What is an Almost Normal Surface?

A major breakthrough in the theory of topological algorithms occurred in 1992 when Hyam Rubinstein introduced the idea of an almost normal surface. We explain how almost normal surfaces emerged naturally from the study of geodesics and minimal surfaces. Patterns of stable and unstable geodesics can be used to characterize the 2-sphere among surfaces, and similar patterns of normal and almost normal surfaces led Rubinstein to an algorithm for recognizing the 3-sphere. 1. Normal Surfaces and Algorithms There is a long history of interaction between low-dimensional topology and the theory of algorithms. In 1910 Dehn posed the problem of finding an algorithm to recognize the unknot [3]. Dehn’s approach was to check whether the fundamental group of the complement of the knot, for which a finite presentation can easily be computed, is infinite cyclic. This led Dehn to pose some of the first decision problems in group theory, including asking for an algorithm to decide if a finitely presented group is infinite cyclic. It was shown about fifty years later that general group theory decision problems of this type are not decidable [22]. Normal surfaces were introduced by Kneser as a tool to describe and enumerate surfaces in a triangulated 3-manifold [12]. While a general surface inside a 3-dimensional manifold M can be floppy, and have fingers and filligrees that wander around the manifold, the structure of a normal surface is locally restricted. When viewed from within a single tetrahedron, normal surfaces look much like flat planes. As with flat planes, they cross tetrahedra in collections of triangles and quadrilaterals. Each tetrahedron has seven types of elementary disks of this type; four types of triangles and three types of quadrilaterals. The whole manifold has 7t elementary disk types, where t is the number of 3-simplices in a triangulation. Kneser realized that the local rigidity of normal surfaces leads to finiteness results, and through them to the Prime Decomposition Theorem for a 3-manifold. This theorem states that a 3-manifold can be cut open along finitely many 2-spheres into pieces that are irreducible, after which the manifold cannot be cut further in a non-trivial way. The idea behind this theorem is intuitively quite simple: if a very large number of disjoint surfaces are all uniformly flat, then some pair of the surfaces must be parallel. Date: July 1, 2012. 1991 Mathematics Subject Classification. Primary 57N10; Secondary 53A10.