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\begin{document}


\begin{center}
\textsc{\small Northeastern University}\\
\textsc{\small Department of Mathematics}\\[1pc]
{\bfseries Prof. Alex Suciu\hspace*{\fill}
{\large MTH 3107 -- TOPOLOGY II} \hspace*{\fill}Winter 1999}\\[0.9pc]
{\bfseries \large Take-Home Final Exam}\\[0.6pc]
Due Monday, March 22\\[0.8pc]
\end{center}

\noindent
{\bf Instructions}: \textsl{
Do at least 5 of the following 6 problems.  Give complete proofs or 
justifications for each statement you make.  Show all your work.}
\smallskip
\hrule
\vskip 1pc


\begin{questions}

\item  Let $X$ be a space.  Show that:

\begin{parts}
\item  $\widetilde{H}_n(X;\Z)=0$ for all $n$ if and only if 
$\widetilde{H}_n(X;\Q)=0$ and $\widetilde{H}_n(X;\Z_p)=0$ 
for all $n$ and all primes $p$.

\item $f: X\to Y$ induces isomorphisms in $H_*(-;\Z)$ if and 
only if it induces isomorphisms in $H_*(-;\Q)$ and $H_*(-;\Z_p)$  
for all primes $p$.

\end{parts}

\item   
Prove the following theorem of Borsuk:  If $f:S^n\to S^n$ 
commutes with the antipodal map, then $f$ has odd degree.  

\noindent 
\small{Remarks:
\begin{types}
\item There is a direct (and rather long) proof 
in Bredon's book (Theorem 20.6, pp.~244-245).  Use 
instead the Borsuk-Ulam theorem (see hint to Problem~7, 
p.~245).
\item Note that this is {\em not} a homotopy-theoretic 
result.  Indeed, $fa\simeq af$, for all $f$ (show that!).
\end{types}
}

\item  Let $X=K\times \RP^{3}$ be the product of the Klein 
bottle with the $3$-dimensional projective space. 
\begin{parts}
\item Find a CW-decomposition of $X$. 
\item Determine the chain complex $(C_{\bullet}(X),d)$ associated to 
that cell decomposition.  
\item Compute the (cellular) homology groups $H_{*}(X)$ and $H_{*}(X; \Z_2)$. 
\end{parts}

\item  Recall that, for a space $X$, and a short exact sequence 
$0\to G' \xrightarrow{\alpha} G\xrightarrow{\beta} G''\to 0$ of 
abelian groups, there is an associated long exact in homology, 
\[
\cdots \to H_i(X; G')\xrightarrow{\alpha_*} H_i(X;G)\xrightarrow{\beta_*} 
H_i(X; G'')\xrightarrow{\partial} H_{i-1}(G')\to\cdots 
\]
Compute explicitly this homology sequence (the terms and the maps), in case 
the coefficients sequence is $0\to \Z \xrightarrow{2} \Z\to \Z_2\to 0$, and 
\begin{parts}
\item $X=\RP^3$.
\item $X=K$, the Klein bottle.
\end{parts}

\item Let $A$ be an integral $3\times 3$ matrix, and let $f:T^3\to T^3$ 
be the induced self-map of the $3$-torus. Compute the Lefschetz number $L(f)$ 
in terms of the entries of $A$.

\item Let $X$ be a finite simplicial complex, and let $G$ be a finite 
group acting simplicially on $X$.  Show that, for every $g\in G$, 
\[
L(g) = \chi(X^g)
\]
where $X^g=\{x\in X \mid gx=x\}$.

\end{questions}
\end{document}