C These are exactly the cases where the surfaces have abelian fundamental group. These are also exactly the cases where the surfaces have a positive-dimensional automorphism group. The complex upper half plane T1,0,0 indexing tori is the second example. Closed loops on these surfaces come in one of four types, according to how they behave under tightening. Trivial loops can be tightened to any point on the surface.
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C These are exactly the cases where the surfaces have abelian fundamental group. These are also exactly the cases where the surfaces have a positive-dimensional automorphism group. The complex upper half plane T1,0,0 indexing tori is the second example.
Closed loops on these surfaces come in one of four types, according to how they behave under tightening. Trivial loops can be tightened to any point on the surface. Puncture loops can be made arbitrarily short about a unique puncture.
Hole loops can be tightened arbitrarily close to the circumference of a unique hole. Generic loops can be tightened to a unique geodesic. There are respectively 1, m, n, and infinitely many homotopy classes of trivial, puncture, hole, and generic loops.
There is a length associated to each homotopy class, zero in the first two cases and positive in the last two cases. Here a marking is certain extra discrete data which gives a label to each homotopy class of closed loops in a natural way.
Intuitively, as one moves on a path through Tg,m,n the corresponding surface is changing its geometry but maintaining a completely fixed topology. By the theory of quasiconformal mappings, any marked Riemann surface can be canonically continuously deformed into any other marked Riemann surface of the same numerical type. Consequently, each Tg,m,n is a contractible metric space with any two points connected by a unique geodesic. Drawing from these natural functions, one can coordinatize Tg,m,n in many ways.
Other approaches to rendering Tg,m,n more explicit highlight other aspects of Tg,m,n. The Mg,m,n are also much more complicated than the Tg,m,n both locally and globally, as they have singularities and non-trivial cohomology groups. However, the spaces Mg,m,0, unlike their covers Tg,m,0, are naturally complex algebraic varieties, and very important ones at that. These moduli spaces Mg,m,0 are often studied by purely algebraic methods.
If we all wrote mathematics in a way which actually took into account how humans understand mathematics, then the task of readers trying to make sense of the literature would be much easier. In his typical way, Thurston advances this point very strongly, writing about "how ineffective and denatured the standard definition theorem proof n remark m style is for communicating mathematics.
He writes, "In mathematics, what is intriguing, puzzling, interesting, surprising, boring, tedious, exciting is crucial," since these attitudes actually shape our cognitive understanding. Good mathematical writing should include some direct appeals to our "spatial and visual senses," as well as the usual appeals to the "linguistic, symbol-handling areas" of our brains. Thurston writes that "John Hubbard approaches mathematics with his whole mind," and indeed he does. Definitions, theorems, proofs, and remarks are embedded in a coherent narrative.
Geometry is made visual whenever possible. The extra narration serves to keep the reader oriented. For example, Hubbard introduces quasiconformal maps on page by explaining that it took him a long time to get used to their paradoxical nature: they are smooth enough for some of calculus to hold but too rough for other parts to hold.
Intuitive preliminaries like this one help readers interpret the rigorous material which follows. The geometrical support comes at all levels. We likewise learn that hyperbolic trousers fit us better than Euclidean trousers Figure 3.
Many figures, such as the cover figure, capture central notions in ways that humans naturally understand them; the corresponding text makes sense only after one has internalized the picture. He aims to be self-contained and appeal to as many readers as possible. Accordingly, there is a two page notation summary, a thirteen page basic glossary, and a fourteen page index. The glossary is particularly handy, with seventy-two definition-based entries, starting with "act freely" and ending with "upper semi-continuous.
These background advanced topics are indeed best isolated in an appendix; the topics are very wide-ranging, from Dehn twists, to holomorphic functions on Banach manifolds, to Serre duality. It should be emphasized that keeping the reader in mind does not at all mean skirting difficult points.
This self-containment at times becomes quite demanding of the reader. For example, much of the text concerns arbitrary Riemann surfaces, not just those of finite topological type, and this necessarily complicates the presentation. He says they are excellent and recommends them highly. The volume under review is already a highly competitive newcomer to the list. It will become even more attractive when its sequel volume appears.
David Roberts is an associate professor of mathematics at the University of Minnesota, Morris.
Volumes 2 through 4 prove four theorems by William Thurston: 1. The classification of homeo-morphisms of surfaces 2. The topological characterization of rational maps 3. The hyperbolization theorem for 3-manifolds that fiber over the circle 4. The hyperbolization theorem for Haken 3-manifolds These theorems are of extraordinary beauty in themselves, and the methods Thurston used to prove them were so novel and displayed such amazing geometric insight that to this day they have barely entered the accepted methods of mathematicians in the field. The results sound more or less unrelated, but they are linked by a common thread: each one goes from topology to geometry.
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