Hopf Invariants for Reduced Products of Spheres

Abstract

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