Prove that the rational cohomology ring requires more than one generator iff $\dim\pi_{\text{odd}}(M)\otimes\mathbb{Q}>1$

Question

Let $M$ be a simply connected closed Riemannian manifold. Then how do we prove that the rational cohomology ring $H^*(M;\mathbb{Q})$ requires more than one generator if and only if the odd homotopy groups tensored with $\mathbb{Q}$, $\pi_{\text{odd}}(M)\otimes\mathbb{Q}$, is more than one-dimensional? (Also, is this true?)

Any help would be much appreciated. Thanks in advance!

Answer 1

The statement is true, and the proof can be found in “The homology theory of the closed geodesic problem” by Sullivan and Vigué-Poirrier, https://projecteuclid.org/download/pdf_1/euclid.jdg/1214433729, in Section 2 (Proof of Theorem), (i)->(ii) (a).