Akshay Venkatesh, Craig Westerland and I have temporarily withdrawn our preprint “Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II,” because there is a gap in the paper which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