Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurfaces, Band 752

We classify 1-connected compact homogeneous spaces which have the same rational cohomology as a product of spheres $\mathbb{S}^{n_1}\times\mathbb{S}^{n_2}$, with $3\leq n_1\leq n_2$ and $n_2$ odd. As an application, we classify compact generalized quadrangles (buildings of type $C_2)$ which admit a point transitive automorphism group, and isoparametric hypersurfaces which admit a transitive isometry group on one focal manifold.