Akshay Venkatesh, Craig Westerland and I have temporarily withdrawn our preprint “Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II,” because there is a gap in the paper which we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.