

\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{}}



\ctln{\bf Math 872 Algebraic Topology}

\ssk

\ctln{Homework \#\ 2}

\ssk

\ctln{Starred (*) problems due Thursday, February 1}

\bsk

\item{\bf 6.} Show that if there are maps $f:X\ra Y$ and $g,h:Y\ra X$
such that $f\circ g:Y\ra Y$ and $h\circ f:X\ra X$ are homotopic to
$I_Y$ and $I_X$, respectively, 
then $f$ is a homotopy equivalence. Show that the same is true if 
$f\circ g$ and $h\circ f$ are homotopy equivalences.

\msk

\item{\bf (*) 7.} Show that if $X$ is a CW complex, $A,B\subseteq X$ are subcomplexes, 
$A\cup B=X$, and $A,B$ and $A\cap B$ are contractible, then $X$ is contractible.

\msk

\item{\bf 8.} Show that if $f:\del D^n\ra X$ is the attaching map of an $n$-cell
$D^n$, with $n\geq 3$, then the inclusion $X\hookrightarrow X\cup_f D^n$ induces
an isomorphism on $\pi_1$. Show that the same is true if we attach any
(finite or infinite) collection of $n\geq 3$ cells. 

\msk

\item{\bf (*) 9.} Let $X$ = the space obtained from a cube $J^3=J\times J\times J$,
$J=[-1,1]$,
by gluing opposite square faces to one another with a 90-degree righthand
twist (e.g., glue $J\times J\times \{-1\}$ to $J\times J\times \{1\}$ by the
map
$(x,y,-1)\mapsto (y,-x,1)$. Describe a CW structure for $X$ and compute a presentation for $\pi_1(X)$. 

\msk

\item{\bf 10.} Starting with a 2-disk, $D^2$, with two small sub-disks deleted,
show that there are essentially only two spaces obtained by identifying 
each of the two interior boundary circles to $\del D^2$ by homeomorphisms. 
Compute presentations
for the fondamental groups of each, and (by abelianizing) show that the
two groups are not isomorphic, so the two spaces are not homotopy
equivalent.


\vfill
\end