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Defn:  $G$ is a {\it topological group} if

\h 1.) $(G, *)$ is a group

\h 2.)  $G$ is a topological space.

\h 3.)  $*: G \times G \rightarrow G$, $*(g_1, g_2) = g_1*g_2$, and 

\centerline{~~~~~~~~~~~~~$In: G \rightarrow G$, $In(g) = g^{-1}$ are both continuous functions.}


Defn:  $G$ is a {\it Lie group} if

\h 1.)  $G$ is a topological group 

\h 2.)  $G$ is a smooth manifold.

\h 3.)  $*$ and $In$ are smooth functions.


Ex:  $(\R, +)$, $(\R - \{0\}, \cdot)$, $(\C - \{0\}, \cdot)$, $(S^1, \cdot)$ where $S^1 \subset \C$,  $(\Z, +)$, $(\Z_p, +)$, 
$(Gl(n, \R), matrix ~multiplication)$ are Lie groups.  For $G_1, G_2$  
lie groups,  $G_1 \times G_2$ is a lie group. 

Defn:  $G$ = group, $X =$ set.
{\it $G$ acts on $X$} (on the left) if $\exists \s:  G \times X \rightarrow X$ such that

\h 1.)  $\s(e, x) = x ~~\forall x \in X$

\h 2.)  $\s(g_1, \s(g_2, x)) = \s(g_1g_2, x)$
\u
Notation: $\s(g, x) = gx$.  \hfil \break
Thus 1) $ex = x$; 2) $g_1(g_2x) = (g_1g_2)(x)$.
\u
If $G$ is a topological group and $X$ is a topological space, then we require $\s$ to be  continuous.
\u
If $G$ is a Lie group and $X$ is a smooth manifold, then we require $\s$ to be  smooth.


Defn:  The {\it orbit} of $x \in X$ = \hfil \break
$G(x) = \{ y \in X ~|~ \exists g$ such that $y = gx\}$

Note: \hfil \break
 1.)  $x \in G(x)$ \hfil \break
 2.)  If $G(x) \cap G(y) \not= \emptyset$, then $G(x) = G(y)$

Thus we can use an action of $G$ to partition $X$ into disjoint subsets.

Hence the action of $G$ on $X$ can be used to define an equivalence relation on $X$:  $x \sim y$ iff $y \in G(x)$ iff $\exists g$ such that $y = gx$.
\u
\centerline{$X/G = X/\sim$.}

If $X$ is a topological space,  then $X/G = X/\sim$ is a topological space with the quotient topology.
\u

When is $X/G = X/\sim$ a manifold?



Ex:  $G= ({\bf Z}, +)$, $M = \R$, $\s(n, x) = n + x$.

$M/G =$

Ex:  $G= ({\bf Z} \times {\bf Z}, +)$, $M = \R^2$, \hfil \break $\s((n, 
m), (x,y)) = 
(n + x, m + y)$.

$M/G =$

Ex:  $G= ({\bf Z_2}, +)$, $M = S^n$, $\s(0, x) = x$, $\s(1, x) = -x$, .

$M/G =$
\eject
Defn:  The action of $G$ on $X$ is {\it free } if $gx = x$ implies $g = e$.

Thm 1.3.9:  If $M$ is a smooth $n$-manifold, and $G$ is a finite Lie group acting freely on $M$, then $M/G$ is a smooth $n$-manifold.  Also, $p:  M \rightarrow M/G$ is smooth.

Cor:

Defn:  $G$ is a {\it discrete group} if 

\h 0.)  $G$ is a group.

\h 1.)  $G$ is countable

\h 2.)  $G$ has the discrete topology


Note a discrete group is a Lie group.


Defn:  The action of $G$ on $M$ is {\it properly discontinuous} if $\forall x \in M$, $\exists U^{open}$ such that $x \in U$ and $U \cap gU = \emptyset ~~\forall g \in G$.

Ex:  $({\bf Z}, +)$ acting on $\R^1$ where $\s(n, x) = n + x$.

Thm 1.3.2:  $M$ smooth $n$-manifold, $G$ discrete group acting properly discontinuously on $M$ implies $M/G$ is a smooth $n$-manifold.  Also, $p:  M \rightarrow M/G$ is smooth.









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