2$ and let $Z\subset X$ be the image of $S^1\times S^1$  under the embedding $j\times j$, where $j$ is the inclusion of a great circle $S^1\subset S^2$.  Compute the fundamental group of $X\setminus Z$, using the van Kampen theorem for fundamental groupoids.
\end{exercise}

\begin{exercise}
Hatcher \S 1.3 Exercises 10, 12, 14. Do but do not hand in 27.
\end{exercise}

\end{document}
