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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 1998}\\[0.9pc]
{\bfseries \large Take-Home Final Exam}\\[0.6pc]
Due Tuesday, March 17\\[0.8pc]
\end{center}

\noindent
{\bf Instructions}: \textsl{
Do at least 6 of the following 10 problems.  Give complete proofs or 
justifications for each statement you make.  Show all your work.}
\smallskip
\hrule
\vskip 0.5pc


\begin{questions}

\item Let $X=[0,1]$ and $A=\{1,\frac{1}{2},\frac{1}{3},\dots ,\frac{1}{n}, \dots\}\cup \{0\}$. 
Is $H_1(X,A)$ isomorphic to $\widetilde{H}_1(X/A)$?  Explain why, or why not.

\item 
\begin{parts}  
\item Let $X$ and $Y$ be two finite CW complexes.  Show that 
$\chi(X\times Y)=\chi(X)\chi(Y)$.
\item Let $A$ and $B$ be two subcomplexes of $X$ such that $X=A\cup B$. 
Show that $\chi(X)=\chi(A)+\chi(B)-\chi(A\cap B)$. 
\end{parts}

\item   
Let $f:(X,A)\to (Y,B)$ be a continuous map.  Assume that 
$f:X\to Y$ and $f|_A: A\to B$ are homotopy equivalences.
\begin{parts}
\item Show that $f_*:H_*(X,A)\to H_*(Y,B)$ is an isomorphism.
\item Give an example where 
$H_*(X,A)\not\cong H_*(Y,B)$, although $X$ is homotopy 
equivalent to $Y$, and $A$ is homotopy equivalent to $B$.
\end{parts}

\item  Let $X=\RP^{3}\times L(4,1)$ be the product of the projective 
space $\RP^3=S^3/\Z_2$ with the lens space $L(4,1)=S^3/\Z_4$.
\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 homology groups $H_{*}(X)$. 
\end{parts}

\item  Let $X=G_3(\R^6)$ be the Grassmanian of $3$-planes in $\R^6$.  
\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 homology groups $H_{*}(X)$. 
\end{parts}

\item Let $T_g$ be the orientable surface of genus $g$.
\begin{parts}
\item  Describe a 2-fold cover $p: T_3 \to T_2$. 
\item  Compute $p_*:H_1(T_3) \to H_1(T_2)$.
\item Compute $p_*:H_2(T_3) \to H_2(T_2)$.
\end{parts}

\item Let $X=(\C^2\setminus \{0\})/(z_1,z_2)\thicksim (2z_1,2z_2)$.  
Compute $H_*(X)$.  

\item Show that $\RP^2$ is not a retract of $\RP^3$. 

\item Problem 1, in Bredon's book, p. 259.

\item Problem 5, in Bredon's book, p. 259.

\end{questions}
\end{document}