The homotopy groups of spaces whose cohomology is a $Z_{p}$ truncated polynomial algebra

Abstract:

On such a space one can define a “Hopf” invariant homomorphism $h:{\pi _{qn - 1}}(K) \to Z$ in two ways. We prove both definitions are equivalent and show that ${}_p{\pi _i}(K) \simeq {}_p{\pi _{i - 1}}({S^{n - 1}}) \oplus {}_p{\pi _i}({S^{qn - 1}})$ if and only if there is an $\alpha \in {\pi _{qn - 1}}(K)$ such that $(h(\alpha ),p) = 1$. As immediate corollaries of this we get a result of Toda on the homotopy groups of the reduced product spaces of spheres and a well-known result of Serre on the odd primary parts of the homotopy groups of spheres.