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Representing 3-manifolds by filling Dehn surfaces
Vigara R., Lozano-Rojo A., World Scientific Publishing Co, Inc., River Edge, NJ, 2016. 350 pp. Type: Book
Date Reviewed: Sep 2 2016

In many applications we encounter manifolds, that is, topological spaces that are locally homeomorphic to the usual Euclidean space. Probably the most well-known example is general relativity, according to which our space-time is a manifold (except for possible singularity points). Less exotic examples come from the analysis of physical systems. In general, the state of a system can be described by listing the values of the corresponding physical quantities. In an idealized description, all possible combinations of these values are possible. However, in real life, there are usually several constraints limiting possible combinations. For example, the orientation of a unit rod AB can be described by the three coordinates x, y, and z of the corresponding vector AB; however, due to the constraint that the length of this vector is one, only tuples (x,y,z) from the surface of a unit sphere are physically possible.

Instead of the usual Cartesian coordinates x, y, and z, we could use, for example, cylindrical or any other coordinates. How can we describe the resulting manifolds? How can we check that two different representations describe the same manifold?

The situation is relatively straightforward in the 2D case, where there is a simple classification of all orientable compact (bounded) 2D manifolds: the simplest is the sphere, the next simplest is a bagel-shaped torus with one hole, then a manifold with two holes, and so on. (Klein bottle--a non-boundary analog of the well-known Mobius strip--is an example of a non-orientable 2D manifold.) For 3D manifolds, there is a similar (very recent) classification, but it is far from being simple and intuitive.

So what is a natural way to describe a general 3D manifold? A traditional way to describe a manifold is via a triangulation, that is, by representing the manifold as a union of simplices, with each simplex sharing bordering triangular faces with its neighbors. A problem with this representation is that the resulting simplicial complex does not provide an intuitively clear description of the manifold: even the 2D sphere can be triangulated in many different ways. There exists an algorithm that checks whether two 3D simplicial complexes represent the same manifold, but this algorithm is also far from being intuitive.

A much more intuitive representation of a 3D manifold M is provided by a “filling Dehn surface” S, an immersion of a 2D surface into M that divides M into appropriate cells, and that has only two types of self-intersections: double points (similar to intersection of surfaces x = 0 and y = 0 in the 3D Euclidean space) and triple points (similar to intersection of surfaces x = 0, y = 0, and z = 0). When the immersed surface is a sphere, the Dehn surface is called the Dehn sphere. It turns out that every orientable 3D manifold has a filling Dehn sphere; this is, by the way, a reasonably recent (2000) result.

The main result of this book is an explanation of how to tell whether two filling Dehn spheres represent the same manifold. It turns out that--in contrast to a more complex comparison of simplicial complexes--equivalent filling spheres can be obtained from each other by a sequence of simple geometric “moves.”

What happens in higher dimensions? Alas, the situation there is much more complicated. Already for dimension four, the problem of checking whether two finite simplicial complexes are homeomorphic is algorithmically undecidable. Interesting, this undecidability was proven in 1958 by A. A. Markov, Jr., the son of A. A. Markov, Sr. who invented Markov chains.

In general, this material is not easy, but the authors try their best to help. They provide all the basic definitions, and plenty of 2D pictures illustrating the corresponding 3D constructions. For example, a 12-page proof of the key lemma is accompanied by 40 pages of pictures. It is tough but possible for a computer scientist to struggle through the text--especially if the reader has some experience in following proofs and a decent geometric intuition. This is a research area in progress, with plenty of challenging open questions--many of which are listed in the book; studying the book’s latest results and ideas will hopefully inspire--and help--readers to solve these questions.

Reviewer:  V. Kreinovich Review #: CR144725 (1611-0779)
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