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\ctln{\bf Math 872 Problem Set 4}

\msk

Starred (*) problems are due Thursday, February 26.


\begin{description}


\item{(*)} 25. [Hatcher, p.19, \#9] Show that if $X$ is contractible,
$A\sset X$, and $r:X\ra A$ is a retraction, then $A$ is contractible.

\msk

\item{26.} [Hatcher, p.19, \#16] Show that the CW-complex $S^\infty=\bigcup_n S^n$
is contractible. 

\ssk

\item{\spc} [Hint: On $S^{\infty}\times[1/n,1/(n-1)]$, assume you have already mapped
into $S^{n-1}\sset S^{\infty}$, and deform it (through $S^{n}$) into (a hemisphere of) $S^{n-2}$ .]

\msk


\item{27.} [Hatcher, p.79, \# 1] 
Show that if $p:\wtl{X}\ra X$ is a covering map and $A\subseteq X$ is
a subspace if $X$, then 

\item{\spc} $p|_{p^{-1}(A)}:p^{-1}(A)\ra A$ is also a covering map.

\msk

\item{(*)} 28. [Hatcher, p.79, \# 2] Show that if $p_1:\wtl{X_1}\ra X_1$ and $p_2:\wtl{X_2}\ra X_2$ are
covering maps, then the map 

\ssk

$p=p_1\times p_2:\wtl{X_1}\times\wtl{X_2}\ra X_1\times X_2$,
given by $p(x_1,x_2)=(p_1(x_1),p_2(x_2))$, is also a covering map.

\msk

\item{(*)} 29. [Hatcher, p80, \# 11] Find a pair of (finite) graphs (= 1-dim'l CW complexes with finitely
many 0- and 1-cells) $X_1$ and $X_2$ that have a common finite-sheeted covering space 
$p_1:X\ra X_1$ , $p_2:X\ra X_2$, but do {\it not} commonly cover another space, i.e., 
they are not both covering spaces of a single space $Y$.

\msk

\item{30.} Show that if $p:X\ra Y$ and $q:Y\ra Z$ are both covering maps, and, for
each $z\in Z$, $q^{-1}(z)$ is {\it finite}, then $q\circ p:X\ra Z$ is also a covering
map.

\ssk

\item{\spc} [N.B. This result can be false, if $q^{-1}(z)$ is infinite.]

\msk

\item{31.} Suppose that $p:\wtl{X}\ra X$ and $q:\wtl{Y}\ra Y$ are finite-sheeted
covering spaces, with all spaces connected. Describe how to use these to build 
connected finite-sheeted covering spaces of $X\vee Y$. What is the smallest number
of sheets that such a covering can have?

\end{description}
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