Without question, low dimensional topology is among the most popular areas of mathematics these days. This is altogether reasonable on several counts, including the fact that it resonates with the world of our ordinary experience (at least to some extent: one doesn’t usually encounter the complement of the trefoil knot on the way to the mall), that it allows wonderful pictures and hence attracts all kinds of spectators, and that a very prominent open problem was recently settled by a marvelously eccentric Russian amidst all forms of controversy involving, for example, a circle of Chinese mathematicians centered on an internationally active Fields Medalist (S.-T. Yao). And the fact that our Russian hero, Misha Perelman, turned down the Fields Medal just adds spice to the tale: the New Yorker covered it, as did the New York Times, as well as a host of other mainstream organs across the social fabric. Low-dimensional topology is high profile indeed.
Among mathematicians of other confessions, number theorists like me or what have you, there has arisen a proportional budding of curiosity (and then some). With two low-dimensional topologists down the hall from me it’s been nothing but fun to attend colloquia covering knot theory, braid theory, hyperbolic geometry (which looks very different in their hands than what I’m used to from modular forms — still, after a bit of reflection (hah!) everything comes into focus…), and even stuff coming from hypermodern physics. It’s all part of the prevailing mathematical culture.
The book under review, being based on lectures presented at the IAS/Park City Mathematics Institute Summer School (2006), is a fantastic introduction to this subject and should prove to be of use to everyone, from novices interested in the subject through more seasoned mathematicians on the outside to topologists interested in what the expert lecturers contributing to this collection have to say. For one thing, the piece, “Fifty Years Ago: Topology of Manifolds in the 50s and 60s,” by none other than John Milnor (who did not turn down his Fields Medal), is alone worth the price of admission. But wait, that’s not all, as the saying goes: there’s a sextette of lectures presented by John Morgan, dealing with (yes!) “Ricci Flow and Thurston’s Geometrization Conjecture,” which is the ambient space, so to speak, for the Poincaré Conjecture and all things Perelman.
By the way, Thurston didn’t pass up his Fields Medal either. Low dimensional topology is possibly the most highly represented Fields field — see e.g. Milnor’s review of the 1950s mentioned above: it all began with Serre’s work, resulting in a Fields Medal, etc.
Well, the book is clearly full of good stuff. Cameron Gordon talks about Dehn surgery and 3-manifolds, David Gabai deals with hyperbolic geometry and surgery on 3-manifolds, link homology is addressed by Marta Asaede and Mikhail Khovanov, Heegard Floer homology is covered by Zoltán Szabó, John Etnyre hits contact geometry, and the closer is a six lecture tour-de-force by Ronals Fintushel and Ronald Stern on 4-manifolds. The very last lecture of this last set is titled, “Putting it all together: The geography and botany of 4-manifolds”: how’s that for scientific ecumenism?!
The lectures are all of a wonderfully high quality even as they are accessible, which is of course consistent with the Park City Summer School goals, and they’re truly fascinating: even an outsider like me is easily captured. This book is a wonderful contribution to the literature.
Michael Berg is Professor of Matematics at Loyola Marymount University in Los Angeles, CA.