

\magnification=1200
\overfullrule=0pt
\parindent=0pt

\input amstex

\input epsf
\loadmsbm

\def\ctln{\centerline}
\def\u{\underbar}
\def\ssk{\smallskip}
\def\msk{\medskip}
\def\bsk{\bigskip}
\def\hsk{\hskip.1in}
\def\hhsk{\hskip.2in}

\def\dsl{\displaystyle}
\def\hskp{\hskip1.5in}
\def\ra{\rightarrow}
\def\lra{$\Leftrightarrow$}
\def\pu{\pi_1}
\def\mpu{$\pi_1$}
\def\bra{$\Rightarrow$}
\def\bbr{{\Bbb R}}
\def\bbz{{\Bbb Z}}
\def\del{\partial}
\def\indt{\item{}}
\def\wtl{\widetilde}



\ctln{\bf Math 872 Algebraic Topology}

\ssk

\ctln{Homework \#\ 3}

\ssk

\ctln{Starred (*) problems due Tuesday, February 13}

\bsk

\item{\bf 11.} Compute the fundamental group of the complement $X$ of the coordinate 
axes in $\bbr^3$, i.e., $X=\{(x,y,z)\in \bbr^3\ :\ |xy|+|xz|+|yz|\neq 0\}$. 
(Hint: Find a simpler space to compute the fundamental group of.)

\msk

\item{\bf 12.} 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{\bf (*) 13.} Let $Y\subseteq \bbr^2$ denote the {\it quasi-circle}, given by
$Y=$

\ssk

$\{(x,\sin(1/x)):x\in(0,{{1}\over{\pi}}]\}\cup\{0\}\times [-1,1]\cup[-{{1}\over{\pi}},0]\times\{0\}
\cup\{({{1}\over{\pi}}\cos(t),-{{1}\over{\pi}}\sin(t))\:0\leq t\leq \pi\}$

\ssk

(See Hatcher, p.79, problem \# 7 for an (approximate) picture.) The quotient map

$q:Y\ra Y/\sim$ that collapses the vertical subinterval $\{0\}\times [-1,1]$ to a point 
gives a space homeomorphic to $S^1$. 
Show that $\pi_1(Y)=\{1\}$ (hint: show that any path in $Y$ is disjoint from $\{(x,y)\in Y\ :\ 0<x<\epsilon\}$
for some $\epsilon>0$), but the map $q$ does not lift to the universal covering
$p:\bbr\ra S^1$. [This demonstrates that we cannot in general eliminate the hypothesis 
that $Y$ be locally path-connected, in the lifting criterion.]

\msk

\item{\bf 14.} Let $p:\wtl{X}\ra X$ be a finite-sheeted covering space, with $\wtl{X}\neq \emptyset$.
Show that $\wtl{X}$ is compact and Hausdorff \lra\ $X$ is compact and Hausdorff.

\msk

\item{\bf (*) 15.} Let $p:\wtl{X}\ra X$ and $q:\wtl{Y}\ra Y$ be covering spaces of path-connected,
locally path connected spaces $X$ and $Y$ with $\wtl{X}$ and $\wtl{Y}$ simply-connected.
Show that if $X$ and $Y$ are homotopy equivalent, then $\wtl{X}$ and $\wtl{Y}$ are homotopy
equivalent. [Problem 6 from Problem Set \# 1 might help.]


\vfill
\end