The Collected Works of William P. Thurston with Commentary

Bill Thurston (1946–2012) was one of the giants of twentieth century mathematics. His best-known work, and most likely the one that had the most profound impact on many branches of mathematics, is centered on his groundbreaking results and insights on what is now known as the Thurston Geometrization Program for $3$-dimensional manifolds. However, this was only a part of his oeuvre, which consistently displayed great breadth and breathtaking originality. In particular, he had a very distinctive way of exploring areas of mathematics through original points of view and exporting these insights to other subjects. The four-volume set of his collected works, published by the AMS, brings together all of Thurston’s writings, many of which were never formally published as books or journal articles. They go from his 1967 senior thesis as an undergraduate at New College to the mathematics that he was still developing at the time of his untimely passing. In addition to an overview of the prodigious scientific output of a great mathematician, these collected works offer a unique glimpse into the originality of how he approached a broad range of problems.

Focusing on $3$-dimensional topology, it is hard to overstate how much, and how suddenly, Thurston’s work changed the field. Up to the mid-1970s, there were essentially two approaches to this topic: one of them used the tools of algebraic topology to show that certain properties could not hold; the other approach involved cutting $3$-dimensional manifolds along surfaces, and deforming these surfaces through cut-and-paste operations to prove uniqueness results. In particular, prior to Thurston’s work, the objects considered were always very flexible (by definition of topology, one could say), and the field was rather self-contained, with very little interaction with nearby areas of mathematics, such as differential geometry. Then, Thurston came along. He was already famous for his fundamental work on foliations, but it was not clear that the topological community was ready to fully embrace results involving rigid geometry. The real eye-opener actually came from a collective work, the proof of the Smith Conjecture MB84. This long-standing conjecture stated that every periodic diffeomorphism of the $3$-dimensional sphere $S^3$ whose fixed point set is 1-dimensional is conjugate to a rotation. The proof was put together by a group of mathematicians with very different backgrounds who combined: Thurston’s construction of hyperbolic metrics on $3$-dimensional manifolds, as a fundamental building block; the recent Bass–Serre theory on the structure of finitely generated groups of 2-by-2 matrices; equally recent existence results of Meeks–Yau for minimal surfaces in $3$-dimensional riemannian manifolds; and more classical techniques of $3$-dimensional topology. After this revolutionary success, everybody had to learn about homogeneous riemannian manifolds, and in particular this weird non-euclidean geometry called hyperbolic geometry. I personally remember my own puzzlement (and skepticism) when, as a graduate student who already had obtained a few results in the classical approach to knot theory, my advisor tried to explain to me the definition of the hyperbolic space.

There followed an explosion of results and, above all, there was a period of intense cross-fertilization between various branches of mathematics⁠¹, much of it spearheaded by Thurston. His approach to $3$-dimensional hyperbolic geometry completely rejuvenated the topic of kleinian groups, moving it from complex analysis to geometry and essentially creating a new field of its own. Conversely, this geometric point of view provided new insights on complex analytic constructions. For instance, Sullivan’s dictionary exploited the dynamical analogies between the action of a kleinian group (coming from a $3$-dimensional hyperbolic manifold) on the Riemann sphere and the action of a holomorphic map on the same Riemann sphere. Thurston himself relied on this analogy with a famous result on the dynamics of rational maps. In fact, dynamical systems were never very far from his work, as a source of tools and inspiration as well as a recipient of geometric insights. The Hyperbolization Theorems provided many examples of negatively curved manifolds; combined with the theory of negatively curved groups that was developed by Gromov at about the same time, this resulted in a fresh impetus for geometric group theory. Thurston’s work also had the effect of bringing topology and differential geometry much closer to each other. Although the methods are very different, the novel Ricci flow techniques developed by Perelman to complete the Thurston Geometrization Program can be partially credited to this expanded interaction between the two fields. Forty years after it started and ten years after Thurston’s passing, this extensive collaboration between different areas of mathematics can still be felt today, for instance with the current trend expanding Thurston’s viewpoints to higher rank Lie groups.

¹

I cannot help mentioning Bon15 for a general audience discussion of Thurston’s work in $3$-dimensional geometry, and its applications to knot theory and low-dimensional topology. There are of course many more such expositions.

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Beyond the impact of his theorems, there was a very characteristic Thurston style of doing mathematics. He always approached a subject with his own point of view, usually relying on his phenomenal geometric intuition. Not only did he use this original approach for himself, but he went to great lengths in his writing and personal communications to communicate these insights to his audience. I myself often relied on what I called the WWTD Principle, standing for “What Would Thurston Do?”, as modeled on the WWJD motto “What Would Jesus Do?” that was popular in certain circles in the early 1990s. When stuck on a problem, I would step back from my original and unsuccessful approach and then invoke the WWTD Principle. This could for instance involve working out an example with very handwavy arguments, or rephrasing the steps of a formal argument in more intuitive terms in order to really understand what made them work, or viewing what originally seemed to be a purely algebraic or analytic argument in a more geometric way, etc.. Some of my best work originated from a very deliberate application of this principle. However, and regrettably, imitation never enabled me to reach the level of creativity of the master.

Famously, Thurston never published a complete exposition of the proof of his Hyperbolization Theorem, as well as of many of his other results. He felt that by writing up his fundamental results on foliations in detail, he had effectively “killed” this subject; he wanted to avoid this in his subsequent work, by leaving space for others to reinterpret his ideas and expand the scope of the field. It is also possible that he was consistently sidetracked by his bubbling creativity, and distracted from the tedious task of formal exposition by whatever exciting new project was occupying his mind. It is certainly a fact that he was quite generous in sharing his insights in preprints, lectures, and informal conversations. In particular, this enabled others to write up his results in a rigorous form; also, his work with co-authors usually appeared in print. In those pre-internet days his preprints and notes on his lectures taken by audience members circulated hand-to-hand and country-to-country akin to the samizdat of the Soviet world. My personal dog-eared copies usually bear the name of the person who lent me their own copy, and wanted to make sure that I would return it after photocopying it. In their often unpolished form, these preprints were actually way more profitable to the reader than an elegant exposition designed to better fit the standards of a journal article. They required a lot more work and commitment from the reader, but in the end one would gain a much better understanding of the subject and of Thurston’s very original insights. Several of these unpublished preprints were eventually TeXed, and made accessible on the arXiv preprint server, and it is now very nice to have them easily accessible in print.

Which brings me to the current edition of Thurston’s collected works. This massive four-volume set, with over 2300 pages, is very different from other collected works. Those tend to feel like a mausoleum, where one can admire the greatness of a mathematician through a body of work that has already appeared elsewhere. This collection includes a lot of unpublished material, often in relatively unpolished form. The editors have grouped these articles by themes. A very nice feature is that each section begins with a few introductory pages on its theme, providing great perspective as well as occasional historic information.

Volume I is split into three parts, each remarkably impressive. The first part covers Thurston’s many contributions to the theory of foliations, from his 1972 PhD thesis to an unfinished 1998 preprint. Among these results, one can single out the following highlights: his fundamental existence results from the 1970s; the Thurston norm on the homology of a $3$-dimensional manifold, characterizing which homology classes can be realized by fibrations over the circle, as well as proving a famous norm-minimizing property for compact leaves of taut foliations; his collaboration with Eliashberg introducing the notion of confoliation of a $3$-dimensional manifold, thereby bridging tight contact structures and taut foliations; his later preprints on slithering and the universal circle for taut foliations of $3$-dimensional manifolds.

The second part of Volume I is equally groundbreaking. It is devoted to Thurston’s work on surfaces. Its most influential component is also the shortest and consists of a very brief discussion of the Thurston compactification of the Teichmüller space of a surface by the space of projective measured foliations (or, equivalently, projective measured laminations), and its application to the Nielsen–Thurston classification of surface diffeomorphisms; full expositions of this work were subsequently written by others, in several books. An article with Kerckhoff shows how this Thurston compactification is more natural than the earlier Bers (complex analytic) compactification, in the sense that the action of the mapping class group on the Teichmüller space continuously extends to the Thurston boundary but not to the Bers boundary. This section also includes the construction of earthquakes (and the Earthquake Theorem) in the Teichmüller space, and the Hatcher–Thurston presentation for the mapping class group of a surface. It concludes with the very rich unpublished preprint on stretch maps.

The last part of Volume I covers various subjects under the heading “Differential Geometry”. This includes the landmark article with Gromov where, in dimension at least $4$, they exhibit negatively curved riemannian manifolds whose sectional curvature cannot be pinched between two given constants, as well as negatively curved manifolds whose curvature is almost constant but do not admit any riemannian metric with constant curvature. Another notable item is Thurston’s beautiful article on spaces of polyhedra, where his trademark originality is in full display. One can also marvel at the two-page 1976 paper where he disproved the long-standing conjecture that every symplectic manifold admits a Kähler metric.

The bulk of Volume II is devoted to Thurston’s articles on $3$-dimensional hyperbolic geometry and topology. It begins with Thurston’s published announcement of his Geometrization Theorems (and conjecture) for $3$-dimensional manifolds, followed by the preprints that he circulated on the proof of these results, only one of which has appeared in print before. The commentary written by the editors here is particularly useful. These solo articles/preprints are followed by articles with various co-authors. A particularly innovative one (with Dunfield) involves the consideration of random $3$-dimensional manifolds.

The second theme of Volume II addresses complexity questions for algorithms involving problems in low-dimensional topology, a topic pioneered by Thurston and his collaborators. This volume concludes with articles on geometric group theory. These again have a very strong algorithmic flavor, in particular with the development of the notion of automatic groups.

The first half of Volume III focuses on Thurston’s writings in dynamics and complex analysis. It begins with his celebrated paper with Milnor on the iteration of piecewise monotonic maps of the interval, with a follow-up on the same topic written thirty years later in the last year of his life (and partially completed by others). The other landmark article in this section is his unpublished preprint on the iteration of complex rational maps. This is another area that he revisited at the end of his life, and his latest results appear in an article completed in part by attendees of his Cornell seminar.

Thurston was always interested in computational issues. In particular, he used insights and methods from geometry to devise algorithms and evaluate their complexity. These computer science contributions form the second part of Volume III. The volume concludes with several articles targeted at a broader audience. One article, written in answer to a polemic started by an earlier article JQ93, offers a remarkably personal reflection on different styles of developing, proving, and communicating mathematics.

Volume IV consists of the celebrated Princeton Lecture Notes. These notes, based on lectures by Thurston in his 1977–79 course and mostly written by Steve Kerckhoff and Bill Floyd, have been extremely influential. Copies circulated very quickly around the world, and they have contributed to the geometric education of a whole generation. The introduction by Kerckhoff offers a very interesting historical perspective, in addition to a quick presentation of the content of each chapter.

This 4-volume set is a fantastic resource. It provides complete access to material that was not as easily available before. For instance, I personally believed that I had at least a general acquaintance with most of Thurston’s work, but after reading his PhD thesis here I realized that its topic was not the one I had thought. There are a few things missing from these collected works for technical reasons. A major one is the book Thu97, issued from a serious editing and expanding of material taken from the original lecture notes, appearing here as Volume IV. Additional missing pieces are, for instance, the electronic resources that were developed with great input from Thurston, such as the software SnapPea/SnapPy CDGW widely used by topologists and hyperbolic geometers, some of the geometry games Wee developed by Jeff Weeks, or the popular videos Not Knot GM91 and Outside In LMM95 developed by the Geometry Center and originally distributed as videotapes, and now available on YouTube. What is not missing, however, is Thurston’s unique style and approach to mathematics. There is much to learn here. One can also be impressed by the amazing breadth of this collection, and by the prodigious number of great results that it includes. It is a great tribute to the genius of one of the greatest mathematicians of all time, as well as a great resource for today’s and tomorrow’s mathematicians.