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  \begin{document}

  \centerline{\Large {Cell Complexes}}

\begin{enumerate}[1.]

\item For $n=1,2,\cdots$ let $f_n:S^1\to S^1$ be the map $z\mapsto
  z^n$. Define $$X(n)=S^1\cup_{f_n} e^2$$ as the cell complex obtained
  by attaching a 2-cell to the circle via the attaching map $f_n$
  (this is an example of a {\it Moore space}). Similarly, for
  $m,n=1,2,\cdots$ define $$X(m,n)=S^1\cup_{f_m}
  e^2_1\cup_{f_n}e^2_2$$ as the cell complex obtained from the circle
  by attaching two 2-cells, with attaching maps $f_m$ and $f_n$.

  \begin{enumerate}[(a)] \item Show that if $m>n$ then the maps
    $$f_m,f_{m-n}:S^1\to S^1\hookrightarrow X(n)$$
    are homotopic as maps into $X(n)$.

    \item Show that for all $m,n$ the space $X(m,n)$ is homotopy
      equivalent to the wedge sum $S^2\vee X(k)$ for a certain
      $k=k(m,n)$. Give an explicit formula for $k$ in terms of $m,n$.
  \end{enumerate}

  \item Let $X$ be a cell complex, $A,B$ two of its subcomplexes such
    that $X=A\cup B$. Suppose $A,B,A\cap B$ are all
    contractible. Prove that $X$ is contractible. N.B.: This is false
    for general spaces. E.g. the comb space $$[0,1]\times \{0\}\cup
    \{1,\frac 12,\frac 13,\cdots\,0\}\times [0,1]$$ is contractible,
      but the union of two copies along $(0,1)$ is not.


  \item Let $X$ be a cell complex and $$X_0\subset X_1\subset
    X_2\subset \cdots$$ a sequence of subcomplexes such that $X=\cup_i
    X_i$. Suppose that each $X_i$ is a retract of $X_{i+1}$. Prove
    that $X_0$ is a retract of $X$. Make sure you give an argument
    that your retraction is continuous.

  \item Let $X$ be the space obtained from the torus $S^1\times S^1$
    by identifying three distinct points to one point. Find an
    explicit cell structure on $X$.


 \item In the following exercise we will show that $SO(3)$ is
   homeomorphic to $\R P^3$.

   \begin{enumerate} [(a)]
     \item Using linear algebra show that every element of $SO(3)$ is
       a rotation about an axis in $\R^3$.
       \item For $x\in S^2$ and $\alpha\in [0,2\pi]$ let
         $\phi_{x,\alpha}$ be the rotation about the axis $\R x$ by
         the angle $\alpha$, where angles are measured using
         orientation of $S^2$.
         \item Observe that $\phi_{x,0}=\phi_{x,2\pi}=\mathbbold 1$ so that
           $\phi$ defines a map $$S^2\times [0,2\pi]/{\sim}\to SO(3)$$
           where $S^2\times\{0\}$ and $S^2\times\{2\pi\}$ are
           collapsed to points. This quotient space can be identified
           with $S^3$.
           \item Show that this map $S^3\to SO(3)$ identifies
             antipodal points and induces a homeomorphism $\R P^3\to
             SO(3)$.
   \end{enumerate}

   \item Viewing the torus $T$ as usual as the square $[0,1]^2$ with
     opposite sides identified, let $X$ be obtained from $T$ by
     removing the open disk centered at $(\frac 12,\frac 12)$ and with
     radius $\frac 14$. Find an explicit cell structure on $X$.


   \item Show that $\R^2$ with $n$ points removed is homotopy
     equivalent to the wedge sum of $n$ circles.
    


 \end{enumerate}

\end{document}
