

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

\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\bbq{{\Bbb Q}}
\def\del{\partial}
\def\indt{\item{}}
\def\wtl{\widetilde}
\def\coker{\text{coker}}
\def\im{\text{im}}
\def\wtih{\widetilde{H}}

\ctln{\bf Math 872 Algebraic Topology}

\ssk

\ctln{Problem Set \#\ 9}

\ssk

\ctln{Starred (*) problems due Tuesday, April 24}

\bsk

\item{\bf 44.} Compute the (cellular) homology groups of the quotient
spaces $\bbr P^n/\bbr P^m$ for $m<n$. (Here we take $\bbr P^m$ to be the
image of $S^n\cap(\bbr^{m+1}\times\{0\})\subseteq \bbr^{n+1}$ under the antipodal map.)

\msk

\item{\bf 45.} Show that if $A\subseteq X$ both have $\oplus_i H_i(A),\oplus_i H_i(X)$
finitely generated, and $A$ has an open neighborhood that deformation retracts to it, then
$\oplus_i H_i(X/A)$ is finitely generated and $\chi(X)=\chi(A) + \chi(X/A)-1$.

\msk

\item{\bf (*) 46.} Show that $\wtih_i(S^n\setminus X)\cong \wtih_{n-i-1}(X)$ when $X$ is homeomorphic to a
finite commected graph. (Hint: prove it first for the case that $X$ is a tree, then (inductively)
add one edge at a time.)

\msk

\item{\bf 47.} For $X$ a finite CW-complex and $F$ a field, show that the Euler characteristic
of $X$ can be computed as $\chi(X) = \sum_i (-1)^i\dim_F(H_i(X;F))$.

\msk

\item{\bf (*) 48.} Show that if $X$ is a space with $H_k(X;\bbz)\cong \bbz_n$,
then $H_{k+1}(X;\bbz_n)\neq 0$. 

\ssk

(Hint: look at the LEHS induced by the SES of coefficient groups 

\ssk

\ctln{$0\ra\bbz\buildrel{\times n}\over\ra \bbz\ra \bbz_n\ra 0$ .)} 


\vfill
\end









\ctln{\vbox{\hsize=4in
\leavevmode
\epsfxsize=4in
\epsfbox{hw6.ai}}}


