William Thurston made many beautiful and fundamental contributions to mathematics. Perhaps best known, in recent years, has been his Geometrization Conjecture (proven by Grigori Perelman). One of Thurston’s important earlier contributions (and one closely related to the Geometrization Conjecture) was a classification of surface diffeomorphisms. Although many of the details are quite technical and require machinery from hyperbolic geometry, measure theory, and dynamical systems, the statement of the classification is both simple and beautiful.

Thurston himself never wrote down the details of his arguments, but he presented many of them in a seminar in Orsay in 1976–1977. Thurston’s lectures were expanded into a collection of “exposés” by Fathi, Laudenbach, and Poénaru and published in 1979 as *Travaux de Thurton sur les surfaces*. It rapidly became a classic and much of the material became standard for mathematicians working in low-dimensional topology and geometry, Teichmüller theory, and Kleinian groups. This new English translation by Kim and Margalit remains faithful to the original while incorporating a few corrections and making a limited number of comments on subsequent developments. The book remains essential reading and hopefully this translation will make Thurston’s ideas even better known outside the research communities already well-acquainted with them.

Before commenting on the text itself, I can’t resist the opportunity to share the essence of Thurston’s classification. For our purposes, we will consider closed (= compact, without boundary), orientable surfaces. Recall that such surfaces are classified by their genus. Let S be such a surface and g(S) its genus and consider a homeomorphism h:S→S. For simplicity, assume also that the homeomorphism is orientation preserving.

If g(S)=0, then S is a sphere and, by classical results from algebraic topology, h has a fixed point and h can be deformed (by “isotopy”) so that it is the identity. This is the classification for the case when S is a sphere (a result which considerably predates Thurston).

When g(S)=1, S is a torus and the homeomorphisms are more interesting. We can build the torus by identifying opposite sides of a parallelogram of, say, unit area. The universal cover of the torus is the plane R2 and the covering map can be used to give the torus a Euclidean geometry. Every homeomorphism of the torus can be deformed so that the images of geodesics (locally length-minimizing paths) are geodesics. A result of Nielsen classifies the homeomorphisms of the torus into those that are reducible, periodic, or *Anosov*. The reducible homeomorphisms are those that (up to deformation) leave a simple closed curve invariant.

For example, cut the torus open along a simple closed curve, twist by 2π, and then reglue. The periodic homeomorphisms are those which have a non-trivial power which can be deformed to the identity map. If the torus is obtained by identifying opposite sides of a square, rotation of the square by π/2 is an example of a periodic homeomorphism. Anosov homeomorphisms stretch the torus by a factor of λ>1 in one direction and shrink it by a factor of 1/λ in the other direction. Anosov homeomorphisms do not leave any simple closed curve invariant but (up to deformation) they do leave invariant two transverse geodesic foliations on the torus. For example, let p:R2→S be the covering map which identifies all points of R2 whose coordinates differ by integers. Let h~:R2→R2 be the linear map h~(x,y)=(1213)(x,y). Let h=p∘h~∘p−1. It is not hard to check that h defines a homeomorphism of the torus. The map h~ stretches R2 by a factor of λ=(2+3√) along the line L1 defined by y=(−1−3√)x. It shrinks R2 by a factor of 1/λ=2−3√ along the line L2 defined by y=(1−3√)x. The covering map p projects the sets of lines parallel to L1 and L2 to transverse foliations on S. Finally, whether a homeomorphism is reducible, periodic, or Anosov can be determined with the trace of the induced isomorphism on the first homology group of the torus.

Thurston’s achievement is to show that Nielsen’s classification of homeomorphisms also applies, with suitable adaptations, to surfaces of higher genus. The homeomorphisms are classified as reducible, periodic, and *pseudo-Anosov*. Reducible and periodic homeomorphisms are defined the same way as for homeomorphisms of the torus. Pseudo-Anosov homeomorphisms, like Anosov homeomorphisms of the torus, stretch and shrink the surface in transverse directions. When g(S)≥2, hyperbolic geometry is used to define geodesics and measure the stretching factor along singular measured foliations.

The book under review thoroughly investigates all three types of homeomorphisms of surfaces and proves Thurston’s classification theorem. In particular, the book carefully explains (singular, measured) foliations, their relationship to surface homeomorphisms, and how they can be used to construct “Thurston’s compactification of Teichmüller space.” Several chapters are devoted to applying the classification inductively by cutting open the surface along a simple closed curve left invariant by a reducible homeomorphism. Later chapters also elucidate some of the dynamics of pseudo-Anosov maps and construct them in different ways. The penultimate chapter gives an account of Thurston’s norm on the first cohomology group of a 3-manifold and its relation to 3-manifolds which are obtained by gluing the boundary components of S×[0,1] together using a pseudo-Anosov homeomorphism S×{0}→S×{1}. The final chapter explains Hatcher and Thurston’s presentation of the mapping class group (i.e. group of homeomorphisms modulo isotopy) of S.

*Thurston’s Work on Surfaces* is very clear and very well-written, although there are many arguments left to the dedicated reader. The prerequisites for the book are a bit of a hodgepodge: certainly some differential topology and geometry is required. Reading [CB] and [FM] simultaneously is also suggested. The first also proves Thurston’s classification theorem, but in a somewhat different and somewhat more linear manner. It, however, does not discuss the connections to Teichmüller space or dynamics. The second is a comprehensive introduction to mapping class groups and gives an expanded explanation of some of the hyperbolic geometry techniques.

In summary, *Thurston’s Work on Surfaces* is a pleasure to study, for it contains beautiful, challenging, and exceedingly useful mathematics developed by one of the great mathematicians of the late twentieth century.

**References:**

[CB] Andrew J. Casson and Steven A. Bleiler, *Automorphisms of Surfaces after Nielsen and Thurston*, Cambridge, 1988.

[FM] Benson Farb and Dan Margalit, *A Primer on Mapping Class Groups*, Princeton, 2012.

Scott Taylor is Assistant Professor of Mathematics at Colby College.