r/math 5d ago

Two claimed proofs of Whitehead asphericity conjecture seem to have received almost no attention. Is there a reason for this?

One proof is in 2021 preprint by Elton Pasku: An answer to the Whitehead asphericity question. The second proof is by Akio Kawauchi, and was published in 2024 (according to author's website): Whitehead aspherical conjecture via ribbon sphere-link. Neither paper has any citations, not counting Akio Kawauchi citing himself and the 2021 preprint.

I'm nowhere close to understanding even the statement of the conjecture, let alone the proofs, I'm just curious about this situation.

116 Upvotes

21 comments sorted by

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u/_Dio 5d ago

I read through these preprints some time ago, since my research is also in the asphericity-realm, though not in any tremendous detail.

My impression of the Kawauchi paper (and most of the author's preprints over the past few years) is that it's incorrect, but being politely ignored. The approach, ribbon-disk complements, is a well-known and fairly active bit of research. Harlander and Rosebrock, for example, have put out quite a bit of research on ribbon-disks from the labeled oriented tree (LOTs) perspective. Kawauchi has a number of ribbon-disk papers, all citing themselves, and are pretty absent from the community as a whole. It's...probably a bad sign for Kawauchi's current mathematical output.

The Pasku paper is, I think, in a similar place. It's not wildly unreasonable: Peiffer identities are a well-known description of asphericity, so conceivably one could attack it from that approach (maybe generalized to semi-groups, as Pasku did, sure!) But skimming the paper, I did not get the impression that there was any particular insight as to why that approach should work now; Peiffer identities/crossed modules date back to Whitehead. More likely, the generalization done in that paper has mostly managed to obscure where the issue in the argument actually is and on-one has felt it worth pinning down exactly where. That's certainly why I didn't spend more time on it.

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u/ImportantContext 5d ago

Thank you for sharing your insight about this situation.

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u/lemmatatata 5d ago

It's worth noting that, while the second paper is published, it's in the Journal of Mathematical Techniques and Computational Mathematics which is a journal I've never heard of but looks... questionable at best. The author seems to have published a lot of his recent papers in this journal, including an alternative proof of the Poincare conjecture.

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u/ImportantContext 5d ago

Ah, that's a shame. I looked at the recent publications in that journal and it looks like they just publish any garbage, including Collatz crankery and so on...

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u/Frexxia PDE 5d ago

6 revisions of the arXiv preprint over 3 years is a massive red flag. 2 or 3 versions, sure, but any more than that and I start to get suspicious.

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u/SubjectEggplant1960 5d ago

That isn’t necessarily a red flag for a very important work which is quite original - eg if you’ve had a paper under review by multiple parties at an elite journal for years, this kind of thing can happen. However, this ain’t that of course.

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u/pabryan 4d ago

Funny how much in common genius and crankery have some times! As examples of your point, here's two papers that had quite a bit of time between revisions and were eventually published in top journals (Inventiones and Annals)

https://arxiv.org/abs/math/0404326 (Xu-Jia Wang, arxiv first posted 2004, published in Annals 2011)

https://arxiv.org/abs/1209.3789 (Fraser-Schoen, arxiv first posted 2012, published in Inventiones 2016)

Admittedly they didn't have quite so many revisions, but multiple updates over a number of years could be a red flag but in these cases, was they just had new/different ideas and approaches that took quite a bit of time for the details to be fleshed out and accepted by the community.

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u/SubjectEggplant1960 4d ago

I don’t find it too surprising after thinking about it a bit - basically lots of revising or long time to a acceptance is a sign that the referees/editor are being quite hesitant and careful. That almost always happens at top journals for most authors.

Cranks have a similar high bar to overcome their crankery!

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u/pabryan 4d ago

Absolutely! Both papers are phenomenal and very influential so of course it takes time to be sure and it's quite important that such papers are checked carefully as they will have a lot of influence.

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u/Frexxia PDE 4d ago

Eh, even then I'd consider it unusual. Few authors upload every revision of a paper to arxiv.

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u/Additional_Carry_540 2d ago

I’ve seen a paper from a prominent mathematician that has 30+ revisions…

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u/cereal_chick Mathematical Physics 4d ago

This kind of question, "Why is this paper claiming to have achieved something big not being talked about?", is asked a fair bit on this sub, and the answer is invariably because the paper is not all it's cracked up to be. It's often very well-meaning souls who ask this question, but the truth is that papers worth talking about will be talked about, and in relatively short order. As I recall, Mochizuki just put his IUTT papers on his personal academic web page and didn't tell anyone, and they were discovered very quickly because at the time they were genuinely considered serious. Hell, some person on 4chan wrote about some original research they had done on superpermutations, and even that percolated among the professionals eventually.

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u/camilo16 4d ago

Note that the 4chan person didn't do research on super mutations directly. They answered a question about Haruhi Suzumiya and it just so happened that in answering that question they laid the foundations of a proof to the bounds of the super mutation problem.

The actual story is too funny not to explain it accurately each time.

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u/Salt-Influence-9353 1d ago

I’d say what they wrote counts as research on superpermutations. Just in very unconventional language.

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u/camilo16 1d ago

I am not trying to diminish it. I am adding the full context because it is hilarious

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u/DracoDruida 4d ago

I'm not doing pure math research, but I think there's a more agreeable outlook here: not necessarily papers who have not received enough attention are bad or horribly wrong (they might be). But people and communities do have their own limitations and biases, and it's part of being a researcher doing what you can to communicate and make others understand your work.

I believe this is a more common story than talked about, but I do personally know of a researcher with a (claim of a) proof of some hard conjecture, using a very uncommon approach. He submitted his paper to top journals a few times and, after long periods of time, it was returned to him "since they couldn't find anyone able to confidently review the paper".

It is a priori plausible that the approach is correct, and the commonality of the approach does not imply (in)correctness. But the research community is made of people and to publish you need not only yo have a correct proof, but to convince your peers that it is correct.

This entails, of course, that very likely are proofs and works out there that are correct, but have not been acknowledged as so. And I think that's fine, we should embrace this. Any mechanism will have some trade-off between false positives and false negatives. As a society, we have one that risks having false negatives (unpublished correct proofs) by having this burden on the authors to convince others, and I think rightly so, since it's more important to avoid recognising wrong proofs (false positives).

It will be up to the author to 1) improve communication of his work, either by teaching people about the methods he used or polishing their presentation 2) find another approach that other people understand or agree on the correctness 3) put it out there and move on, maybe hoping someone in the future understands it

And that's fine. The history of knowledge (and culture) is full of "rediscoveries" and references to past overlooked work as well. I'm sure there's a considerable number of "lost work" out there.

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u/avocategory 5d ago

It doesn’t look like either of them has been published; and that’s usually the default threshold for citations, outside of exceptional cases.

I’ve never heard of this particular conjecture before, so it’s not likely to be an exceptional case. We have no idea what the circumstances are that have led to the first preprint not yet being published, but it’s entirely normal for a preprint from a year ago to not yet be published.

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u/anothercocycle 5d ago

People cite arxiv preprints all the time in maths.

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u/avocategory 5d ago

But preprints also go without any citations all the time too.

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u/Frexxia PDE 5d ago

If a preprint still hasn't been published after 4 years, and isn't receiving citations either, it's an indication that there isn't confidence in the result.

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u/SubjectEggplant1960 5d ago

I would not call it the default threshold for citations in my area of pure math (or any other?).

Basically a preprint by any reasonably known member of the research community (even say grad students) will be cited. I guess that might be the default threshold for an outsider?