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 : π q n − 1 ( K ) → Z h:{\pi _{qn - 1}}(K) \to Z in two ways. We prove both definitions are equivalent and show that p π i ( K ) ≃ p π i − 1 ( S n − 1 ) ⊕ p π i ( S q n − 1 ) {}_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 α ∈ π q n − 1 ( K ) \alpha \in {\pi _{qn - 1}}(K) such that ( h ( α ) , p ) = 1 (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.