Akshay Venkatesh, Craig Westerland and I have temporarily withdrawn our preprint “Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II,” because there is a gap in the paper which we do not, at present, see how to remove. There is no reason to think any of the theorems stated in the paper aren’t true, but because some of them are not proved at this time, we’ve pulled back the whole paper until we finish preparing a revised version consisting just of the material that does in fact follow from the arguments in their current form, together with some patches we’ve come up with. We are extremely grateful to Oscar Randall-Williams for alerting us to the problem in the paper.
I’ll explain where the gap is below the fold, and which parts of the paper are still OK, but first a few thoughts about the issue of mistakes in mathematics. Of course we owe a lot of people apologies. All three of us have given talks in which we told people we had a proof of (a certain version of) the Cohen-Lenstra conjecture over F_q(t). But we do not. I know several people who had work in progress using this theorem, and so of course this development disrupts what they were doing, and I’ve kept those people up-to-date with the situation of the paper. If there are others planning immediately to use the result, I hope this post will help draw their attention to the fact that they need to go back to treating this assertion as a conjecture.
One thing I found, when I talked to colleagues about this situation, is that it’s more common than I thought. Lots of people have screwed up and said things in public or written things in papers they later realized were wrong. One senior colleague told me an amazing story — he was in the shower one day when he suddenly realized that a paper he’d published in the Annals four years previously, a result he hadn’t even thought about in months, was wrong; there was an induction argument starting from a false base case! Fortunately, after some work, he was able to construct a repaired argument getting to the same results, which he published as a separate paper.
Naturally nobody likes to talk about their mistakes, and so it’s easy to get the impression that this kind of error is very rare. But I’ve learned that it’s not so rare. And I’m going to try to talk about my own error more than I would in my heart prefer to, because I think we have to face the fact that mathematicians are human, and it’s not safe to be certain something is true because we found it on the arXiv, or even in the Annals.
In a way, what happened with our paper is exactly what people predicted would happen once we lost our inhibitions about treating unrefereed preprints as papers. We wrote the paper, we made it public, and people cited it before it was refereed, and it was wrong.
But what would have happened in a pre-arXiv world? The mistake was pretty subtle, resting crucially on the relation between this paper and our previous one. Would the referee have caught it, when we didn’t? I’m not so sure. And if the paper hadn’t been openly shared before publication, Oscar wouldn’t have seen it. It might well have been published in its incorrect form. On balance, I’d guess wide distribution on arXiv makes errors less likely to propagate through mathematics, not more.
Sociology of mathematics ends here; those who want to know more about the mistake, keep reading past the fold.
(Note: in the interest of space this discussion will assume in places that the reader already knows the basic ideas of the paper.)
So in our first paper we proved that the sequence of Hurwitz spaces has stable cohomology; that is, that there are stabilization maps from to , for some d, which induce an isomorphism on the homology group H^i for n big enough relative to i.
In the second paper, we showed that the limit of as is 0 for all i > 1.
Those two theorems together — we said — show that when n is large relative to i.
But here’s the wrinkle. The stabilization map in the first paper is a map called U. And the stabilization map in the second paper is a map called V. They are not the same — and this matters. In other words, if you write X_n for the ith homology of the n-th Hurwitz space, then you have two sequences:
They look the same, right? But no: in the first sequence, the maps are U, and in the second sequence, the maps are V. We prove (and there is no problem here) that U is an isomorphism for n large enough, and that V^k is 0 for k large enough. But these two theorems don’t fit together, as Oscar pointed out to us. If we knew that V were an isomorphism for n large enough, then that would show X_n = 0 for all sufficiently large n. Or if we knew that U^k = 0 for k large enough, we would have the same. But unfortunately, when we thought about this, we realized there was a real issue. The technique for proving stability in the first paper really only works for U, not V. And the technique for proving vanishing in the limit in the second paper really only works for V, not U. So while the main theorems of both papers are correct, the way we put them together is not; we do not know, as we claimed, that the ith cohomology of the Hurwitz space vanishes when n is sufficiently large relative to i, and thus we don’t have proofs of the arithmetic statements claimed in sections 6 and 12 of the original version of the paper.
On the other hand, the proof of the central theorem, that V^k is 0 for k large enough (sections 2-5) is correct. So is the computation of the structure of the stable component group of the Hurwitz space as a set with Galois action (sections 7-8). And the material in section 11 is fine, too — as it happens, here we carefully work out the arithmetic consequences of what we expect to be true about the stable homology of Hurwitz spaces. The problem doesn’t come until section 12, where we assert that we have actually proved these expectations correct in the situation relevant to Cohen-Lenstra.
Please feel free to contact me if you are or have been using this paper and want to discuss which parts of Cohen-Lenstra we think we can still prove, and which ones we think we may be able to prove soon. Again (and on behalf of all three of us) I apologize for the confusion and inconvenience we have caused by publicly asserting a theorem with an incorrect proof.
I am sad to hear of this problem, and hope you will quickly find a solution. From the description, it seems this will certainly lead to new insights.
I admire this post tremendously. As a profession, we hide from view the rough edges, false starts, and mistakes in a way that is unhealthy, especially for young people who compare themselves to an unrealistic ideal. Your honesty about this issue is refreshing.
Regarding the issue, “to arXiv or not to arXiv”, that you raise: even if mistakes do propagate faster in the internet age, to me this is just a part of faster propagation of all mathematics. Eventually mistakes are corrected. But, without free, rapid dissemination of mathematics, much wonderful research would never happen. Mathematician A, who is looking for just the right tool to solve a problem, would not learn about Mathematician B, who has developed that tool. Without free, rapid dissemination, mathematics knowledge must flow through “gatekeepers” — editors, conference organizers, etc. Obviously this brings its own set of sociological problems.
Ouch! I’ve had to issue a few errata myself with some non-trivial fixes in them, though I’ve never yet had to withdraw a published paper completely because of such gaps (knock on wood…)
Bravo for being so open about this. It makes the world a better place.
I applaud you for being so open. Computer proof checking technology may eventually provide a good solution to the wider (sociological) problem, and tools such as Coq are already nearly good enough.
Unfortunately, there are old published papers still floating around out there as booby traps. A few years ago I stumbled on a paper using Google Scholar. It had been published in a respected journal, and was old enough that it was scanned from paper into a PDF file. The main result was possibly relevant to what I was doing, and I wondered why I didn’t recall running into this one in bibliographies. I thought that it might have been something I just overlooked. I emailed the link to a trusted expert who has been around for longer than I and asked him if he’d seen this. He had, and he was able to recall that the paper was in error. (Eventually the person with that memory will no longer be here.) I was able to pluck this thing that came with no warning label right out of the ether. It would be nice, but probably impossible, to completely weed out these historical goofs.
We can only learn from each others mistakes and failed strategies, and it would be better for all if people discussed their own more often, as happened here.
If the likes of Coq were always the ultimate arbiter of truth, then my Coq would be talking directly to your Coq, and I would be out of the loop, not learning much of anything, and bored.
@Richard: I believe that the MathSciNet reviews will update their review with notes about corrections, retractions, and errors. That is one way the community can keep a record (most useful in cases where authors do not submit errata themselves).
@Jason: MathSciNet is licensed to institutions only. I’m an AMS member but have no official affiliation, so this very important and nearly necessary tool is not available to me. I’m not sure why this is the case—I’ve never seen an official justification for this restriction. (I was in grad school many years ago before my career path went astray. However, I still dabble in this stuff.)
I was told that, some 20 years ago, a mathematician was giving a talk at a seminar at the IAS. Borel was in the audience and very quickly stopped the speaker, telling him that the result he was claiming was false as it was contradicting some other thing that Borel knew to be true. The speaker paused, reviewed the proof in his head and asked whether he could deliver it to the audience as he saw no place where he could have made a mistake. Borel agreed and everyone concluded that the proof was perfectly correct; the only thing that could not be checked on the spot was a reference to a paper by Borel and Serre…
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