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 HIρ on [A,Y] which extends Hopf’s definition when ρ is a comultiplication. Then HI = HIρ 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 An+1, the second only on d, the third only on ρ, 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).