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\begin{document}
\noindent Name:\\
\noindent Date:\\
\noindent Due: Friday, December 7

\noindent Homework 10 \\
\noindent Do all of the problems.

These problems all come from prelim exams. The semester and year are indicated above the question. 
\vspace{10mm}


\noindent 2007 Spring\\ 
3. Let $Z\subset \R^3$ denote the z-axis, and let $C_1, C_2,$ and $C_3$ denote the pictured simple closed curves in $\R ^3$:

\begin{figure}[h]
\begin{center}
\includegraphics[width = 3 in]{C1C2C3}
\end{center}
\end{figure}

\noindent (a) Prove that there is no homeomorphism $f: \R^3 \rightarrow \R^3$ sending $Z$
 onto inself and $C_1$ onto $C_2$.

\noindent  (b) Prove that there is no homeomorphism $f: \R^3 \rightarrow \R^3$ sending $Z$
 onto inself and $C_1$ onto $C_3$. (Hint: Consider the unversal cover of $\R^3 -Z$).

\vspace {10 mm}

\noindent 2006 Fall\\
\noindent A2. Let W be the space obtained from a 2-simplex by identifying its edges as pictured.

\begin{figure}[h]
\begin{center}
\includegraphics[height = 1 in]{threesidesglued}
\end{center}
\end{figure}

\noindent (a). Find all covering spaces of W. Draw pictures for each and prove that you have found them all. (hint: $\pi_1$)\\
\noindent (b). An edge of the triangle becomes a closed curve in W. Show there is no retraction of W onto this curve.

\vspace{10 mm}

\noindent Fall 2004\\
\noindent 1. Let's define the 3-hole connected sum of two closed connected surfaces, $M_1^2$  $M^2 _2$ as follows: Let $D_1$, $D_2$, $D_3$ be three disjoint disks in 
$M^2_1$ and let $E_1$, $E_2$, $E_3$ be three disjoint
disks in $M^2_2$. Then the ''3-hole connected sum'' of $M^2_1$ and 
$M^2_2$ is the space created by
removing the interiors of $D_1$, $D_2$, and $D_3$ from $M^2_1$, 
removing the interiors of $E_1, E_2$,
and $E_3$ from $M^2_2$, and identifying $\partial D_1$ with $\partial E_1$, $\partial D_2$ with $\partial E_2$, and $\partial D_3$ with $\partial E_3$ via
homeomorphisms (''$\partial $'' denotes boundary).\\
\noindent (a) Is the 3-hole connected sum well-defined up to homeomorphism? Explain.\\
\noindent (b) Describe all the surfaces that can result from a 3-hole connected 
sum of a Klein bottle $K^2$ with a genus 2 ''double torus'' $M^2$ . Explain your answers.

(Hint: First consider the two holed connected sum operation. )

\end{document}