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\begin{document}
\noindent Name:\\
Date:\\
Group number mod 6:\\
Due: 10/5/07

Homework 5 (Chapter 3 3.32-3.51)\\
Everyone must do the starred problems and you must do your own problems mod 6.
\setcounter{chapter}{3}

\setcounter{section}{2}

\section{Van Kampen's Theorem, I}

\setcounter{thm}{31}
\begin{thm}
Let $X=U\cup V$, where $U$ and $V$ are open and $U\cap V$ is path connected, 
and let $p\in U\cap V$.  Then any element of $\pi_1(X,p)$ has a representative
$\alpha_1\beta_1\cdots\alpha_n\beta_n$, where each $\alpha_i$ is a loop in $U$ based at
$p$ and each $\beta_i$ is a loop in $V$ based at $p$.
\end{thm}

\subsection{Van Kampen's Theorem: simply connected intersection case}

\setcounter{thm}{32}
\begin{thm*}[Van Kampen's Theorem, simply connected intersection case]
\index{Van Kampen's Theorem!simply connected intersection}
Let $X=U\cup V$, where $U,V$ are open, path connected subsets of $X$,
$U\cap V$ is path connected and  simply connected, and $x\in U\cap V$.
Then $\pi_1 (X,x)\cong \pi_1(U,x) * \pi_1(V,x)$.
\end{thm*}

\setcounter{thm}{33}
\begin{cor}*
Let $\infty$ denote the wedge of two circles. Then $\pi_1(\infty) \cong \Z *\Z$.\
\end{cor}

\setcounter{thm}{34}
\begin{question}
Let $X$ be the wedge of $n$ circles. What is $\pi_1(X)$?
\end{question}

\setcounter{thm}{35}
\begin{thm}
If $A$ and $B$ are each connected, then $A\vee B$ is connected. If $A$ and $B$ are each path
connected, then $A\vee B$ is path connected.
\end{thm}

\setcounter{thm}{36}
\begin{thm}
Let $X$ be the wedge of two cones over two Hawaiian earrings, where they are identified at the points
of tangency of the circles of each Hawaiian earring, as
in the figure below. Then $\pi_1 (X) \not\cong 1$.

\begin{figure}[h]
\begin{center}
\includegraphics[width = 3in]{wedgeofconeshawaiian}
\caption{\label{wedgehawaiian} Wedge of cones over Hawaiian earrings }
\end{center}
\end{figure}

\end{thm}

\setcounter{thm}{37}
\begin{question}
State conditions
that suffice to ensure that $\pi_1 (A\vee B)\cong \pi_1 (A) * \pi_1(B)$.
\end{question}

\setcounter{thm}{38}
\begin{thm}
Show that $\pi_1($Hawaiian earring$)$ is not finitely generated, in fact, 
$\pi_1($Hawaiian earring$)$ is not countably generated.]
\end{thm}

\subsection{Van Kampen's Theorem: simply connected pieces case}

\setcounter{thm}{39}
\begin{thm}[Van Kampen's Theorem, simply connected pieces case]
\index{Van Kampen's Theorem!simply connected parts}
Let $X= U\cup V$ where $U$ and $V$ are open, path connected, and simply
connected subsets of $X$ and $U\cap V$ is path connected. Then $X$ is
simply connected.
\end{thm}

\setcounter{thm}{40}
\begin{exercise}{\rule{0cm}{0.1cm}}\newline\vspace*{-0.5cm}
\be
\item[1.] $\pi_1 (\Sph^2) = 1$
\item[2.] $\pi_1 (\Sph^n) = 1$
\ee
\end{exercise}

\setcounter{thm}{41}
\begin{question}
Can you find an example where $U$ and $V$ are simply connected, 
but $X=U\cup V$ is not simply connected?
\end{question}

\section{Fundamental groups of surfaces}

\setcounter{thm}{42}
\begin{exercise}
Describe a strong deformation retract, together with its fundamental group,
of a once-punctured compact, connected, triangulated
$2$-manifold.
\end{exercise}

\setcounter{thm}{43}
\begin{exercise}
Let $M^2$ be a compact, connected, triangulated $2$-manifold, and assume we write $M^2$ as
$U\cup D^2$, where $U$ and $D^2$ are open subspaces of $M^2$, $D^2$ is an open disk,
 $U\cap D^2\simeq A^2$ is 
an open annulus, and $p \in A^2$. Describe the non-trivial elements of $\pi_1(U,p)$ that are trivial in $\pi_1(M^2,p)$. 
\end{exercise}

\setcounter{thm}{44}
\begin{exercise}
State and prove a theorem that 
allows you to calculate $\pi_1(M^2)$ for any compact, connected, triangulated $2$-manifold $M^2$. 
\end{exercise}

\setcounter{thm}{45}
\begin{exercise}*
{\rule{0cm}{0.1cm}}\newline\vspace*{-0.5cm}
\be
\item[1.]Describe a group presentation of $\num_{i=1}^k\T^2_i$.
\item[2.]Describe a group presentations of $\num_{i=1}^k\PP^2_i$.
\ee
\end{exercise}

\setcounter{thm}{46}
\begin{exercise}*
Explicitly determine, using the Classification of Finitely Generated Abelian Groups, what
the abelianizations of the fundamental groups found in the previous exercises are. What, if anything,
distinguishes orientable from non-orientable surfaces? Are any of these abelianized
groups isomorphic? Is this invariant (the abelianized fundamental group) a complete invariant for closed
surfaces---{\em i.e.}, is it sufficient to distinguish between any two surfaces?
\end{exercise}

 \setcounter{thm}{47}
\begin{exercise}
Suppose that $M^2=\T_1 \# \T_2$ where $\T_1$ and $\T_2$ are tori and $M^2=U \cup V $ where $U$ is an open set of $\T_1$ homeomorphic to $T_1-$(a disk),  $V$ is an open set of $\T_2$ homeomorphic to $T_2-$(a disk), and $U \cap V$ is homeomorphic to an open annulus. Let $p \in U \cap V$.  We  know from a previous exercise that $\pi_1(U,p)$ is generated by two loops $\alpha$ and $\beta$. Likewise, $\pi_1(V,p)$ is generated by two loops $\gamma$ and $\delta$. Consider the loop $\mu$ that generates $\pi_1(U\cap V,p)$. Represent $\mu$ in terms of the generators of  $\pi_1(U,p)$. Now represent $\mu$
in terms of the generators of  $\pi_1(V,p)$. So the single loop $\mu$ is equivalent to two different loops in $M^2$. $\pi_1(M^2,p)$ is generated by $\{\alpha,\beta,\gamma,\delta\}$. What relations exist among these generators?  Give a presentation of $\pi_1(M^2,p)$ whose generators are $\{\alpha,\beta,\gamma,\delta\}$.
\end{exercise}


\section{Van Kampen's Theorem, II}

  \setcounter{thm}{48}
\begin{thm*}[Van Kampen's Theorem; group presentations version]
\index{Van Kampen's Theorem!general!group presentations}
Let $X=U\cup V$, where $U,V$ are open and path connected and $U\cap V$
is path connected and non-empty. Let $x\in U\cap V$.

Let
$\pi_1 (U,x)=\langle g_1,\ldots,g_n | r_1, \ldots , r_m \rangle$,
 $\pi_1 (V,x)=\langle h_1,\ldots,h_t | s_1, \ldots , s_u \rangle$ and
$\pi_1 (U\cap V,x)=\langle k_1,\ldots,k_v | t_1, \ldots , t_w \rangle$
then
\begin{eqnarray*}
\pi_1 (X,x) & =  \left\langle g_1,\ldots,g_n , h_1,\ldots,h_t \right.|& r_1, \ldots , r_m, s_1, \ldots , s_u,\\
             &                         & \left. i_*(k_1)=j*(k_1),\ldots , i_*(k_v)=j*(k_v)\right\rangle
\end{eqnarray*}
where $i$, $j$ are the inclusion maps
of  $U\cap V$ into $U$ and $V$ respectively.
\end{thm*}

Without the language of group presentations, Van Kampen's Theorem is stated as follows:

 \setcounter{thm}{49}
\begin{thm*}[Van Kampen's Theorem]
\index{Van Kampen's Theorem!general}
Let $X=U\cup V$ where  $U,V$ are open and path connected and $U\cap V$
is path connected and non-empty. Let $x\in U\cap V$.  Then
\[
\pi_1 (X,x) \cong \frac{\pi_1(U,x) * \pi_1(V,x)}{N}
\]
where $N$ is the
smallest normal subgroup containing
$\{ i_* (\alpha)j_* (\alpha^{-1})
\}_{\alpha\in\pi_1 (U\cap V,x)}$ and $i$, $j$ are the inclusion maps
of $U\cap V$ in $U$ and $V$ respectively. Note that $N$ is the set of products of conjugates of $i_*(\alpha)j_*(\alpha^{-1})$.

\end{thm*}

 \setcounter{thm}{50}
\begin{exercise}*
Use Van Kampen's theorem to explicitly calculate the group presentation of the double torus
$\T^2\num \T^2$.
\end{exercise}


\end{document}