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UF MTG 4303/5317\\
Introduction to Topology 2\\
Spring 2024
\begin{center}
{\large \bf Homework 1}\\
Due Wednesday, January 24, anytime, on Canvas
\end{center}

{\bf Reading.}
Munkres \S51--53.


{\bf Problems.}
\begin{enumerate}

\item
Let $X$ be a topological space.
Let $f$ be a loop in $X$ based at $x_0\in X$.
Show that $f * e_{x_0} \simeq_p f$ by \emph{explicitly} writing down a path homotopy $H\colon I\times I \to X$ between $f * e_{x_0}$ and $f$.

\item \S51 \#1.
Show that if $h,h'\colon X\to Y$ are homotopic and $k,k'\colon Y\to Z$ are homotopic, then $k\circ h$ and $k'\circ h'$ are homotopic.

\item \S52 \#4.
Let $A\subset X$; suppose $r\colon X\to A$ is a continuous map such that $r(a)=a$ for each $a\in A$.
(The map $r$ is called a \emph{retraction} of $X$ onto $A$.)
If $a_0\in A$, show that $r_*\colon \pi_1(X,a_0) \to \pi_1(A,a_0)$ is surjective.

%\emph{Remark: This problem has appeared on the first year exam before; see for example (\footnote{\url{https://gma.math.ufl.edu/wp-content/uploads/sites/130/FY-Topology-1-2022_05.pdf}}).}




\end{enumerate}

%\vfill

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

%\item \S32 \#2.

%\item \S43 \#5.

%\item \S44 \#2.

%\item \S48 \#11.

%\end{itemize}

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