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%\firstpageheader{Topology Ph.D. Qualifying Exam} 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 modify the question so that in your view it is correctly stated, but not in such a
%way that it becomes trivial. {Page \thepage\ of \numpages}{Fall 2006}
\author{Gerard Thompson \and Mao-Pei Tsui }
\date{January 12, 2007}
\runningheader{Topology Ph.D. Qualifying Exam}{Page \thepage\ of \numpages}{January 2007}
\firstpagefooter{Topology Ph.D. Qualifying Exam}{}{} \runningfooter{}{}{}
\runningheadrule \firstpageheadrule
\begin{document}
\title{Topology Ph.D. Qualifying Exam}
\author{Gerard Thompson \and Mao-Pei Tsui }
\date{January 12, 2008}
\maketitle \noindent This examination 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 modify the question so that in your view it is correctly stated, but not in such a way that it
becomes trivial. If you feel that the examination is on the long side do not panic. The grading will be adjusted
accordingly.
\setcounter{page}{1}
\section{Part One: Do six questions}
\begin{questions}
\question
If $(X,d)$ is a metric space then $\{x\in X: d(x,x_0) < \epsilon \}$ is said to be the open ball of radius
$\epsilon$. Prove that an open ball is an open set.
\question
Define what it means for $(X,d)$ to be a metric space. Then $d:X \times X:\rightarrow \R$: is $d$ continuous?
Discuss. If you cannot answer in general do it for $X=\R$ with the usual topology. %Choose a metric $d_1$ on $X$
%defined by $d_1((x,a),(y,b))=(d((x,y)^2+d((a,b)^2)^{\frac{1}{2}}$ and put $\delta=\frac{\epsilon}{2}$. Then if
%$d_1((x,a),(y,b)) < \frac{\epsilon}{2}$, $d((x,y) < \frac{\epsilon}{2}$ and $d((a,b) < \frac{\epsilon}{2}$. Now
%$|d((x,y)-d((a,b)| < d((x,y)+d((a,b) < \frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon. $
\question
%A set is said to have the finite complement topology if the closed sets are the finite sets together with the
%empty set. If $X$ is a topological space with an infinite number of points show that the diagonal $\Delta:=
%\{(x,x) | x \in X\}$ is not closed in the finite complement topology.\\
%Suppose that $x,y \in X$ with $x \neq y$ and that $x \in U$ and $v \in V$ with $U, V$ open in $X$. Then $X-U$ and
%$X-V$ are both finite but since $X$ is infinite we can only have that $U \cap V \neq \emptyset$.
A set is said to have the countable-closed topology if the closed sets are the countable sets together with the
empty set. If $X$ is a topological space with an uncountably infinite number of points is the diagonal $\Delta:=
\{(x,x) | x \in X\}$ closed in the finite complement topology? Justify your answer carefully.
%Suppose that $x,y \in X$ with $x \neq y$ and that $x \in U$ and $v \in V$ with $U, V$ open in $X$. Then $X-U$ and
%$X-V$ are both finite but since $X$ is infinite we can only have that $U \cap V \neq \emptyset$.
\question
Let $B$ be an open subset of a topological space $X$. Prove that a subset $A \subset B$ is relatively open in $B$
if and only if $A$ is open in $X$.
%\question
%A topological space $X$ is said to be \emph{locally connected} if any neighborhood of any point contains a
%connected neighborhood. Prove that the connected components of a locally connected space are both open and closed.
\question
Let $D$ be the open unit disk in the complex plane that is $D:=\{z | \,\, |z| < 1\}$. Let $\sim{}$ be an
equivalence relation on $D$ defined by $z_1 \sim{ z_2}$ if $|z_1 |=|z_2|$. Is the quotient space $D/\sim{}$
Hausdorff? Prove or disprove.
\question
%Define compactness for a topological space. A collection of subsets is said to have the finite intersection
%property if every finite class of those subsets has non-empty intersection. Prove that a topological space is
%compact iff every collection of closed subsets that has the finite intersection property itself has non-empty
%intersection.
Define compactness for a topological space. Prove from your definition that the closed interval $[0,1]$ is
compact.
\question
Prove that $\R$ is not compact.
%2
%\item Prove that a function $f:W \rightarrow \Phi$ from a space
%$W$ into the topological product $\Phi$ is continuous iff, for each $\mu \in
%M$, the composition $p_{\mu} \circ f$ is continuous.\\
%\vspace{.45in}
%Let $\gamma$ be a given cover of a topological space $X$. Assume that for each member $A\in\gamma$, there is given
%a continuous map $f_A:A \rightarrow Y$ such that
%\[f_A \ | \ A\cap B=f_B \ | \ A \cap B\]
%\noindent for each pair of members $A$ and $B$ of $\gamma$. Then we may define a function $f:X \rightarrow Y$ by
%taking
%\[f(x)=f_A(x), \qquad \mbox{(if $x \in A \in \gamma$)}.\]
%Prove that if $\gamma$ is a finite closed cover of $X$, then the function $f$ is continuous.
%\vspace{.25in}
%4
%\item Prove that every compact space is normal.
\question
Let $X=\Pi_{\mu \in M}X_{\mu}$ and $Y=\Pi_{\mu \in M}Y_{\mu}$ be the Cartesian products of the topological spaces
$(X_{\mu})_{\mu \in M}$ and $(Y_{\mu})_{\mu \in M}$ and let $X$ and $Y$ have the product topologies, respectively.
Prove that if for each $\mu \in M$ the maps $f_{\mu}:X_{\mu} \rightarrow Y_{\mu}$ are continuous then $f:X
\rightarrow Y$ defined by $f(x)_{\mu}=f_{\mu}(x_{\mu})$ is continuous.
%\question
%Prove that the continuous image of a connected set is connected.
%Prove that a path-connected topological space is connected.
%Prove that if two connected sets $A$ and $B$ in a space $X$ have a common point $p$, then $A \cup B$ is connected.
%9
%\item Prove that a point $p$ of a space $X$ belongs to the closure $\overline{E}$ of a set $E$ in $X$ iff there
%exists a net $\Phi$ in $E$ which converges to $p$.
\question
A Hausdorff topological space is known to be \emph{locally compact} if every point has a compact neighborhood.
Prove that every closed subspace of a locally compact Hausdorff space is locally compact.
%\vspace{.25in}
%\question
%It is a fact that every compact subset of a Hausdorff space is closed. Moreover a topological space is said to be
%\emph{normal} if every pair of disjoint closed sets can be separated by disjoint open sets. Prove that a compact
%Hausdorff space is normal.
\question
Let $X$ be a topological space. Let $A \subset$ X be connected. Prove that the closure $\overline{A}$ of $A$ is
connected.
%\question
%Prove that if $f:X \mapsto Y$ is continuous and surjective and $X$ is compact and $Y$ Hausdorff then $f$ is an
%identification map.
% Let $f:X \mapsto Y$ be a continuous map between compact Hausdorff spaces. Show that f is a homeomorphism if
%and only if it is injective and surjective.
\question
%Define the terms {\it topological space} and {\it neighborhood space}. Explain carefully how a topological space
%gives rise to a neighborhood space and conversely a neighborhood space gives rise to a topological space.
%Prove or disprove: in a compact topological space every infinite set has a limit point. If you cannot answer the
%question for a compact topological space answer it for a metric space.
% Let $X$ be a compact space and let $A_1 \supset A_ 2 \supset \cdots A_k \cdots$ be a descending chain of
%non-empty closed subsets of $X$. Show that the intersection $\cap_{k =1}^{\infty} A_k$ is not empty.
%Let $A$ be a subspace of a topological space $X$. Prove that $A$ is disconnected if and only if there exist two
%closed subsets $F$ and $G$ of $X$ such that $A \subset F \cup G$ and $F \cap G \subset X-A$.
Define the term ``connected component" for a topological space. Prove that a connected component is connected.
%Prove that a subset of $\R^n$ is compact if and only if it is closed and bounded.
%Is an arbitrary product of T$_1$ or Hausdorff spaces T$_1$ or Hausdorff?
%A closed subset of a locally compact Hausdorff space is locally compact.
\question
A topological space $X$ is said to be \emph{regular} if disjoint singleton and closed sets can be separated by
disjoint open sets. Prove that in a regular space disjoint closed and compact sets can be separated by disjoint
open sets.
\end{questions}
\section{Part Two: Do three questions}
\begin{questions}
%\question
%Let $$S^1 = \{(x,y)|x^2+y^2=1\}$$ and
%$$S^4=\{(x_1,x_2,x_3,x_4,x_5)|x_1^2+x_2^2+x_3^2+x_4^2+x_5^2=1\}.$$
%(i) Let $A$ be the antipodal map on $S^1$ defined by $A(x,y)=(-x,-y)$. Show that $A$ is homotopic to the identity
%map on $S^1$.
%(ii) Let $B$ be the map on $S^4$ defined by $B(x_1,x_2,x_3,x_4,x_5)=(-x_1,-x_2,x_3,-x_4,-x_5)$. Show that $B$ is
%homotopic to the identity map on $S^4$.
\question
Recall that $S^n$ is defined to be $\{(x_1,x_2,...,x_n,x_{n+1}) \in
\R^{n+1} |x_1^2+x_2^2+...+x_n^2+x_{n+1}^2=1\}$. We shall be
interested in the values $n=1,2$. Then define $f,g: S^1 \rightarrow
S^2$ by $f(\cos(s),\sin(s))=(\cos(s),\sin(s),0)$ and
$g(\cos(s),\sin(s))=(\cos(s),-\sin(s),0)$. Show that $f$ is
homotopic to $g$.
%\question
%Recall that $S^k = \left\{(x_1, \cdots, x_{k+1} )\Big| x_1^2+ \cdots x_{k+1}^2 = 1 \right\} \subset \R^{k+1}$.
%The antipodal map $A_k : S^k \mapsto S^k$ is the smooth map defined by $(x_1, \cdots, x_{k+1} ) \mapsto (-x_1,
%\cdots, -x_{k+1} )$.
%(i) Show that $A_1 : S^1 \mapsto S^1$ is homotopic to the identity
% map.
%(ii) Show that $A_k : S^k \mapsto S^k$ is homotopic to the identity map if $k$ is odd.
%We identify (x_1,x_2) with x_1+i x_2. One can construct the homotopy easily by f(t)= exp(i t\pi) (x_1+ i x_2) .
%Then f(0)=x_1+ i x_2 f(1)= -(x_1+ i x_2).
\question
(i) Let $X$ be a topological space. Let $f$, $g:I \mapsto X$ be two paths from $p$ to $q$. Show that $f\sim g$,
that is, $f$ is homotopic to $g$ if and only if $f \cdot g^{-1} \sim c_p$ where $g^{-1}$ is the inverse path to
$g$, $c_p$ denotes the constant path based at $p$ and the $``\cdot"$ denotes the product of (compatible) paths.
(ii) Show that $X$ is simply connected if and only if any two paths in $X$ with the same initial and terminal
points are path homotopic.
%\question
%Let X and Y be topological spaces.
%(i) Define what it means for X and Y to have the same homotopy type.
%(ii) A space is contractible if it is homotopy equivalent to the one-point space. Prove that X is contractible if
%and only if the identity map $id_X : X \mapsto X$ is homotopic to a map $r : X \mapsto X $ whose image is a single
%point.
%(iii) Suppose that $Y \subset X$. Define what it means for $Y$ to be a retract of $X$.
%(vi) Prove that a retract of a contractible space is contractible.
\question
(i) The polygonal symbol of a certain surface without boundary is $xyzx^{-1}zy^{-1}$. Identify the surface. What
is its Euler characteristic?
(ii) Explain how polygons with an even number of sides may be used to classify surfaces without boundary. You do
not need to give detailed proofs.
\question
Let $X=[0,1] \times [0,1]$ denote the rectangle in $\R^2$. Let $\sim$ be the equivalence relation generated by
$(0,p) \sim (1, 1-p)$ wher $0 \le p \le 1$. The quotient space $X/{\sim}$ is called the M\"obius band. Show that
$S^1$ is a retract of the M\"obius band.
\question
Compute the first three homology groups of the \emph{hollow} sphere $S^2$. You may use simplicial theory and a
triangulation but do not simply say that $H_1(S^2)$ is the abelianization of $\pi_1(S^2)$.
%\question
%Let $D=\{ (x,y) \in \R^2 | x^2+ y^2 =1 \}$ be the closed unit disk. Prove that $D$ in cannot be retracted to the
%unit circle $S^1$. Deduce that any continuous map $f : D \longrightarrow D$ has a fixed point. ({\it Hint:}\
%Consider the line joining $x$ to $f(x)$ where $x \in D$.)
\question
\noindent Compute the fundamental group of the (surface of) a sphere when three points on it are removed.
%\question
%Recall the definition of $S^n$ from question 1. By dividing $S^n$ into two ``hemi-spheres" use the Sefeirt van
%Kampen Theorem to find $\pi_1(S^n)$ for $n \geq 1$.
\question
For the sake of this problem a manifold of dimension $n$ will be defined as a topological space in which each
point has a neighborhood that is homeomorphic to $\R^n$. If $M$ is a connected manifold of dimension at least $3$
and $q \in M$, show that $\pi_1(M - \{q\})$ is isomorphic $\pi_1(M )$.
\end{questions}
\end{document}
%\question
%(i) Compute the fundamental group of the \emph{solid} torus in the figure below when the points $x$ and $y$ are
%identified.
%(ii) Compute the fundamental group of the \emph{hollow} torus in the figure below when the points $x$ and $y$ are
%identified.
%\begin{center}
%\begin{figure}[hhh]
% Requires \usepackage{graphicx}
% \includegraphics[scale=.8]{topology1.jpg}
%\includegraphics[scale=.8]{3_hole_torus.jpg}
% \end{figure}
%\end{center}
\begin{center}
\begin{figure}[hhh]
\includegraphics[scale=.75]{3holetorus1.jpg}
\end{figure}
\end{center}
%\question
%Let $P^2$ be the two-dimensional real projective space and $T^2$ be the two-dimensional torus.
%(i) What is $\pi_1(P^2)$? Explain your answer.
%(ii) The space $P^2$ can be obtained from the disk $D^2$ by identifying $x \sim -x$ if $\|x\| = 1$. Let $p\in
%\hbox {int}(D^2)$, the interior of $D^2$. Find the fundamental group of $P^2 -\{[p]\}$.
%(iii) Let $f:P^2 \mapsto T^2$ be a continuous map. Show that $f$ is null homotopic.
%\question
%It is well-known that the punctured plane $\R^2-(0,0)$ has the structure of a topological group with
%multiplication defined by $(x,y) \cdot (p,q) =(xp-yq,xq+yp)$ (induced by multiplication of complex numbers). Can
%one define a topological group structure on $\R^2 - \{(1,0),(-1,0)\}$? Explain.
%%Outline the main points in the construction of the fundamental group of a topological space.
%\item
%The space $G$ is a topological group meaning that $G$ is a group and also a Hausdorff topological space such that
%the multiplication and map taking each element to its inverse are continuous operations. Given two loops based at
%the identity $e$ in $G$, say $\alpha (s)$ and $\beta (s)$, we have two ways to combine them: $\alpha \cdot \beta$
%(product of loops as in the definition of fundamental group) and secondly $\alpha \beta$ using the group
%multiplication. Show, however, that these constructions give homotopic loops.
%\item Show that the fundamental group of a path-connected topological group is abelian. (Hint: Show that $\alpha
%\beta \sim \beta \alpha $.)
%\item Let $D^2=\{ (x,y) \in R^2 | x^2+ y^2 =1 \}$. Prove that the closed unit disk $D^2$ in ${\Bbb R}^2$ cannot be
%retracted to the unit circle $S^1$. Deduce that any continuous map $f : D \longrightarrow D$ has a fixed point.
%({\it Hint:}\ Consider the line joining $x$ to $f(x)$ where $x \in D$.)
%\item For two continuous maps $f$, $g: X \mapsto S^n$ such that $f(x)\neq -g(x)$ for all $x\in S^n$, show that
%$f\simeq g$.
\end{document}
{\lambda}^{3}-d \left( {d}^{2}{c}^{2}{a}^{4}{y}^{2}-{b}^{2}{a}^{4}{z}^
{2}{c}^{2}+{a}^{2}{c}^{4}{b}^{4}-{c}^{2}{a}^{4}{y}^{2}{b}^{2}+{d}^{2}{
a}^{2}{y}^{2}{c}^{4}-{b}^{2}{x}^{2}{a}^{2}{c}^{4}+{c}^{2}{b}^{4}{a}^{4
}+{b}^{2}{a}^{4}{c}^{4}-{a}^{2}{y}^{2}{c}^{4}{b}^{2}+{d}^{2}{b}^{2}{a}
^{4}{z}^{2}+{d}^{2}{b}^{2}{x}^{2}{c}^{4}-{a}^{2}{b}^{4}{z}^{2}{c}^{2}-
{c}^{2}{x}^{2}{b}^{4}{a}^{2}+{d}^{2}{a}^{2}{b}^{4}{z}^{2}+{d}^{2}{c}^{ 2}{x}^{2}{b}^{4} \right)
{\lambda}^{2}+{d}^{2} \left( -{x}^{2}{b}^{4}{
a}^{2}{c}^{4}+{x}^{2}{b}^{4}{c}^{4}{d}^{2}+{b}^{4}{a}^{4}{c}^{4}-{b}^{
4}{a}^{4}{z}^{2}{c}^{2}+{b}^{4}{a}^{4}{d}^{2}{z}^{2}-{a}^{4}{y}^{2}{c} ^{4}{b}^{2}+{d}^{2}{a}^{4}{y}^{2}{c}^{4}
\right) \left( {x}^{2}{b}^{2 }{c}^{2}{d}^{2}-{b}^{2}{x}^{2}{a}^{2}{c}^{2}+{a}^{2}{c}^{2}{b}^{4}+{b}
^{2}{a}^{4}{c}^{2}+{a}^{2}{b}^{2}{c}^{4}+{z}^{2}{a}^{2}{b}^{2}{d}^{2}-
{a}^{2}{b}^{2}{c}^{2}{z}^{2}+{y}^{2}{a}^{2}{c}^{2}{d}^{2}-{a}^{2}{b}^{ 2}{c}^{2}{y}^{2} \right)
\lambda-{d}^{3}{b}^{2}{c}^{2}{a}^{2} \left( -
{x}^{2}{b}^{4}{a}^{2}{c}^{4}+{x}^{2}{b}^{4}{c}^{4}{d}^{2}+{b}^{4}{a}^{
4}{c}^{4}-{b}^{4}{a}^{4}{z}^{2}{c}^{2}+{b}^{4}{a}^{4}{d}^{2}{z}^{2}-{a
}^{4}{y}^{2}{c}^{4}{b}^{2}+{d}^{2}{a}^{4}{y}^{2}{c}^{4} \right) ^{2}