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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 In contrast, Donaldson's invariants have been superseded by more easily computed invariants.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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The answer to this question might have changed since it was first asked nine years ago: a book is now available whose goal it is to give a detailed elaboration on Freedman's work:

The Disc Embedding Theorem, ed. Behrens, Kalmar, Kim, Powell, Ray (Oxford University Press, 2021).

Some excerpts from the Preface:

We choose to follow the proof from [FQ90], using gropes, which differs in many respects from Freedman's original proof using Casson towers [Fre82a]. The infinite construction using gropes, which we call a skyscraper, simplifies several key steps of the proof, and the known extensions of the theory to the non-simply connected case rely on this approach. …

We briefly indicate, for the experts, the salient differences between the proof given in this book and that given in [FQ90]. First, there is a slight change in the definition of towers (and therefore of skyscrapers). …

Additionally, the statement of the disc embedding theorem in [FQ90] asserts that immersed discs, under certain conditions including the existence of framed, algebraically transverse spheres, may be replaced by flat embedded discs with the same boundary and geometrically transverse spheres. The proofs given in [Fre82a, FQ90] produce the embedded discs but not the geometrically transverse spheres. We remedy this omission by modifying the start of the proof given in [FQ90], as in [PRT20]. … Besides these points, the proof of the disc embedding theorem given in this book only differs from that in [FQ90] in the increased amount of detail and number of illustrations.

In Section 1.5 they mention some things that are not covered in the book. In addition to bypassing the part of Freedman's original proof that "consisted of embedding uncountably many compactified Casson handles within the original Casson handle and then applying techniques of decomposition space theory and Kirby calculus," they say:

Note that the ambient manifold is required to be smooth in the statement of the disc embedding theorem. There exists a category preserving version of the theorem, where ‘immersed’ discs in a topological manifold are promoted to embedded ones. However, the proof requires the notion of topological transversality and smoothing away from a point (see Section 1.6). These facts, established by Quinn, in turn depend upon the disc embedding theorem in a smooth 4-manifold stated above. The fully topological version of the disc embedding theorem is beyond the scope of this book, since we will not discuss Quinn's proof of transversality.

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    $\begingroup$ Quanta magazine just published an article about this book: New Math Book Rescues Landmark Topology Proof $\endgroup$
    – Jason Rute
    Commented Sep 9, 2021 at 18:01
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    $\begingroup$ @JasonRute With a reference to remarks in mathoverflow.net/a/117144 :) $\endgroup$
    – Yemon Choi
    Commented Sep 9, 2021 at 18:28
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    $\begingroup$ Came here because of the Quanta article, which indicates that Freedman's 4 dimensional proof is maybe even harder than Perelman's 3 dimensional proof. Can it really be like that? I read over Terence Tao's article about Perelman's proof and didn't understand it, but it gave a constant sense of having to spot and control issues that were very hard to visualize. I'd have no idea how to check such a proof even if I had the technical background. It seems too easy to miss something. $\endgroup$
    – none
    Commented Sep 10, 2021 at 2:07
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    $\begingroup$ @none I don't fully understand the details of Perelman's proof, but it is more well understood than Friedman's work. Part of this is that people have spent a lot of time writing expositions of the proof. Also, the overarching strategy of the proof is fairly clear and once you see the monotonicity of the $\mathcal{W}$ functional and why it implies non-collapsing, the approach seems promising. There are many tough details along the way (e.g., ruling out the cigar soliton or showing that surgeries don't accumulate), but these are concrete steps which can be checked by experts. $\endgroup$
    – Gabe K
    Commented Sep 12, 2021 at 17:43
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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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    $\begingroup$ 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! $\endgroup$
    – Steven Gubkin
    Commented Oct 2, 2012 at 13:37
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    $\begingroup$ I recently gave a talk discussing the motivation and ideas behind this question and two other linked mathoverflow questions: mathoverflow.net/questions/108631/fake-versus-exotic and mathoverflow.net/questions/252563/the-freedman-dichotomies . The talk was recorded and can be viewed at: youtube.com/watch?v=VZs1UG2Wtn8 $\endgroup$
    – Brendan Guilfoyle
    Commented Jan 27, 2020 at 15:14
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    $\begingroup$ Promotion committee: "What have you done in the last 5 years?" B.G.: "I've been writing a book about a paper published in 1982." Promotion committee: "How is that groundbreaking overachieving hyped research?" B.G.: "It is not, but I am preserving the knowledge of the elders." Promotion committee: "How told you that was a good idea?" B.G.: "A guy on the internet, and 24 people liked his suggestion." Promotion commitee: "We will call you, don't call us. Good bye." $\endgroup$
    – Andrej Bauer
    Commented May 26, 2021 at 17:51
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    $\begingroup$ @BrendanGuilfoyle Depending on your opinion of the cited book, consider changing the accepted answer to this one. If nothing else, it does appear to provide evidence that is independent to the 1982 paper, which is the core of the OP. $\endgroup$
    – Emilio Pisanty
    Commented Sep 17, 2021 at 16:11
  • $\begingroup$ Would you accept as evidence the fact that even higher dimensional manifolds (for example the seven-sphere) have a more complicated smooth theory than topological? $\endgroup$
    – Sam Nead
    Commented Mar 1, 2022 at 17:23
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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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    $\begingroup$ Freedman and Quinn's book does not discuss the critical "convergence criteria" needed to show that infinite Casson handles (or capped gropes) are standard. $\endgroup$
    – Brendan Guilfoyle
    Commented Feb 7, 2012 at 13:10
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    $\begingroup$ The latter comment isn't right: the Freedman-Quinn book does contain an alternative proof of the disk embedding theorem, constructing infinite convergent towers and shrinking the holes and the gaps in the design. It is not easy going, but it is there. See also arxiv.org/abs/2006.05209. $\endgroup$
    – Dave Davidson
    Commented May 27, 2021 at 15:25
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    $\begingroup$ Re Ian Agol's comment, indeed Stong's corrections to FQ are about Chapter 10, the applications of the disk embedding theorem to classifications of 4-manifolds, and are not related to the proof of the theorem. In fact there are four such papers, one joint with Wang. jstor.org/stable/… sciencedirect.com/science/article/pii/… sciencedirect.com/science/article/pii/… sciencedirect.com/science/article/pii/… $\endgroup$
    – Dave Davidson
    Commented May 27, 2021 at 15:32
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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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    $\begingroup$ 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. $\endgroup$
    – Andrew Lobb
    Commented Dec 24, 2012 at 16:43
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    $\begingroup$ 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. $\endgroup$
    – Brendan Guilfoyle
    Commented Feb 6, 2013 at 13:02
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    $\begingroup$ A discussion wiki for questions (and hopefully answers) on the proof is now running at mpim-bonn.mpg.de/node/4494. $\endgroup$
    – Brendan Guilfoyle
    Commented Feb 18, 2013 at 14:25
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    $\begingroup$ And, thankfully, the lectures gave rise to more lectures, and a big team effort, resulting in the book with a fully worked-out proof. $\endgroup$
    – David Roberts ♦
    Commented Sep 16, 2021 at 1:59
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    $\begingroup$ @KarlLuttinger This answer is quoted in this article on Quanta magazine. $\endgroup$
    – Emilio Pisanty
    Commented Sep 17, 2021 at 15:50
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Maybe these notes (entitled "THE 4 DIMENSIONAL POINCARÉ CONJECTURE") of Danny Calegari are useful for your interests?: https://math.uchicago.edu/~dannyc/courses/poincare_2018/4d_poincare_conjecture_notes.pdf

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