Recall the following conjecture about aspherical manifolds (i.e., manifolds whose universal cover is contractible):
If M is a closed, aspherical manifold, then M does not admit any Riemannian metric of positive scalar curvature.
In January I saw a preprint being posted on the arXiv (2001.02644) claiming to have resolved this conjecture. If it turns out to be correct, that would be a major breakthrough.
Update (2020-04-10): Jon Wolfson wrote me an eMail stating: “Unfortunately, I recently found an error in the proof coming from an earlier paper […] The error occurs in the limiting procedure (Section 2, Corollary 2.15). When the constraint functions go to zero, by scaling these functions one can see that the limit map need not be a geodesic.”
Below you will find my original post, if you are still interested in its content.
Now if you look into the introduction of that preprint, you won’t see the author mentioning this conjecture, which is a bit strange. He only states that he resolves a conjecture of Gromov and you have to know that this conjecture of Gromov implies the above one about aspherical manifolds.
The conjecture of Gromov is the following. We first define the notion of macroscopic dimension: a metric space X has macroscopic dimension at most k if there is a k-dimensional polyhedral space P and a continuous map f from X to P such that the diameters of the fibres of points are uniformly bounded. Gromov conjectured that if an n-dimensional complete Riemannian manifold M has positive scalar curvature, then the macroscopic dimension of M must be at most n-2, i.e., we should see a drop of 2 for the macroscopic dimension compared to the topological dimension. Gromov's conjecture implies the above one about aspherical manifolds since one can show that their macroscopic dimension coincides with the topological dimension. Of course, for this it suffices to prove Gromov's conjecture in the weaker form that you witness only a drop of 1 for the macroscopic dimension. And this is actually what the above cited preprint claims to have proven.
The whole introduction of the preprint is actually a bit of a mess: the author tries to define asymptotic dimension, but instead writes something unintelligible, Theorem 0.1 is not what he actually proves in the paper, and he recalls redundantly the notion of Uryson width (it is the same as macroscopic dimension if you carry around with you the information about the diameters of the fibres of the continuous map).
But going quickly through his idea of the proof, it actually sounds reasonable. We first rewrite having positive scalar curvature as the condition that the Ricci endomorphism has at every point of the manifold at least one positive eigenvalue, and then:
- The eigenspaces corresponding to the positive eigenvalues of the Ricci endomorphism form a distribution on M. Let us assume that it is integrable for this blog post, i.e., we get a foliation of M.
- He writes down certain functionals for paths in M and minimizes them. By this he then shows that for any two points in a single leaf there is horizontal geodesic joining them. But since the tangent spaces of the leaves correspond to the positive eigenvalues of the Ricci endomorphism, the Bonnet-Myers theorem tells us that the length of this geodesic is bounded by a constant depending only on the curvature. Hence the leaves of the foliation have uniformly bounded diameters.
- The final idea is to use at the end the quotient map to the space of leaves to witness the drop of macroscopic dimension.
Note that Step 2 is the actual meat of the preprint. Unfortunately, I am not an expert on variational problems, which means that I’m basically out of the task of checking the validity of these arguments (I would have to spend several months to learn the mathematics of variational problems before I could even think about starting to check the details of his proof).
But what I tried to do was to understand Step 3. And here I have to admit that I failed. Since the space of leaves need not be Hausdorff, the author introduces a further quotient map out of the space of leaves (see the end of Section 2 of his preprint). But to me the definition of this quotient map is unclear, especially since there are choices involved resulting in honestly different maps. Further, for me it is completely unclear why this new quotient space should be a polyhedral space, especially since there is, as already said, a gazillion of choices involved in its definition. Also, note that the foliation that we get in Step 1 of the proof is in general not regular, i.e., it might have singularities. I do not completely understand how to induce from a neighbourhood which contains a singular point of the foliation a locally Euclidean structure in the final quotient space or how to make the image of the singularity a face of the putative polyhedron.
But given all my reservations about Step 3, if Step 2 turns out to be correct, then I think we could maybe get the final step done. I can imagine that one could, e.g., work around Step 3 by directly working with the foliation somehow. That is to say, one could try to prove that aspherical manifolds do not admit a foliation with all these properties and therefore circumventing the need to form quotient spaces.
So anyway, I am curious whether this preprint will turn out to be correct. Especially since I am myself working on such kind of problems. So if you happen to be an expert on variational problems and if you have taken a look at this preprint, please drop me a message and tell me your opinions about it.