Essential Laminations and Haken Normal Form

0. Introduction. The notion of (Haken) normal form w.r.t. a triangulation of a 3-manifold traces back to Kneser's work in the 1930's on surfaces in 3-manifolds. Haken studied it extensively in the 1960's, and showed [8] how to use it to create nite algorithms for the determination of various properties of embedded surfaces. This has since culminated, in the work of Jaco and Oertel [10], in an algorithm to determine if an irreducible 3-manifold is a Haken manifold, i.e., if it contains a 2-sided incompressible surface. In [7] a generalization of the incompressible surface, the essential lamination, was introduced. There it was shown that a 3-manifold M containing an essential lamination has some of the same desirable properties of a 3-manifold containing an incompressible surface, the most notable property being that M has universal cover R. Since then, it has also been shown [6] that, in some sense, `most' 3-manifolds contain essential laminations. The purpose of this paper is to prove a Haken normal form result for essential laminations. The reader is referred to [7] for de nitions and basic properties concerning essential laminations. In this paper the word `lamination' will mean a lamination which is carried by a branched surface, i.e., it has `air' between its leaves. Since we will ultimately be interested only in the existence of an essential lamination with certain properties, this additional restriction will cause no di culties; we can `blow air' between leaves the leaves of a foliation (see [7]) to obtain a lamination in our sense. Generalizing the de nition for a compact surface [8], we will say that a lamination L M is in Haken normal form w.r.t. a triangulation of M if L is in general position w.r.t. , and for every 3-simplex of , L\ 3 is a lamination consisting of compact disks, each of which meets the 1-skeleton