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

{\bf Reading.}
Munkres \S69--72.


{\bf Problems.}
\begin{enumerate}

\item \S68 \#2.
Let $G=G_1 * G_2$, where $G_1$ and $G_2$ are nontrivial groups.

(a) Show $G$ is not abelian.

(b) If $x\in G$, define the \emph{length} of $x$ to be the length of the unique reduced word in the elements of $G_1$ and $G_2$ that represents $x$.
Show that if $x$ has even length (at least 2), then $x$ does not have finite order.
Show that if $x$ has odd length (at least 3), then $x$ is conjugate to an element of shorter length.

(c) Show that the only elements of $G$ that have finite order are the elements of $G_1$ and $G_2$ that have finite order, and their conjugates.

\item Let $(Y,y_0)$ and $(Z,z_0)$ be path-connected pointed spaces such that $Y$ has a contractible neighborhood $N_Y\ni y_0$ and $Z$ has a contractible neighborhood $N_Z\ni z_0$.
Prove that $\pi_1(Y\vee Z)$ is isomorphic to $\pi_1(Y)*\pi_1(Z)$.

[Recall that the \emph{wedge sum} of two pointed spaces $(Y,y_0)$ and $(Z,z_0)$ is the quotient space $(Y\coprod Z)/(y_0\sim z_0)$.]

\item The connected sum of two surfaces $S$ and $S'$ is obtained by cutting out a disk from each surface
and gluing the resulting surfaces along their circle boundaries:
$S \# S' = (S\setminus D^2)\cup_{S^1}(S'\setminus D^2)$.
Let $T$ be the torus.
Using the Seifert-van Kampen theorem on the natural decomposition into two pieces given by the connected sum, calculate $\pi_1(T \# T)$.




\end{enumerate}



\vfill

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

\item \S59 \#2

\item \S60 \#2

%\item \S67 \#4(a,b)

\item \S68 \#3

\item \S70 \#3

\end{itemize}

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