Show that the rational cohomology ring $H^*(M;\mathbb{Q})$ needs at least two generators

Question

Let $M$ be a simply connected closed Riemannian manifold. How does one find a condition that may be imposed on $M$ (perhaps on the curvature of $M$ and on torsion) which guarantees that the rational cohomology ring $H^*(M;\mathbb{Q})=\bigoplus_{k\in\mathbb{N}}H^k(M;\mathbb{Q})$ needs at least two generators? That is, how does one force $M$ not to have rational cohomology that is the quotient of a polynomial ring?

Cross-posting on MO: https://mathoverflow.net/questions/306490/show-that-the-rational-cohomology-ring-hm-mathbbq-needs-at-least-two-ge

Any help would be much appreciated. Thanks in advance!

Answer 1

I can't imagine any kind of curvature condition which would work. I will use the phrase "standard examples" to refer to a space diffeomorphic to a sphere or projective space over $\mathbb{C},\mathbb{H}$ or $\mathbb{O}$. This are certainly the most well known examples of manifolds having the kinds of cohomology rings you are trying to avoid.

For example, one could demand negative/zero/positive sectional curvature. But the first two cases can't arise on a closed simply connected manifold, by Cartan-Hadamard Theorem. This leaves the case of positive sectional curvature - but then all of the standard examples admit such a metric.

So, perhaps one could demand negative/zero/positive Ricci curvature. Again, in the positive case, all the standard examples admit such a metric. Further, Lohkamp has shown that every manifold of dimension at least $3$ admits a metric of negative Ricci curvature. And it seems to be open whether or not $S^n$ admits a Ricci flat metric (see this MO question). I do not know about Ricci flat metrics on any other space with singly generated rational cohomology.

Finally, in the realm of scalar curvature, all the standard examples admit metrics on positive scalar curvature. By Kazdan-Warner, they each admit scalar flat and scalar negative metrics.

To further complicate matters, there are known examples of exotic spheres which do not admit metrics of positive scalar curvature (so don't admit metrics of positive sectional or Ricci curvature). So it doesn't seem like any of the usual curvature conditions can limit the rational cohomology ring like you would like.

Answer 2

The following is Ian Agol's answer on my cross-post on MO: If $M$ is simply-connected and has reducible holonomy, then a theorem of de Rham implies that $M$ is a product, and hence does not have homology generated by one element. Moreover, coupling Berger's classification with de Rham's decomposition theorem, one obtains a classification of reducible holonomy groups by requiring that each factor is one of the examples coming from the list.