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\title{Homework 5}
\author{Due Wednesday, February 23, 2011}
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\item Let $X = S^2/\{(0,0,\pm1)\}$ be the sphere with the north and south poles glued together.  Compute $\pi_1(X)$.

\item Let $X$ be a torus minus an open disc.
\begin{enumerate}
\item Argue that $X$ is homotopy equivalent to $S^1 \vee S^1$.
\item From part (a), $\pi_1(X)$ is the free group on two generators $a$ and $b$.  Let $\partial X$ be the boundary of $X$, which is a circle, $i: \partial X \to X$ the inclusion, and $c$ a generator of $\pi_1(\partial X) = \mathbb Z$.  What is $i_*(c)$?
\item Let $Y$ be the two-holed torus obtained by gluing two copies of $X$ together along their boundary circles.  (This operation of taking two surfaces, cutting a disc out of each, and gluing the boundary circles is called the ``connected sum.'')  Draw a picture.  Use van Kampen's theorem to compute $\pi_1(Y)$ from this description.
\item What is the Abelianisation of the group in part (c)?
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\item Use van Kampen's theorem to show that $\pi_1(S^n) = 1$ for $n \ge 2$.  Why does the same argument not show that $\pi_1(S^1) = 1$?

\end{enumerate}

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