Homotopy groups of some homogeneous spaces and some remarks on their sections over spheres

Abstract

Stable homotopy groups and a few of the unstable homotopy groups of the homogeneous spacesSO(2n)/U(s) andU(2n)/Sp(s) fors<n, and those ofSO(4n)/Sp(s) fors≤n are computed. The enumeration of homotopy classes of normalized almost complex (n−1) and (n−2) substructures with oriented framed complements are given. Some results about the existence of the equivariant cross sections of these homogeneous spaces over the spheres with respect to certain unitary group actions are proved.