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UF MTG 6346\\
Topology 1\\
Fall 2024
\begin{center}
{\large \bf Homework 3}\\
Due Monday, October 7, anytime, on Canvas
\end{center}

{\bf Reading.}
Hatcher pages 56--87.


{\bf Problems.}
\begin{enumerate}

\item Let $M$ be an $n$-dimensional manifold, with $n\ge 3$. Let $p\in M$ be any point in the manifold $M$. There is a nice relationship between the fundamental groups $\pi_1(M)$ and $\pi_1(M\setminus\{p\})$ --- how are they related? Prove your answer is correct.

\item \S1.3, Exercise 9.
Show that if a path-connected, locally path-connected space $X$ has $\pi_1(X)$ finite, then every map $X\to S^1$ is nullhomotopic. [Use the covering space $\R\to S^1$.]

\item \S1.3, Exercise 12.
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 Let $\tilde{X}$ be the 6-fold cover of $S^1\vee S^1$ drawn below.
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\includegraphics[width=3in]{SixFoldCover.png}
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\begin{enumerate}
\item Use Proposition~1.32 and Proposition~1.39 to deduce the size of the group $G(\tilde{X})$ of deck transformations. Use the symmetries of $\tilde{X}$ to identify the group $G(\tilde{X})$.
\item Alter $\tilde{X}$ by reversing the direction of the three $\alpha$ arrows on the inner circle only. What is the size of the group $G(\tilde{X})$ of deck transformations? What is the group $G(\tilde{X})$?
\end{enumerate}

\end{enumerate}

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{\bf Recommend Problems (not to turn in).}
\begin{itemize}

%\item \S1.2, Exercise 21.

\item \S1.3, Exercise 10.
%Find all the connected 2-sheeted and 3-sheeted covering spaces of $S^1\vee S^1$ up to isomorphism of covering spaces without basepoints.
%You do not need to prove your answer is correct.

\item \S1.3, Exercise 17.
%Given a group $G$ and a normal subgroup $N$, show that there exists a normal covering space $\tilde{X}\to X$ with $\pi_1(X)\approx G$, $\pi_1(\tilde{X})\approx N$, and deck transformation group $G(\tilde{X})\approx G/N$.

\end{itemize}

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