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{\bf Math 444/539 HW\#4, due Monday 9/23/19} \hspace{.5in} \textbf{NAME:} \hspace{4in}
%outlier malcom
\begin{enumerate}
\item Show that $\RP^2 \# \RP^2 \# \RP^2$ is homeomorphic to $T^2 \# \RP^2.$
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\item Let $X \subset \R^3$ be the union of $n$ lines through the origin. Compute $\pi_1(\R^3 \setminus X)$.
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\item Show that the free product $G*H$ of nontrivial groups $G$ and $H$ has trivial center, and that the only elements of $G*H$ of finite order are the conjugates of finite-order elements of $G$ and $H$.
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\item Give an example of a local homeomorphism $f: X \to Y$ and a subset $A \subset X$ such that $f|_A$ is not a local homeomorphism of $A$ onto $f(A)$.
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\item[*math:] $\pi_1(\mbox{The Hawaiian Earring})$ \\
The Hawaiian earring $H$ is the topological space defined by the union of circles in the Euclidean plane $\R^2$ with center $(\frac{1}{n},0)$ and radius $\frac{1}{n}$ for $n = 1, 2, 3...$. The space $H$ is homeomorphic to the one-point compactification of the union of a countably infinite family of open intervals. The Hawaiian earring can be given a complete metric and it is compact. It is path connected but not semilocally simply connected.
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At first glance the Hawaiian earring looks very similar to the wedge sum of countably infinitely many circles, but the two spaces are not homeomorphic.
\begin{enumerate}
\item Show that $H$ is not homeomorphic to $\bigvee_{i=1}^\infty S^1$.
\item Show that $\pi_1(H)$ is uncountably generated.
\item Tell me your favorite peculiar feature of $\pi_1(H)$.
\end{enumerate}
Before googling the Hawaiian earring (you inevitably will need to do so) try to sort out what is going on with its fundamental group by talking amongst each other.
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\item[everyone:] How difficult was this assignment? How many hours did you spend on it?
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\end{enumerate}
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