We classify 1-connected compact homogeneous spaces which have the same
rational cohomology as a product of spheres Sni x S™2, with 3 n\ ri2 and ri2
odd. As an application, we classify compact generalized quadrangles (buildings of
type C2) which admit a point transitive automorphism group, and isoparametric
hypersurfaces which admit a transitive isometry group on one focal manifold.
Received by the editor February 22, 2001.
1991 Mathematics Subject Classification. Primary 51H, 53C; Secondary 51E12, 57T15.
Key words and phrases. Buildings, generalized quadrangles, topological geometry, isopara-
metric submanifolds, polar representations, homogeneous spaces, Stiefel manifolds.
Supported by a Heisenberg Fellowship by the Deutsche Forschungsgemeinschaft.
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