\magnification 2000
\parskip 10pt
\parindent 0pt
\hoffset -0.3truein
\hsize 7truein

\voffset -0.3truein
\vsize 9.8truein



\def\N{{\bf N}}
\def\S{\Sigma_{i=1}^n}

\def\Z{{\bf Z}}
\def\R{{\bf R}}
\def\C{{\bf C}}
\def\x{{\bf x}}
\def\a{{\bf a}}
\def\y{{\bf y}}
\def\e{{\bf e}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\p{\psi}
\def\f{\phi}
\def\s{\sigma}
\def\ep{\epsilon}
\def\f{\vskip 10pt}
\def\u{ \vskip -5pt }
\def\h{\vskip -5pt \hskip 20pt}


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

$M/G =$


Lens spaces, $L(p, q)$


 $M = S^3 = \{(\x, \y) \in \C ~|~ ||(\x, \y)|| = 1 \}$





m), (x,y)) = 
(n + x, m + y)$.

$M/G =$

\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.









\end

 


 


 

\end