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UF MTG 6346\\
Topology 1\\
Fall 2024
\begin{center}
{\large \bf Homework 2}\\
Due Friday, September 20, anytime, on Canvas
\end{center}

{\bf Reading.}
Hatcher \S1.1--1.2, pages 21--55.


{\bf Problems.}
\begin{enumerate}

\item \S1.1, Exercise 3.
For a path-connected space $X$, show that $\pi_1(X)$ is abelian iff all basepoint-change homomorphisms $\beta_h$ depend only on the endpoints of the path $h$.

\item \S1.1, Exercise 16(a,b,c,f).
Show that there are no retractions $r\colon X\to A$ in the following cases:
\begin{enumerate}
\item $X=\R^3$ with $A$ any subspace homeomorphic to $S^1$.
\item $X=S^1\times D^2$ with $A$ its boundary torus $S^1\times S^1$.
\item $X=S^1\times D^2$ with $A$ the circle shown in the figure.

\includegraphics[width=1.5in]{1-1-16.pdf}
\item[(f)] $X$ the M\"obius band and $A$ its boundary circle.
\end{enumerate}

%\item \S1.2, Exercise 6.
%Suppose a space $Y$ is obtained from a path-connected subspace $X$ by attaching $n$-cells for a fixed $n\ge 3$. Show that the inclusion $X\hookrightarrow Y$ induces an isomorphism on $\pi_1$. [See the proof of Proposition~1.26.] Apply this to show that the complement of a discrete subspace of $\R^n$ is simply-connected if $n\ge 3$.

\item \S1.2, Exercise 8.
\begin{enumerate}
\item ``\emph{Compute the fundamental group of the space obtained from two tori $S^1\times S^1$ by identifying a circle $S^1\times\{x_0\}$ in one torus with the corresponding circle $S^1\times\{x_0\}$ in the other torus.}" Let's call this identification space $X$. Compute $\pi_1(X)$ by using van Kampen's theorem on the natural union of two tori along a circle.
\item Write this identification space $X$ as a product $X=Y\times Z$ (where neither $Y$ nor $Z$ are just a single point), and use this to give an alternate computation of $\pi_1(X)$.
\end{enumerate}

\item \S1.2, Exercise 10.
Consider two arcs $\alpha$ and $\beta$ embedded in $D^2 \times I$ as shown in the figure. The loop $\gamma$ is obviously nullhomotopic in $D^2\times I$, but show that there is no nullhomotopy of $\gamma$ in the complement of $\alpha \cup \beta$.

\includegraphics[width=1.5in]{1-2-10.pdf}

\emph{[Hint: What is the fundamental group of $(D^2\times I)\setminus(\alpha\cup\beta)$?]}

\end{enumerate}

\vfill

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

\item \S1.1, Exercise 5.

\item \S1.1, Exercise 6.

%\item \S1.1, Exercise 8.

\item \S1.1, Exercise 9.

%\item \S1.1, Exercise 10.

%\item \S1.1, Exercise 12.

%\item \S1.1, Exercise 13.

\item \S1.2, Exercise 6.

\item \S1.2, Exercise 9.

\end{itemize}

\end{document}