\magnification=\magstep1
\parskip=5pt
\def\nd{{\noindent}}
\def\P{{\cal{P}}}
\def\R{\mathop{{\rm I}\kern-.2em{\rm R}}\nolimits}
%**end of header
\centerline{\bf Algebraic Topology}
\centerline{Homework 3: Due September 15}      % e.g. September 13
%
%  Now simply type text and problems.  For small print use \small.  To
%  make a problem write
%  \problem ... \endproblem.
%  For a problem with many sections, use
%  \probsec ... \endprobsec.
%  Start each section of the problem with \sec.
%  For a series of true/false questions on a test, use
%  \truefalse ... \endtruefalse
%  Start each problem with \tf.
%  Reset the numbering using \probno=0.

 
1. Page 42, problem 3.2

%2. Page 44, problem 4.2  %% Not an interesting problem. Point-set. 

2. Page 46, problem 4.9.  You may want to do 4.8 as a warm-up. Also, you 
should assume that $S$ is {\it not} a sphere. 

3. Page 49, problem 5.1

4. Page 50, problem 5.2

5. Page 50, problem 5.3.


{\bf Problem 6}.  Let $X$ be any topological space.  (If you want, you
can restrict your attention to Hausdorff spaces, but it really doesn't matter.)
let $Y = [0,1] \times X/\sim$, where $(0,x) \sim (0,y)$ for all $x,y \in X$.
$Y$ is called the {\it cone} of $X$, and the equivalence class of $(0,x)$ is
called the {\it cone point}.  Prove that $Y$ is path-connected (easy) and
simply connected. 

{\bf Problem 7}.  Let $X$ be any topological space, and let $Y = [0,1]
\times X/\sim$, where $(0,x) \sim (0,y)$ and $(1,x)\sim (1,y)$ for all
$x,y \in X$.  $Y$ is called the {\it (free) suspension} of $X$, and is
sometimes denoted $SX$.  (The {\it reduced suspension} $\Sigma X$ is a
slightly different space that doesn't concern us here. Look up
Suspension (topology) on Wikipedia if you want the definition.) For
instance, $S^{k+1}$ is the suspension of $S^k$.  Prove that if $X$ is
path-connected, then $Y$ is simply-connected.

\bye
