On the higher order exterior and interior Whitehead products

Abstract

We extend the notion of the exterior Whitehead product for maps αi:ΣAi→Xi for i= 1, …, n, where Σ Ai is the reduced suspension of Ai and then, for the interior product with Xi=Jmi(X), the mith-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 X1, …, Xn of simply connected co-H-spaces to obtain a higher Gray–Whitehead product map (Formula presented.), where T1(X1, …, Xn) is the fat wedge of X1, …, Xn.