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 ) − { ϕ } which is strictly functorial if G is abelian. Given a class ρ : X → Y , there is a Hopf invariant HI ρ on [ A , Y ] which extends Hopf's definition when ρ is a comultiplication. Then HI = HI ρ is a functorial sum of HI L over L ⊂ K , ‖ L ‖ ≥ 2 . Each HI L is a functorial composition of four functors, the first depending only on A n + 1 , the second only on d , the third only on ρ , and the fourth only on Y n . There is a connection here with Selick and Walker's work, and with the Hilton matrix calculus, as described by Bokor (1991).