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{\bf Math 444/539 HW\#5, due Tuesday 10/1/19} \hspace{.5in} \textbf{NAME:} \hspace{4in}
%outlier malcom
\begin{enumerate}
\item Let $a$ and $b$ be the generators of $\pi_1(S^1 \vee S^1)$ corresponding to the two $S^1$ summands. Draw a picture of the covering space of $S^1 \vee S^1$ corresponding to the normal subgroup generated by $a^2$, $b^2$, and $(ab)^4$, and prove that this covering space is indeed the correct one.
\item Construct a simply-connected covering space of the space $X \subset \R^3$ that is the union of a 2-sphere and a diameter. Do the same when $X$ is the union of a sphere and a circle intersecting it in two points.
\item (Massey V Exercise 7.2)
Determine the group of automorphisms of the covering spaces described in Examples 2.2, 2.4, 2.7, and 2.9 of Massey V Section 2.
\item Let $X := T^n = \R^n/\Z^n.$ Let $\tilde{X}$ be a path connected covering space of $X$. Show that $\tilde{X}$ is homeomorphic to $T^m \times \R^{n-m}$ for some $m \in \{0,...,n\}.$
\item[*math:] Find all the connected covering space of $\RP^2 \vee \RP^2$.
% Given a covering space action of a group $G$ on a path-connected, locally path-connected space $X$, then each subgroup $H \subset G$ determines a composition of covering spaces $X\to X/H \to X/G.$ Show:
%\begin{enumerate}
%\item Every path-connected covering space between $X$ and $X/G$ is isomorphic to $X/H$ for some subgroup $H \subset G.$
%\item Two such covering spaces $X/H_1$ and $X/H_2$ of $X/G$ are isomorphic iff $H_1$ and $H_2$ are conjugate subgroups of $G.$
%\item The covering space $X/H \to X/G$ is normal iff $H$ is a normal subgroup of $G,$ in
%which case the group of deck transformations of this cover is $G/H$.
%\end{enumerate}
\item[everyone:] How difficult was this assignment? How many hours did you spend on it?
\end{enumerate}
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