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\title{Topology} 
\author{Hamilton College Topology Class, '05}
\begin{document} 

\setcounter{section}{5}
\section{Manifolds, Group Presentations}
Here we look at a special class of spaces that share many properties of Euclidean spaces.  We also introduce some useful algebraic notation.

\begin{defn}
Let $\E^n_+ = \{(x_1, \cdots ,x_n) \in \E^n | x_n \geq 0\}$.
\end{defn}

\begin{defn}
A Hausdorff space $X$ is an \emph{$n$-manifold} $\Leftrightarrow \forall x \in X,  \exists U_x$ open in $X$ containing $x$ and a homeomorphism $f_x: \E^n \rightarrow U_x$ or $\E^n_+ \rightarrow U_x$ such that $f_x(\mathbf{0})  =  x$.  The set of points for which $f_x : \E^n \rightarrow U_x$ is called the \emph{interior} of $X$ and is denoted int($X$). The set of $x$ for which $f_x : \E^n_+ \rightarrow U_x$ we call the \emph{boundary} of $X$ and denote it by bnd($X$).  A 2-manifold is usually called a \emph{surface}.
\end{defn}

\begin{thm}
If $X$ is a surface then int($X$) $\cap$  bnd($X$) = $\emptyset$ .  
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Prove \dots let $x \in  \text{int}(X) \cap \text{bnd}(X)$, there exist two open sets $U_x$ and $V_x$ and  homeomorphisms $f_x, g_x$  \dots think about something like $h_x=g^{-1}_x \circ f_x$ which doesn't  quite make sense but .... think about what this implies about $E^2  \setminus \{0\}$  and $E^2_+ \setminus \{0\}$ \dots
\end{xbox}
\end{proof}

\begin{thm}
Let $X$ and $Y$ be surfaces such that $X \approx Y$.  Then int($X$) $\approx$  int ($Y$) and 
bnd($X$) $\approx$  bnd($Y$).   
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Prove.
\end{xbox}
\end{proof}

\begin{ex}\ \\
\begin{xbox}
Does the theorem above help with classifying any of our spaces?
\end{xbox}
\end{ex}

\begin{defn}
Given a finite set $X$, we define $[X]$ as follows:
\begin{itemize}
\item The elements of $X$ are called generators.
\item A word in $X$ is a finite product $x_1^{n_1} x_2^{n_2} \cdots x_n^{n_n}$ where each $x_i$ is in $X$ and each $n_i$ is an integer.  
\item A word is reduced if $x_i$ never equals $x_{i+1}$ and all the $n_i$ are non-zero.
\item Given any word we can reduce it by collecting powers of adjacent elements and replacing the word $x_i^0$ with the empty word $\emptyset$. 
\item $[X] = \{ \text {All reduced words in }X\}$.  
\item We multiply elements of $[X]$ by simply writing them next to each other and reducing the resulting word. 
\end{itemize}
\end{defn}

\begin{thm}
$[X]$ is a group.   
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Prove.
\end{xbox}
\end{proof}

\begin{defn}
We call $[X]$ the  \emph{free group generated by $X$}.
\end{defn}
\begin{rem}
$[x_1,x_2 \cdots x_n] \cong [y_1,y_2 \cdots y_n]$.  \
\end{rem}

\begin{defn}
A \emph{group with relators $[X | R]$} is defined in an analogous way to a free group:
\begin{itemize}
\item Start with a set of generators $X =\{x_1,x_2, \cdots x_n\}$.
\item Let $R = \{r_1,r_2, \cdots r_m\}$ be a set of words in $[X]$.  These are called \emph{relators}.
\item Elements of $[X | R]$ are again reduced words in $X$ except that when a relator $r_i$ appears in a word we can set $r_i  = \emptyset$ or we can insert $r_i$ at any point.
\item Note that if $x \in X$ and $r = x^3 \in R$ we can replace $x^3$ by $\emptyset$  or $x^2$ by $x^{-1}$. 
\item The symbol $[x_1,x_2, \cdots x_n | r_1,r_2, \cdots r_m]$ is called a \emph{presentation for the group}. 
\item Note that if $R$ is empty then the group is just the free group $[X]$.
\end{itemize}
\end{defn}

\begin{ex}
$[x], [x | x^3], [x,y | x], [x,y], \text { and } [x,y | xyx^{-1}y^{-1}]$ are all presentations for (somewhat) familiar groups.\\ 
\begin{xbox}
Identify these groups.
\end{xbox}
\end{ex}

\begin{ex}\ \\ 
\begin{xbox}
Give presentations for the fundamental groups of each of the following spaces.  In each case draw a picture of the space and on the picture draw the image of a representative loop from the equivalence class of each generator.  $\S^1, \S^2, \text{ Mobius Band }, \S^1 \times [0,1], \E^2 \setminus \{0\}, \text{ and } \S^1 \times \S^1$.
\end{xbox}
\end{ex}

\begin{defn}
If $G= [X | R]$ and $H=[Y | S]$ are two groups then the \emph{free product of $G$ and $H$}, denoted by $G \star H =[X \uplus Y | R \uplus S]$.  In the case where $X$ and $Y$ contain some of the same symbols, we usually rename them first so that $X \uplus Y = X \cup Y$.
\end{defn}

\begin{thm}
If $X$ is a space with $X = A \cup B$ where $A \cap B = \{x\}$  then $\Pi_1(X,x) \cong \Pi_1(A,x) \star \Pi_1(B,x)$.
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Prove.  Actually an intuitive argument is OK here
\end{xbox}
\end{proof}

\begin{ex}\ \\ 
\begin{xbox}
Give presentations for the fundamental groups of each of the following spaces.  In each case draw a picture of the space and on the picture draw the image of a representative loop from the equivalence class of each generator.  A figure eight,  $\B^2 \setminus \{2 \text{ copies of }\B^2\}$,  and $\S^1 \times \S^1 \setminus \{1 \text { copy of } \B^2\}$ .
\end{xbox}
\end{ex}

\begin{ex}Recall that if $A \subset X$ and $i : A \rightarrow X$ is the inclusion map $i(a) = a,  \forall a \in  A$  then $\hat{i}:\Pi_1(A) \rightarrow  \Pi_1(X)$ is a group homomorphism.\\
\begin{xbox}
For each of the following pairs $X,A$ 
\begin{itemize}
\item Find a presentation of $\Pi_1(A)$ and $\Pi_1(X)$. Hint:  If the two groups are the same, use different names for the generators.
\item Draw a picture of the image under $i$ of a loop from each generating class in $\Pi_1(A)$.
\item Write the image under $\hat{i}$ of each generator of $\Pi_1(A)$ in terms of the generators of $\Pi_1(X)$.
\end{itemize}
\begin{center}
\begin{tabular}{|c|c|}\hline
$X$ &	$A$\\   \hline\hline
$\S^2$	&The equator\\
$\E^2 \setminus \{0\}$ & 	$\S^1$\\
$\B^2$	& The boundary circle\\
$\S^1 \times [0,1]$ &	One of the boundary circles\\
Mobius Band &	The boundary circle\\
$(\S^1 \times \S^1)\setminus \B^2$ & The boundary circle\\ \hline
\end{tabular}
\end{center}
\end{xbox}
\end{ex}

\begin{thm}
(Van Kampen) Let $X = A \cup B$, $A$,$B$ open in $X$, $A \cap B$ be path connected, and   
$i : A \cap B \rightarrow A$, and   $j : A \cap B \rightarrow B$ be the inclusion maps.  Then $\Pi_1 (X) \cong \Pi_1 (A) \star \Pi_1 (B)$ with the additional relations that $\hat{i} (\langle \alpha \rangle) = \hat{j} (\langle \alpha \rangle)$ for all $\langle \alpha \rangle$ generators of $\Pi_1 (A \cap B)$. 
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Prove.  Give an intuitive argument for the proof of this.
\end{xbox}
\end{proof}

\begin{ex}\ \\ 
\begin{xbox}
Use the Van Kampen Theorem to compute the fundamental group for each of the following spaces:  Torus, Projective Plane, Klein Bottle, Double Torus.
\end{xbox}
\end{ex}

\begin{ex}\ \\ 
\begin{xbox}
Do we have any pairs of spaces left that we can't distinguish?
\end{xbox}
\end{ex}




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