Hopf invariants for reduced products of spheres

Abstract

Let S m n S_m^n be the m m th reduced product complex of the even dimensional sphere S n {S^n} . Using ’cup’-products, James defined a Hopf invariant homomorphism \[ H m n : π m n − 1 ( S m − 1 n ) → Z H_m^n:{\pi _{mn - 1}}(S_{m - 1}^n) \to {\mathbf {Z}} \] such that H 2 n H_2^n is the classical Hopf invariant. Extending the result of Adams on H 2 n H_2^n we determine the image of H m n H_m^n . Partial calculations were made by Hardie and Shar.