Homotopy groups of some homogeneous spaces and some remarks on their sections over spheres (Q803581)

The authors compute the stable homotopy groups and a few of the unstable homotopy groups of the homogeneous spaces SO(2n)/U(s), U(2n)/Sp(s) for \(s<n\) and those of SO(4n)/Sp(s) for \(s\leq n\). These groups are important for the study of almost complex and almost-quaternion substructures on the spheres. By an almost-complex s-substructure on an n-dimensional manifold M with a Riemannian metric is meant a pair (I,\(\eta\)) consisting of a 2s-dimensional subbundle \(\eta\) of the tangent bundle \(T_ M\) and a complex structure I on \(\eta\). Further the concepts of an almost complex s-substructure on M with framed complement, of a normalized almost complex s-substructure with framed complement and of a normalized almost complex s-substructure with oriented framed complement are introduced. As an application, the authors enumerate the homotopy classes of normalized almost complex (n-1) and (n-2) substructures with oriented framed complements on the spheres.