One of the triumphs of early twentieth century mathematics is the uniformization theorm of complex analysis:
Theorem: The interior of every compact connected 2-dimensional manifold is the quotient of the sphere, the euclidean plane, or the hyperbolic plane by a discrete group of isometries.
Informally, the uniformization theorem says that every compact connected surface has an intrinsic geometry that is spherical, euclidean, or hyperbolic. If we restrict ourselves to closed surfaces (i.e. compact surfaces without boundary) we can group the surfaces into three collections: 1) the sphere, which is the unique closed surface having a spherical geometry; 2) the torus and Klein bottle, which are the only closed surfaces having a Euclidean geometry; and 3) all other closed surfaces, each of which has a hyperbolic geometry. We call the sphere, euclidean plane, and hyperbolic plane the 2-dimensional model spaces.
Are there higher dimensional versions of the uniformization theorem? That is,
In dimension n ≥ 2, can we find a finite collection of model spaces such that the interior of each compact, connected n-dimensional manifold is the quotient of a model space by a discrete group of isometries?
For dimension n ≥ 4, the differences between topological, piecewise-linear, and smooth manifolds make the question, as stated, problematic. Even in dimension n = 3, where the three categories of manifolds coincide, the question is too naive. However, William Thurston proposed:
Geometrization Conjecture: There are eight 3-dimensional model spaces and every compact 3-manifold can be canonically decomposed into 3-dimensional geometrizable submanifolds, each of which is the quotient of a model space by a discrete group of isometries.
Thurston proved his conjecture in the case when M is a so-called “Haken” 3-manifold. The general case was proved in a much celebrated series of preprints by Grigori Perelman in 2006. The affirmative solution to the Geometrization Conjecture for non-Haken manifolds implies the 3-dimensional Poincaré conjecture. (The drama associated with Perelman’s solution of the Geometrization and Poincaré conjectures has been chronicled elsewhere. The hitherto oblivious reader who enjoys salacious gossip can find many articles and books recounting the juicy details. For a more complete introduction to the geometrization conjecture, see Peter Scott’s paper [S].)
It is hard to overestimate the impact of Thurston’s geometrization conjecture on 3-manifold topology. Prior to Thurston’s work, few people dreamed that there was any hope of such a result. It also promised a re-uniting of topology and geometry in dimension 3 and created a number of new research directions in the field.
Although Thurston proved the geometrization conjecture for Haken manifolds, he never wrote down a complete proof. Instead, as he describes in [T2], he focused on training mathematicians in his revolutionary new ways of thinking. The main forum for this education was a substantial collection of widely distributed notes, now available online [T1]. As a result of his efforts, 3-dimensional topology, rather than being killed off as a discipline, is a thriving field of mathematics.
Mathematicians, however, like complete written proofs, so there was a need for someone to write a complete account of Thurston’s proof of the Geometrization Conjecture for Haken manifolds. The meat of Thurston’s proof can be found in two books: Michael Kapovich’s Hyperbolic Manifolds and Discrete Groups and J.P. Otal’s Le théorème d`hyperbolisation pour les vari\’tés fibrées de dimension 3 [O]. The precise theorem that these books (jointly) prove is:
Hyperbolization Theorem: Suppose that M is a compact atoroidal Haken 3-manifold that has zero Euler characteristic. Then the interior of M is the quotient of hyperbolic 3-space by a discrete group of isometries.
The adjective “atoroidal Haken” means that the 3-manifold contains an “essential” surface and that there are no essential spheres, discs, annuli, or tori. (The definition of “essential” is somewhat technical and so we won’t repeat it here.) Haken manifolds have the property that they can be successively cut along essential surfaces until they become a collection of 3-balls. Such a succession of cuts is called a hierarchy of the manifold. The proof of the hyperbolization theorem is by induction on the length of the hierarchy.
Kapovich’s and Otal’s books treat two different cases of the proof of the hyperbolization theorem and so they should be treated as complementary rather than competing. Otal’s book proves the hyperbolization theorem in the case when the first surface cut along in the hierarchy is a so-called “virtual fiber”. Kapovich’s treats the case when the first surface is not a virtual fiber, although he does sketch the proof of the fibered case as well.
The new edition of this book is in the “Modern Birkhäuser Classics” series. This is indeed a classic. MathSciNet shows that 109 papers and books reference it.
Hyperbolic Manifolds and Discrete Groups is an essential text for anyone working in the topology and geometry of 3-manifolds. It is largely self-contained in that it defines all the needed concepts and machinery and often provides proofs of facts that can be found elsewhere in the literature. This book is most valuable for compiling all the needed concepts in one place. This collection is breath-taking in scope; it includes the theories of Kleinian and Fuchsian groups, Teichmüller theory, classical 3-manifold theory, orbifolds, representation varieties, the Rips machine, ending laminations, and more. Most of the book is taken up with introducing these ideas. The actual proof of the hyperbolization theorem (in the non-fibered case) accounts for only 26 pages of the book.
The writing style is exceptionally clear, but terse. Readers encountering some ideas for the first time may want to use this book as a guide to find other, more expansive introductions. The reader is also advised to make continual use of the excellent preface which provides an outline of the proof.
Kapovich’s approach to the hyperbolization theorem differs slightly from Thurston’s original approach by using group actions on trees in a number of places where Thurston used other methods. Thus, geometric group theorists will also find much of interest here.
Thurston’s work has had and will continue to have a lasting impact on mathematics. Kapovich’s book is an excellent, substantial exposition of the varied aspects of the mathematics present.
[O] Otal, Jean-Pierre. “Le théorème d`hyperbolisation pour les variétés fibrées de dimension 3”. Astéisque No. 235 (1996).
[S] Scott, Peter. “The geometries of 3-manifolds”. Bull. London Math. Soc. 15 (1983), no. 5, 401-487.
[T1] Thurston, William P. “The geometry and topology of 3-manifolds”. Available at http://library.msri.org/books/gt3m/.
[T2] Thurston, William P. “On proof and progress in mathematics”. Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 2, 161-177. Available at arXiv:math/9404236.
Scott Taylor is a knot theorist and 3-manifold topologist who would love to spend a lot more time with Kapovich’s book. He is an assistant professor at Colby College.