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\begin{document}


\begin{center}
\textsc{\small Northeastern University}\\
\textsc{\small Department of Mathematics}\\[10pt]

{\bfseries\large Prof. A. Suciu\hfill  
MTH 3481 --- TOPOLOGY 3 \hfill Spring 1998}\\[7pt]
{\bfseries \Large{F}\large{INAL} \Large{E}\large{XAM}}\\[7pt]
\end{center}

\noindent\textsl{This is a take-home exam, due Monday, June 15, at 9AM.  
Do 6 of the following 7 problems.  Give complete proofs or justifications 
for each statement you make.  Show all your work.}
\vskip 3pt
\hrule
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\begin{questions}

\item  Let 
$Y=\{(x,y)\in \R^2\mid x>0,\, y=\sin\frac{1}{x}\}\cup 
\{(x,y)\in \R^2\mid x=0,\, -1\le y \le 1\}$ and $X=\{0,1\}$. 
Let $f:X\to Y$, given by $f(0)=(0,0)$ and $f(1)=(\frac{1}{\pi},0)$. 
\begin{parts}
\item Show that $f_*:\pi_n(X)\to \pi_n(Y)$ is an isomorphism, for all $n\ge 0$. 
\item Show that $f$ is {\em not} a homotopy-equivalence.
\item Does this contradict Whitehead's theorem?  Why, or why not?
\end{parts}

\item Let $X$ be a connected, finite CW-complex, with $\pi_1(X)$ 
having a non-trivial element of finite order.  Let 
$Y=\widetilde{X}\times K(\pi_1(X),1)$, where $\widetilde{X}$ 
is the universal cover of $X$.
\begin{parts}
\item Show that $\pi_n(X)\cong\pi_n(Y)$, for all $n\ge 0$.  
\item Show that $X$ is {\em not} homotopy-equivalent to $Y$.
\item Does this contradict Whitehead's theorem?  Why, or why not?
\end{parts}  

\item Let $f: T^3 \to S^2$ be the composite of the Hopf bundle map 
$p:S^3\to S^2$ and the quotient map $q:T^3\to S^3$, which collapses the 
$2$-skeleton of the $3$-torus to a point.
\begin{parts}
\item Show that $f_*=0:\pi_n(T^3)\to \pi_n(S^2)$, for all $n\ge 0$.
\item Show that $f_*=0:H_n(T^3)\to H_n(S^2)$, for all $n> 0$.
\item And yet $f$ is {\em not} homotopic to a constant map.
\end{parts}

\item Let $X$ be a connected CW-complex, with $\pi_i(X)=0$ for $1< i <n$, 
for some $n\ge 2$.  Let $h:\pi_n(X)\to H_n(X)$ be the Hurewicz homomorphism. 
Show that $H_n(X)/h(\pi_n(X)) \cong H_n(K(\pi_1(X),1)$. 

\item Let $G$ be an abelian group.
\begin{parts}
\item Show that $H_{n+1}(K(G,n))=0$, for $n>1$. 
\item Show that there is a Moore space $M(G,1)$ if and only if $H_2(K(G,1))=0$.  
\item For what values of $n$ does there exist a Moore space of type $M(\Z^n,1)$?
\end{parts}

\item  Let $X=\CP^2\cup e^3$, with attaching map 
$S^2\xrightarrow{\times p} S^2=\CP^1\subset \CP^2$, 
and $Y=M(\Z_p,2)\vee S^4$. 
\begin{parts}
\item  Show that $M(\Z_p,2)$ can be chosen so that $X$ and $Y$ have the 
same $3$-skeleta.
\item  Show that $H^*(X;\Z)\cong H^*(Y;\Z)$ (as graded rings). 
\item  Show that $H^*(X;\Z_p)\not\cong H^*(Y;\Z_p)$ (as graded rings). 
\end{parts}

\item  Let $G$ be a group, and let $\{M_n\}_{n=1}^{\infty}$ be a sequence 
of $\Z G$-modules. 
\begin{parts}
\item  Construct a CW-complex $X$ with $\pi_1(X)=G$, and $\pi_n(X)=M_n$ 
(as $\Z G$-modules).   
\item  If $X=K(G,1)\times Y$, where $\pi_1(Y)=0$, show that $\pi_n(X)$ 
is trivial as a $\Z G$-module, for all $n>1$. 
\item  If $X=\RP^n$, show that $\pi_n(X)=\Z$ is trivial as a $\Z\Z_2$-module 
if and only if $n$ is odd. 
\end{parts}

\end{questions}
\end{document}