Abstract: In this work, some consequences of the existence of almost complex and almost-quaternion 

substructures on the sphere are studied. Existence problem of almost-complex substructures on the real 

projective space Pⁿ for  n # 15(mod16) and the existence problem of almost quaternion substructures 

on Pⁿ for n 31 (mod 32) are solved. Some calculations of the stable and unstable homotophy groups 

of the spaces SO(2n) / U(s) are given. 

Key words: Almost complex substructures, almost-quaternion substructures, homotophy groups, 

homotopy groups of homogeneous spaces.