Jonathan Rosenberg introduced the following conjecture: A closed manifold \(M\) admits a Riemannian metric of positive scalar curvature if and only if the product \(M \times S^1\) admits one.
One direction of the conjecture is trivial: If \(M\) admits a psc-metric, then the product metric on \(M \times S^1\) will also have psc. The other direction is the highly non-trivial one and is in its full generality still open, at least in dimensions at least 5 (in dimension 4 one can construct a counter-example using Seiberg-Witten theory). There are two ways to attack the converse direction:
- If \(M\) does not admit a psc-metric, then one tries to show that \(M \times S^1\) does also not admit a psc-metric. This works well if there is a “reason” for \(M\) to not admit a psc-metric, for example based on index theory (in the spin case) or on the minimal hypersurface method (in low dimensions). But if \(M\) only “accidentally” does not admit a psc-metric, it is not clear at all why \(M \times S^1\) should also not admit one.
- The other way, which is of course logically equivalent to the previous one, is to construct a psc-metric on \(M\) given one on \(M \times S^1\). The problem with this approach is that it is hard to construct psc-metrics! But nevertheless this approach was now successfully implemented by Steven Rosenberg and Jie Xu: They proved that if \(M\) is an oriented closed manifold such that \(kM\) (the \(k\)-fold disjoint union) is an oriented boundary, then a psc-metric on \(M\times S^1\) can be used to construct a psc-metric on \(M\) (arXiv:2412.12479).
Note that the oriented bordism ring is very well understood. Especially, the condition that \(kM\) is an oriented boundary for some natural number \(k\) is always satisfied if the dimension of \(M\) is not divisible by \(4\).
edit: As always with fresh preprints on the arXiv one should take these results with a grain of salt …