The Haken Method

Abstract

Recall that a closed surface F in a three-dimensional manifold M is normal relative to a given decomposition ß of the manifold M into handles if

• 1) it does not intersect handles of index 3;

• 2) it intersects each plate D ² × I (a handle of index 2) along several parallel copies of the disc D ² × {*};

• 3) its intersection with each cylinder D ² × I (a handle of index 1) has the form L×I, where L is a finite set of arcs in D ² with ends on ∂D ²;

• 4) its intersection with each ball (a handle of index 0) consists of discs called elementary;

• 5) its intersection of each elementary disc with each bridge is either empty or consists of exactly one segment (see section 7.3).