What prerequisites do I need to read the book Ricci Flow and the Poincare Conjecture, published by CMI

Question

As mentioned in the title, I want to understand the proof of Poincare Conjecture by Perelman, what prerequisites do I need?

Answer 1

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's "Knots and Links". I remember being very happy when I worked out why $S^1\times S^2$ 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 (a) a Heegaard splitting, (b) 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 book on mapping classes for an excellent presentation).

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's "How 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 paper 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.

Answer 2

My humble advice for learning about Ricci flow generally, after obtaining some background in Riemannian geometry, would be to start with a book which gets you to important results quickly. An excellent book is the one by Peter Topping. (The only typo I observed there is the one regarding backwards uniqueness, which is now due to Brett Kotschwar.) After that, there are excellent books on the differentiable spherical space form theorem by Brendle and Andrews--Hopper; see also the original papers of Boehm--Wilking and Brendle--Schoen.

What is irreplaceable is to read and to reread the original works by the masters, i.e., Hamilton and Perelman. A collection of Ricci flow papers, mostly by Hamilton, is edited by H.D. Cao, etal.; this is a convenient place to get Hamilton's papers in one place. Perelman's papers are on arXiv. There are a number of excellent expositions of their work (focused on Perelman's work), which actually go beyond expositions and include various degrees of original work, namely in alphabetical order: Bessieres--Besson--Boileau--Maillot--Porti, Cao--Zhu, Kleiner--Lott, and Morgan--Tian.

The above remarks only pertain to Riemannian Ricci flow.