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

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I think there's 7 or more books on the topic now. There's Morgan-Tian. There's Topping. There's Chow (2 different books). There's Cao-XiPing. And there's Kleiner. Simon Brendle. Zhang. On and on. I've got a vague memory of a few others. I think different approaches demand different backgrounds. Some are more traditionally 3-manifolds-ish, some are more DG/PDE-ish in flavour. –  Ryan Budney Feb 28 '12 at 9:28
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What prior background do you have in this or related areas? –  Yemon Choi Feb 28 '12 at 9:29
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Also, there's a rather slender set of John Morgan's lecture notes from a lecture series he gave at Stanford. Authors are Morgan and Frederick Fong. IMO as far as pencil-sketch "warm up" type notes go, they seem to be some of the friendliest reading. This is in the University Lecture Series, Volume 53. –  Ryan Budney Feb 28 '12 at 9:31
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Why not start reading and look up unfamiliar stuff when you come across it? I find that's a good way to read anything. Say they use Hamilton's maximum principle for tensors and you look up Hamilton's paper and are completely lost so you go back to some basic PDE book and learn about the maximum principle for scalars under parabolic flows. You learn something and eventually you get the feel for what's going on and can move on to the next perplexing point. This kind of reverse-engineering is a good way to decide what you need to learn. –  Jonny Evans Feb 28 '12 at 9:57
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This should be retagged. –  Sean Tilson Feb 28 '12 at 14:29
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2 Answers 2

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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.

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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. –  Deane Yang Feb 29 '12 at 9:04
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I also endorse Jonny's endorsement of the books written by Ben Chow with others (disclaimer: Ben and I have been friends since graduate school). I think Ben and his co-authors have gone to great pains to present the foundations of the Ricci flow very carefully and in great detail. –  Deane Yang Feb 29 '12 at 9:59
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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 Ni "Hamilton'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. –  Thomas Richard Feb 29 '12 at 12:26
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Your answer is quite encouraging!!!..I will start my learning journey!.. –  user18717 Mar 1 '12 at 9:00
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Excellent! Good luck. –  Jonny Evans Mar 1 '12 at 16:49
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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.

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