Newsgroups: sci.math From: dy@shire.math.columbia.edu (Deane Yang) Subject: Re: Freedman-Donaldson Theorem Date: Thu, 10 Sep 1992 16:12:45 GMT Keywords: Exotic R^4, differential structures, topology In article <4173@seti.UUCP> hussein@bora.inria.fr (Hussein Yahia) writes: > I'm interested in understanding the proof of the famous Freedman-Donaldson >theorem about "fake R^4", that is the existence of exotic diffentiable >structures on R^4. (For others R^n , n different of 4, all differentiable >structures are always diffeomorphic to the standard one). I tried to read >a book "Connections, Definite Forms, and Four-manifolds" by T. Petrie and >J. Randall (Oxford Science Publications) but it is quite a little bit >too difficult for me. > >Could someone give me a list of references leading to a proof of that theorem >for somebody like me who is just at the level of Volumes 1-2 of Spivak's >Course on Differential Geometry and Hirsh's Differential Topology (Springer >Verlag) ? > The proof divides into two very different parts, the topological theory developed by Freedman and the study of the moduli space of Yang-Mills fields developed by Donaldson. Neither part is particularly easy to understand, although, as a differential geometer, I find the latter far more understandable. My advice would be to focus on one or the other, depending on your taste. I believe that Freedman has written a few surveys on his work on 4-manifolds. The Donaldson theory is presented in the book cited above, a book by Freed-Uhlenbeck, and a set of lecture notes by Lawson. Any of these references should be used only as a starting point. You will need to consult other books and papers to fill in the details. Given your background, it might be better to pursue more fundamental material first. For the Freedman stuff, you should learn more topology (here, I can't help much). For the Donaldson stuff, you should learn more about analysis and elliptic PDE's. Deane Yang Polytechnic University ============================================================================== From: sec@otter.hpl.hp.com (Simon Crouch) Date: Thu, 10 Sep 1992 15:08:43 GMT Subject: Re: Freedman-Donaldson Theorem Newsgroups: sci.math In sci.math, hussein@bora.inria.fr (Hussein Yahia) writes: [Concerning Donaldson/Freedman theory] > Could someone give me a list of references leading to a proof of that theorem > for somebody like me who is just at the level of Volumes 1-2 of Spivak's > Course on Differential Geometry and Hirsh's Differential Topology (Springer > Verlag) ? Well, a reasonable place to start is (a) Donaldson and Kronheimer's book "The geometry of four manifolds" and (b) Freedman and Quinn's book "The topology of four manifolds". (Sorry if the titles are inaccurate but my copies are at home). You will get hopelessly stuck :-) but the authors are considerate enough to give good overviews of what is happening and to explain (implicitly) what is required of the reader. (And the references are pretty good). You can then work back from there to fill in your knowledge, but be warned, there's a lot to get familiar with! In general terms you will need a fairly solid background in differential geometry and analysis for Donaldson theory (I've found that Kobayashi/Nomizu's "Differential Geometry" and Griffiths/Harris's "Algebraic Geometry" are helpful references [though I'm not sure I'd want to learn from them.....] and Lawson's "Spin Geometry" has the avowed aim of helping the reader understand Donaldson theory. [oh, and is a good book!]). For Freedman's stuff, you need to understand geometric topology....I seem to remember review articles in the AMS Bulletins being very useful, but I can't offhand remember any references. Oh, alternatively, you could go and study at Oxford for a year :-) Hope this is useful, Simon.