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UF MTG 4303/5317\\
Introduction to Topology 2\\
Spring 2024
\begin{center}
{\large \bf Homework 2}\\
Due Monday, February 5, anytime, on Canvas
\end{center}

{\bf Reading.}
Munkres \S54--55, 35, 38.


{\bf Problems.}
\begin{enumerate}

\item \S51 \#3(a,b,c).
A space $X$ is said to be \emph{contractible} if the identity map $i_X\colon X\to X$ is nullhomotopic.
\begin{enumerate}
\item Show that $I$ and $\R$ are contractible.
\item Show that a contractible space is path connected.
\item Show that if $Y$ is contractible, then for any $X$, the set $[X,Y]$ has a single element.
\end{enumerate}
\emph{Remark: Note that $[X,Y]$ is defined in the prior problem, \S51 \#2.}

\item \S52 \#2.
Let $\alpha$ be a path in $X$ from $x_0$ to $x_1$; let $\beta$ be a path in $X$ from $x_1$ to $x_2$.
Show that if $\gamma=\alpha * \beta$, then $\hat{\gamma}=\hat{\beta}\circ\hat{\alpha}$.

\item \S53 \#3.
Let $p\colon E\to B$ be a covering map; let $B$ be connected.
Show that if $p^{-1}(b_0)$ has $k$ elements for some $b_0\in B$, then $p^{-1}(b)$ has $k$ elements for every $b\in B$.
In such a case, $E$ is called a \emph{$k$-fold covering} of $B$.

\item \S54 \#3.
Let $p\colon E\to B$ be a covering map.
Let $\alpha$ and $\beta$ be paths in $B$ with $\alpha(1)=\beta(0)$; 
let $\tilde{\alpha}$ and $\tilde{\beta}$ be liftings of them such that $\tilde{\alpha}(1)=\tilde{B}(0)$.
Show that $\tilde{\alpha}*\tilde{\beta}$ is a lifting of $\alpha*\beta$.

\end{enumerate}



\vfill

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

\item \S52 \#1

\item \S54 \#1

\end{itemize}

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