On the higher order exterior and interior Whitehead products

We extend the notion of the exterior Whitehead product for maps $\alpha_i:\Sigma A_i \to X_i$ for $i=1,\dots,n$, where $\Sigma A_i$ is the reduced suspension of $A_i$ and then, for the interior product with $X_i=J_{m_i}(X)$ as well. The main result stated in Theorem 3.10 generalizes Theorem 1.10 in K.\ A.\ Hardie, \textit{A generalization of the Hopf construction}, Quart.\ J.\ Math.\ Oxford Ser.\ (2) \textbf{12} (1961), 196--204. and concerns to the Hopf invariant of the generalized Hopf construction. We close the paper applying the Gray's construction $\circ$ (called the Theriault product) to a sequence $X_1,\dots,X_n$ of simply connected co-$H$-spaces to obtain a higher Gray--Whitehead product map \[w_n:\Sigma^{n-2}(X_1\circ\dots\circ X_n)\to T_1(X_1,\dots,X_n),\] where $T_1(X_1,\dots,X_n)$ is the fat wedge of $X_1,\dots,X_n$.