Renaud Gauthier from KSU posted this preprint on ArXiv a few days ago, in which he claims to have found a serious problem with the construction of the LMO invariant, the universal finite-type invariant for rational homology spheres (it’s defined for all 3-manifolds, but I think of it as an invariant of homology spheres). What a headline that makes! A possible hole in the definition of the LMO invariant, with the potential to wash large swaths of quantum topology of 3-manifolds down the drain! Indeed, this is the topic of his second preprint.
Tomotada Ohtsuki was my PhD advisor, and he’s a careful mathematician with tremendous technical ability, who checks his answers against computational data to make absolutely sure no errors creep into his work. Le and Murakami are similar. Gauthier’s claim is that they made a fatal error calculating the effect of the second Kirby move on the framed Kontsevich invariant, which is used to construct the LMO.
Without having read Gauthier’s preprint (which is 81 pages long), my bias is to be skeptical of his claim. Maybe he found a typo, but surely not more. But what a headline it would make if a substantial error was there! This is math drama in the making.
I think it might be fun and educational to crowdsource peer review Gauthier’s claim. It’s important if it’s right, and if it’s wrong, at least it’s an opportunity to get into the kishkes of the LMO.
UPDATE: A MathOverflow question about this.
UPDATE: A third paper by Gauthier was uploaded on Thursday.
UPDATE: Gwenael Massuyeau shows a major flaw in Gauthier’s arguments in the comments. Another two large flaws are noted by Dylan Thurston. Due to these problems, Gauthier’s claim of an error in LMO does not appear to hold up.