WHAT IS AN ALMOST NORMAL SURFACE? 3

INSTANCE: A triangulated compact 3-dimensional manifold M

QUESTION: Is M simply connected?

The 3-Sphere recognition problem has important consequences. Note for exam-

ple that the problem of deciding whether a given 4-dimensional simplicial complex

has underlying space which is a manifold reduces to verifying that the link of each

vertex is a 3-sphere, and thus to 3-Sphere Recognition.

In dimension two, the corresponding recognition problem is very easy. Deter-

mining if a surface is homeomorphic to a 2-sphere can be solved by computing its

Euler characteristic. In contrast, for dimensions five and higher there is no algo-

rithm to determine if a manifold is homeomorphic to a sphere [25], and the status

of the 4-sphere recognition problem remains open [15]. The related problem of fun-

damental group triviality is not decidable in manifolds of dimension four or higher.

Until Rubinstein’s work, there was no successful approach to the triviality problem

that took advantage of the special nature of 3-manifold groups.

For 3-sphere recognition one needs some computable way to characterize the 3-

sphere. Unfortunately all 3-manifolds have zero Euler characteristic, and no known

easily computed invariant that can distinguish the 3-sphere among manifolds of

dimension three. Approaches developed to characterize spheres in higher dimen-

sions were based on simplifying some description, typically a Morse function. The

simplification process of a Morse function in dimension three, as given by a Hee-

gaard splitting, gets bogged down in complications. Many attempts at 3-sphere

recognition, if successful, imply combinatorial proofs of the Poincare Conjecture.

Such combinatorial proofs have still not been found. A breakthrough occurred in

the Spring of 1992, at a workshop at the Technion in Haifa, Israel. Hyam Rubin-

stein presented a characterization of the 3-sphere that was suitable to algorithmic

analysis. In a series of talks at this workshop he introduced a new algorithm that

takes a triangulated 3-manifold and determines whether it is a 3-sphere. The key

new concept was an almost normal surface.

2. What is an almost normal surface?

Almost normal surfaces, as with their normal relatives, intersect each 3-simplex

in M in a collection of triangles or quadrilaterals, with one exception. In a single

3-simplex the intersection with the almost normal surface contains, in addition to

the usual triangles or quadrilaterals, either an octagon or a pair of normal disks

connected by a tube, as shown in Figure 2. For Rubinstein’s 3-sphere recognition

algorithm, it suﬃces to consider almost normal surfaces that contain an octagon

disk. Later extensions also required the second type of local structure, two normal

disks joined by an unknotted tube, one that is parallel to an edge of the tetrahedron.

Rubinstein argued that an almost normal 2-sphere had to occur in any trian-

gulation of a 3-sphere, and in fact that the search for the presence or absence of

this almost normal 2-sphere could be used to build an algorithm to recognize the

3-sphere. Shortly afterwards, Abigail Thompson combined Rubinstein’s ideas with

techniques from the theory of thin position of knots, and gave an alternate approach

to proving that Rubinstein’s algorithm was valid [24]. The question we address here

is the geometrical background that motivated Rubinstein’s breakthrough.

To describe the ideas from which almost normal surfaces emerged, we take a

diversion into differential geometry and some results in the theory of geodesics and

minimal surfaces. A classical problem asks which surfaces contain closed, embedded