\magnification 2000
\parindent 0pt
\parskip 10pt
\hsize 7.4truein
\vsize 9.7 truein
\hoffset -0.45truein
\def\u{\vskip -15pt}
\def\v{\vskip -6pt}
\input colordvi.tex

A handle body is a 3-manifold homeomorphic to a connected sum of solid tori.

A Heegard splitting (of genus g) of a 3-manifold $M$ consists of a surface $F = \#_1^g T^2$  which separates $M$ into two handlebodies.  I.e. $M = V_1 \cup_f V_2$ where $V_i$ are handlebodies of genus $g$.

Every closed orientable 3-manifold has a Heegard splitting (use triangulation and let $V_1$ be thickened 1-skeleton and $V_2$ corresponds to the dual triangulation). 


If $F$ is an orientable surface in orientable 3-manifold $M$, then $F$ has a collar neighborhood $F \times I \subset M$.  $F$ has two sides.  Can push $F$ (or portion of $F$) in one direction.


$M$ is prime if every separating sphere bounds a ball.

$M$ is irreducible if every sphere bounds a ball.  $M$ irreducible iff $M$ prime or $M \cong S^2 \times S^1$.

A disjoint union of 2-spheres, $S$, is independent if no component of $M - S$ is homeomorphic to a punctured sphere ($S^3$ - disjoint union of balls).

$F$ is properly embedded in $M$ if  $F \cap \partial M = \partial F$.


Two surfaces $F_1$ and $F_2$ are parallel in $M$ if they are disjoint and $M - (F_1 \cup F_2)$ has a component $X$ of the form $\overline{X} = F_1 \times I$ and $\partial \overline{X} = F_1 \cup F_2$.
\eject

A compressing disk for surface $F$ in $M^3$ is a disk $D \subset M$ such that $D \cap F = \partial D$ and $\partial D$ does not bound a disk in $F$ ($\partial D$  is essential in $F$).

\vfill

Defn:  A surface $F^2 \subset M^3$ without $S^2$ or $D^2$ components is incompressible if for each disk $D \subset M$ with $D \cap F = \partial D$, there exists a disk $D' \subset F$ with $\partial D = \partial D'$

\vskip 4truecm
\eject
Lemma:  A closed surface $F$ in a closed 3-manifold with triangulation $T$ can be isotoped so that $F$ is transverse to all simplices of $T$ and for all 3-simplices $\tau$, each component of $F \cap \partial \tau$ is of the form:


\vskip 2.6truecm
Defn:  $F$ is a normal surface with respect to $T$ if

1.)  $F$ is transverse to all simplices of $T$.

2.)  For all 3-simplices $\tau$, each component of $F \cap \partial \tau$ is of the form:

\vskip 2.6truecm
3.)  Each component of $F \cap \tau$ is a disk.

\vfill

Lemma 3.5:  (1.) If $F$ is a disjoint union of independent 2-spheres
then $F$ can be taken to be normal.

 (2.)  If $F$ is a closed  incompressible surface in a closed irreducible 
3-manifold,
then $F$ can be taken to be normal.


Thm 3.6 (Haken)  Let $M$ be a compact irreducible 3-manifold.  If $S$ is a closed incompressible surface in $M$ and no two components of $S$ are parallel, then $S$ has a finite number of components.

\end