Study of Hopf spaces and p-compact groups

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Spaces are assumed to be simply connected and localized at a fixed odd prime p. We assume that the homology of each Hopf space has no p torsion, and the mod p cohomology ring of it is finite. We first completed the classification of irreducible Hopf spaces with rank less than or equal to five. Then, by using this result, we determined the irreducible decomposition of Hopf spaces satisfying the condition that the difference between the maximum and the minimum dimensions of the generators is less that 8(p-1). In the course of studying this problem we showed that some low dimensional unstable homotopy groups of spheres are generated by elements given by long Toda bracket of alpha elements. Moreover we extended a known fact on quasi p regularity of quaternionic Stiefel manifolds and real Stiefel manifolds.