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  \newtheorem*{rrt}{Rank Rigidity Theorem}{}

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  \begin{document}

  \centerline{\Large {Fundamental Group}}

\begin{enumerate}[1.]

\item Prove that a retract of a contractible space is contractible.

\item Prove that the torus minus one point deformation retracts to the
  wedge sum of two circles. Look up the definition of ``wedge sum'' in
  Hatcher.

\item Prove that $S^m*S^n=S^{m+n+1}$. Here $X*Y$ is the join of two
  spaces (look up the definition in Hatcher). Hint 1: Think about
  $\R^{m+1}\times \R^{n+1}=\R^{m+n+2}$ and the unit spheres in these
  spaces. Hint 2: Prove that join is associative and then induct on
  $m+n$. For associativity, identify both $(X*Y)*Z$ and $X*(Y*Z)$ with
  a certain quotient of $X\times Y\times Z\times \Delta$ where
  $\Delta$ is a triangle.

\item Show that the following statements are equivalent for a space
  $X$:
  \begin{enumerate}[(a)]
    \item Every map $S^1\to X$ is null-homotopic (i.e. homotopic to a
      constant map).
    \item Every map $S^1\to X$ extends to $D^2\to X$.
    \item $\pi_1(X,x_0)=1$ for every $x_0\in X$.
  \end{enumerate}

\item A {\it topological group} is a set $G$ which is both a
  topological space and a group, and the two structures are compatible
  in the sense that the group operations
  $$G\times G\to G, \qquad (x,y)\mapsto xy$$
  and
  $$G\to G, \qquad x\mapsto x^{-1}$$
  are continuous. For example, Lie groups are topological groups. If
  $G$ is a topological group then the set of path-components
  $\pi_0(G)$ is also a group (while for a general topological space it
  is just a set).

  Prove that if $G$ is a topological group then $\pi_1(G,1)$ is
  abelian. Hint: For given $[f],[g]\in G$ construct a map $I\times
  I\to G$ of the square that agrees with $f$ on the top and the bottom
  side, and agrees with $g$ on the left and the right side.

  For example, $\pi_1(SO(2))=\pi_1(S^1)=\Z$, $\pi_1(SO(n))=\Z/2\Z$ for
  $n>2$. Also, $\pi_1(U(n))=\Z$ while $\pi_1(SU(n))=1$ for all $n$. We
  may see this later in the class.

  \item Prove that the following statements are equivalent for a
    path-connected space $X$:
    \begin{enumerate}[(a)]
      \item $\pi_1(X)$ is abelian. (Suppressing the basepoint here,
        they are all isomorphic.)
      \item for any two paths $h,h'$ with the same endpoints we have
        $\beta_h=\beta_{h'}$. Thus $\pi_1(X,x_0)$ and $\pi_1(X,x_1)$
        can be {\it canonically} identified for any two basepoints.
    \end{enumerate}

    In general, $\beta_h$ and $\beta_{h'}$ differ by conjugation.

  \item Let $X$ be a path-connected space with basepoint $x_0$. Consider the function
    $$\Phi:\pi_1(X,x_0)\to [S^1,X]$$
    to the set of homotopy classes of maps $S^1\to X$ given by sending
    $[f]$ to the class of $S^1=[0,1]/0\sim 1 \to X$ induced by $f$. Show
    that
    \begin{enumerate}
      \item this is well-defined, i.e. does not depend on the choice
        of a representative $f$,
      \item $\Phi$ is onto, and
        \item $\Phi([f])=\Phi([g])$ iff $[f]$ and $[g]$ are conjugate
          in $\pi_1(X,x_0)$.
    \end{enumerate}

    Thus $[S^1,X]$ can be thought of as the set of conjugacy classes
    in $\pi_1(X,x_0)$. Note that that's not necessarily a group.

 \end{enumerate}

\end{document}
