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\begin{document}
\title{Topology Ph.D. Qualifying Exam}
\author{Mao-Pei Tsui \and G. Martin}
\date{April 22, 2006}
\maketitle
\noindent
This exam has been checked carefully for errors. If you find what
you believe to be an error in a question, report this to the proctor.
If the proctor's interpretation still seems unsatisfactory to you, you may
alter the question so that in your view it is correctly stated, but
not in such a way that it becomes trivial.
\bigskip
\section*{Section 1}
Do 3 of the following 5 problems.
\begin{enumerate}
\item Prove that $(0,1)$ is homeomorphic to $\mathbb{R}$.
\bigskip
\item Let $X$ be a topological space which is connected and locally path connected.
Prove that $X$ is path connected.
\bigskip
\item Prove that a connected space is path connected if and only if every path component
is open.
\bigskip
\item Prove that a space $Y$ is Hausdorff if and only if for every space
$X$ and every pair of continuous functions $f : X \mapsto Y$ and $g
: X \mapsto Y$ , the set $\{x \in X|f(x) = g(x)\}$ is closed in $X$.
\bigskip
\item Let $f : X \mapsto Y$ be a quotient map, with $Y$ connected. Show that if
$f^{-1}({y})$ is connected for all $y \in Y$ , then $X$ is
connected.
\bigskip
\end{enumerate}
\newpage
\section*{Section 2}
Do 3 of the following 5 problems.
\begin{enumerate}
\item Prove
\begin{enumerate}
\item Define $f, g: X \longrightarrow S^n$ are continuous and \\
$f(x)\neq -g(x)$ for all $x \in X$, then $f$ is homotopic to
$g$.
\item A continuous $f :S^n \mapsto S^n$ either has a fixed point or is homotopic to the antipodal
map.
\end{enumerate}
\bigskip
\item Let $X$ be a connected, locally path-connected space. Suppose $\pi_1(X)$ is finite. Show that
every continuous map $f : X \mapsto S^1$ is homotopic to a constant
map.
\bigskip
\item Let $X_{1}$ and $X_{2}$ be two copies of $S^{2}$ and let $N_{1},
S_{1}$ and $N_{2}, S_{2}$ be the north and south poles of $X_{1}$ and
$X_{2}$, respectively. Define $X$ to be the quotient space obtained by
identifying $N_{1}$ with $N_{2}$ and $S_{1}$ with $S_{2}$. Compute
the fundamental group of $X$ by using the Seifert-van Kampen theorem.
\bigskip
\item
Let $S^1$ be the unit circle in $\mathbb{R}^2$ and $p : X \mapsto
S^1$ be a covering map with finitely many sheets. Prove that $X$ is
homeomorphic to $S^1$.
\bigskip
\bigskip
\item Let $p: \widetilde{X} \mapsto X$ be the universal covering space of a space $X$ and let $f: X \mapsto
X$
be a continuous map.\begin{enumerate}
\item Prove that there exist lifts of $f$ to $\widetilde{X}$,
that is, maps $\widetilde{f}: \widetilde{X} \mapsto \widetilde{X}$
such that $p\circ \widetilde{f} = f \circ p$.
\item Suppose $\widetilde{f_1}$, $\widetilde{f_2}$ are lifts of
$f$ and there exists $\widetilde{x_1}$, $\widetilde{x_2}$ such that
$\widetilde{f_1}(\widetilde{x_1}) =\widetilde{x_1}$,
$\widetilde{f_2}(\widetilde{x_2}) =\widetilde{x_2}$ and
$p(\widetilde{x_1}) = p(\widetilde{x_2})$. Prove that there exists a
covering transformation $\sigma: \widetilde{X} \mapsto
\widetilde{X}$ such that $\widetilde{f_2} = \sigma \widetilde{f_1}
\sigma^{-1}$.
\end{enumerate}
\bigskip
\end{enumerate}
\end{document}