Contemporary Mathematics

Volume 597, 2013

http://dx.doi.org/10.1090/conm/597/11777

What is an Almost Normal Surface?

Joel Hass

This paper is dedicated to Hyam Rubinstein on the occasion of his 60th birthday.

Abstract. A major breakthrough in the theory of topological algorithms oc-

curred in 1992 when Hyam Rubinstein introduced the idea of an almost nor-

mal 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 [23].

Normal surfaces were introduced by Kneser as a tool to describe and enumer-

ate surfaces in a triangulated 3-manifold [13]. While a general surface inside a

3-dimensional manifold M can be floppy, and have fingers and filligrees that wan-

der 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

2010 Mathematics Subject Classification. Primary 57N10; Secondary 53A10.

Key words and phrases. Almost normal surface, minimal surface, 3-sphere recognition.

Partially supported by NSF grant IIS 1117663.

c 2013 American Mathematical Society

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