Rational toral ranks in certain algebras (Q1777890)

The rational toral rank of a simply connected space \(X\) is the largest integer \(r\) such that the \(r\)-torus can act continuously on a CW-complex \(Y\) such that \(X\simeq_{\mathbb{Q}} Y\). By definition, the toral rank is a rational homotopy invariant. The purpose of this paper is to prove that it is not a cohomology invariant. Some relevant questions are indicated.