On the higher order exterior and interior Whitehead products

We extend the notion of the exterior Whitehead product for maps α i : Σ A i → X i for i = 1 , … , n , where Σ A i is the reduced suspension of A i and then, for the interior product with X i = J m i ( X ) , the m i th-stage of the James construction J ( X ) as well. The main result stated in Theorem 4.10 generalizes (Hardie in Q J Math Oxford Ser 12(2):196–204, 1961 , Theorem 1.10) and concerns to the Hopf invariant of the generalized Hopf construction. We close the paper applying Gray’s construction ∘ (called the Theriault product) to a sequence X 1 , … , X n of simply connected co- H -spaces to obtain a higher Gray–Whitehead product map w n : Σ n - 2 ( X 1 ∘ ⋯ ∘ X n ) → T 1 ( X 1 , … , X n ) , where T 1 ( X 1 , … , X n ) is the fat wedge of X 1 , … , X n .