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Is there any evidence for the classification of topological 4-manifolds, aside from Freedman's 1982 paper "The topology of four-dimensional manifolds", Journal of Differential Geometry 17(3) 357–453? The argument there is extraordinarily complicated and a simpler proof would be desirable.

Is there evidence from any other source that would suggest that topological 4-manifolds are so much simpler than smooth 4-manifolds, or does it all hinge on Freedman's proof that Casson handles are homeomorphic to standard handles?

My question is motivated from a number of points of view:

  1. The classification of topological 4-manifolds is now 30 years old and an easier version of the proof has not emerged. In contrast, Donaldson's invariants have been superseded by more easily computed invariants. This is a very unsatisfactory state of affairs for such a far-reaching topological result, particularly as it is so regularly used in proof-by-contradiction arguments against results in smooth 4-manifold theory.

  2. As the Bing topologists familiar with these arguments retire, the hopes of reproducing the details of the proof are fading, and with it, the insight that such a spectacular proof affords. I am delighted to see that the MPIM, Bonn is running a special semester on this topic next year. Hopefully this will introduce these techniques to a new generation of mathematicians (and save them from having to reinvent them!)

  3. It may be possible to refine the proof to gain more control over the resulting infinite towers - and perhaps get Hoelder maps rather than homeomorphisms, for example. This would require either a better exposition of the fundamental result or some new independent insight, which was the basis of my question.

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I've rephrased this question to remove the (perhaps unintended) impression that the original proof may have issues. I do not think such innuendos are appropriate on MO. Feel free to revert if you disagree with me. –  Noah Snyder Feb 6 '12 at 17:18
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@Brendan: I thought a more elementary argument is given in front.math.ucdavis.edu/0001.5063, which is a paper of Krushkal-Quinn "Subexponential groups in 4-manifold topology". I gather you are not satisfied with their approach, would you explain why not? –  Igor Belegradek Feb 8 '12 at 19:47
    
Thanks for the reference, I wasn't aware of it. However, it does not deal with the central question I'm interested in: the convergence conditions for infinite towers of Casson handles (or gropes). This paper gives a simplified proof that the original work extends from simply connected to other fundamental groups. It assumes what we'd like to see reproven. –  Brendan Guilfoyle Feb 9 '12 at 8:47
    
Isn't it best at this point to ask the experts via email? –  Igor Belegradek Feb 9 '12 at 16:59
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The Bonn semester web page is here now: people.mpim-bonn.mpg.de/teichner/Math/4-Manifolds.html –  Ian Agol Sep 11 '12 at 12:42
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3 Answers 3

There's a somewhat different exposition in Freedman and Quinn's book. I think the main difference is that they use gropes instead of Casson handles. Gropes are made of embedded surfaces instead of singular disks, and introduce some technical simplifications to the proof (they originated with Stan'ko). Richard Stong gave a correction to one of the arguments in the book, although I think it isn't relevant to the proof of the disk theorem.

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Freedman and Quinn's book does not discuss the critical "convergence criteria" needed to show that infinite Casson handles (or capped gropes) are standard. –  Brendan Guilfoyle Feb 7 '12 at 13:10
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up vote 12 down vote accepted

After 7 months, over 800 MO views and (as suggested) emails to experts, the answer to the question is "No": other than Freedman's 1982 paper, there is no evidence what-so-ever that topological 4-manifolds are so much simpler (i.e. determined up to homeomorphism by their intersection form) than smooth 4-manifolds. Subsequent research (capped gropes etc) hinge on the key step in the paper - the removal of "gaps" in the "design". While this is a highly unsatisfactory state of affairs for the reasons mentioned in the original question, it is what it is.

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Since you seem quite interested in making sure this knowledge is preserved, and in making sure the arguments are understandable, it seems that you are the perfect choice for the person who will write such a book! –  Steven Gubkin Oct 2 '12 at 13:37
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There is no other evidence. In fact there is absolutely no evidence what so ever. I have never met a mathematician who could convince me that he or she understood Freedman`s proof. I attempted to read that monstrosity of a paper a number of times by myself and quite a few times with a group of other mathematicians. We never were able to finish checking all of the details. Such seminars always ended before we could make it through even half of his paper. No other expositions on the subject seem to be any better. It is truely an odd state of affairs that after all of these years no one has managed to write a clear exposition of this so called proof,and that no one seems to question the claim that there ever was a proof. I remember thinking as a young mathematician either this "proof" is sheer nonsense or someone will eventually write out a clear and detailed explanation. As of April of 2011 I have understood that the so called proof is full of errors and they can not be fixed. I mentioned this to several mathematicians during the summer of 2011 and I believe these conversations are directly linked to the dialogue seen here on math overflow.

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It would be good if the Bonn semester produces such a clear exposition. One reason that such an exposition has not been written might be that Freedman's Theorem somehow killed the major question. This wasn't the case with Donaldson's work of the same time, which showed that there were interesting questions still to be asked in the smooth category. I think it's recognized that this lack of understanding of Freedman's work is regrettable: this is one of the motivational reasons behind having this semester. I think people should hold fire on this thread until after the Bonn semester is done. –  Andrew Lobb Dec 24 '12 at 16:43
    
Freedman´s arguement is critically flawed and the work can not be salvaged. Your seminar is a waste of time! How much more time must mathematicians waste on this foolishness? I find your suggestion that "people should hold fire on this thread until after the Bonn semester is done" idiotic. –  Karl Luttinger Dec 27 '12 at 17:57
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For those interested in coming to their own conclusions, videos of Freedman's lectures are available at mpim-bonn.mpg.de/node/4436 . The lectures are taking place twice a week (Tues and Thurs) and are uploaded the following day. –  Brendan Guilfoyle Feb 6 '13 at 13:02
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A discussion wiki for questions (and hopefully answers) on the proof is now running at mpim-bonn.mpg.de/node/4494. –  Brendan Guilfoyle Feb 18 '13 at 14:25
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