On Pierce-like idempotents and Hopf invariants

Given a set K with cardinality □K□ =n, a wedge decomposition of a space Y indexed by K, and a cogroup A, the homotopy group G= [A, Y] is shown, by using Pierce-like idempotents, to have a direct sum decomposition indexed by P (K)-(□) which is strictly functorial if G is abelian. Given a class p:X→Y, there is a Hopf invariant HIp on [A, Y] which extends Hopf's definition when p is a comultiplication. Then HI=HIp is a functorial sum of HIL over L⊂K, □L□ ≥2. Each HIL is a functorial composition of four functors, the first depending only on A n+1, the second only on d, the third only on p, and the fourth only on Yn. There is a connection here with Selick and Walker's work, and with the Hilton matrix calculus, as described by Bokor (1991).