Vanabel/ 8月 3, 2013/ 数学笔记/ 0 comments

今天看到有人在MO上问读田刚J.MorganRicci flow and the Poincaré conjecture需要看那些书作为准备。这里我转载下(你也许也有兴趣看看Bruce Keiner关于Poincare猜想类似的讨论):

If I were going there I wouldn’t start from here.

If you’re new to 3-manifolds, it might better to familiarise yourself with them intimately before starting on Perelman’s work. In fact, learning some knot theory (in particular Dehn surgery) would be a good first step. I don’t remember where I first learned this stuff, but I do remember sitting on the floor in the library in front of the low-dimensional topology section and looking at lots of books (perhaps a better search mechanism than Google when you’re not quite sure what you’re looking for). One good such book is Rolfsen’sKnots and Links“. I remember being very happy when I worked out why $S^1×S^2$ is the result of doing 0-surgery on $S^3$ (there’s a nice picture).

Maybe using the Wirtinger presentation and van Kampen’s theorem to compute the fundamental group of the Poincaré sphere would be a good exercise to convince yourself you understand what’s going on with Dehn surgery.

The basic observation in all of this is that the 3-sphere is the union of two solid tori (or indeed of two handlebodies of arbitrary genus).

If that grabs your imagination then a good step would be to convince yourself that every 3-manifold can be presented as

  1. a Heegaard splitting,
  2. a sequence of Dehn surgeries on the 3-sphere.

This uses the Lickorish theorem (that the mapping class group of a surface is generated by Dehn twists) and that will lead you into studying 2-manifolds (see Farb and Margalit’s book1 on mapping classesfor an excellent presentation, A PDF copy here).

When you have convinced yourself that the classification of 3-manifolds is an interesting and worthwhile subject then there are Hatcher’s survey, Allen Hatcher’s notes on 3-manifolds and Hempel’s book (amongst other places). You could have a look at Stalling’sHow not to prove the Poincaré conjecture” (available on his website) and maybe at the proof of the Poincaré conjecture in high dimensions (either Smale’s original paper2 or Milnor’s wonderful h-cobordism theorem book) to get an idea of what you’re missing by living in three dimensions.

Perelman’s approach comes from a completely different world to any of this: the world of Thurston’s geometrisation conjecture. Thurston’s book introduces some of these ideas (with an emphasis on the hyperbolic) and his papers are full of beautiful insights. Once you have at least some familiarity with this stuff you could reasonably crack open a book on Ricci flow and start learning about that, but be warned that it won’t necessarily bear much resemblance to anything else you’ve read about 3-manifolds.

Of course you don’t need all this background to understand Ricci flow, but at least you’ll know what a 3-manifold is.

I also stand by my comment that the best way to learn something is to pick up a difficult book containing something you would like to understand and then look stuff up as and when you need it. Google and Wikipedia are wonderful for quick reference but they are not an easy place to learn a subject thoroughly for the first time.

Edit: As Deane Yang points out below, if you’re more interested in Ricci flow itself, there may be better learning approaches. For instance, Chow and Knopf have a nice book in which they introduce Ricci flow and use it to prove the uniformisation theorem in two dimensions. They also cover Hamilton’s theorem that a positively curved 3-manifold admits a metric of constant positive sectional curvature. These are both strictly easier than Perelman, while still involving hard differential geometry. Of course, you need to learn some differential geometry but there are plenty of good books about that.

Another comments by Deane Yang: As you say, Perelman’s proof is completely different from Thurston-style topology. I am under the impression that you don’t need to know much of the latter to understand Perelman’s proof, which is all either analytic estimates or a careful geometric analysis of regions of the 3-manifold where the flow breaks down. It is useful to know at least a little bit of 3-manifold topology. In addition to Thurston’s book, I recall a nice Bulletin of the AMS survey by Peter Scott.

Comments add by Thomas Richard: The book by Chow and Knopf is in my opinion the best available place to learn about Ricci Flow for itself, and its 3 sequels entitled “Ricci and its applications” are excellent references for any one working in the field. I should add two more references for Ricci flow for itself, there is the book by Chow, Lu and NiHamilton’s Ricci flow” which in my opinion is a bit harder than Chow and Knopf but covers a bit more material. There are also notes by P. Topping, available on his website, which goes up to the beginning of Perelman’s work (F and W functionals) in a nice self-contained way.

  1. Farb, Benson, and Dan Margalit. A Primer on Mapping Class Groups (PMS-49). Princeton University Press, 2011.
  2. Smale, Stephen. “Generalized Poincaré’s conjecture in dimensions greater than four.” The Annals of Mathematics 74.2 (1961): 391-406.
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