Recently I was having a look at the preprint *Essentiality and simplicial volume of manifolds fibered over spheres* by Thorben Kastenholz and Jens Reinhold (arXiv:2107.05892).

In Theorem C therein they construct a closed, totally non-spin manifold (i.e., one whose universal cover is even not spin) whose space of Riemannian metrics of positive scalar curvature has infinitely many path components. But what surprised me was the sentence above that theorem: This is the first such example (besides in dimension 4 using Seiberg-Witten invariants)! Since the corresponding statement for spin manifolds is known for more than 30 years by now, I assumed that the totally non-spin case should be also known since many years – but apparently not.

One should remark, that for non-Spin manifolds with Spin universal cover, corresponding statements have been known for a few years: Alvaro-Gonzales–Dessai have shown that the space of psc-metrics on RP^5 has infinitely many path components (see https://zbmath.org/?q=an%3A7288848 or https://arxiv.org/abs/1902.08919). The same remains true even for the moduli space, which is not the case with Kastenholz–Reinhold’s result.

Interestingly, to the best of my knowledge it is not known if the space of psc-metrics on the 5-sphere has more than one component.