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
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