On the higher order exterior and interior Whitehead products

Abstract

We extend the notion of the exterior Whitehead product for maps $$\alpha _i{:}\,\Sigma A_i \rightarrow X_i$$ for $$i=1,\ldots ,n$$ , where $$\Sigma A_i$$ is the reduced suspension of $$A_i$$ and then, for the interior product with $$X_i=J_{m_i}(X)$$ , 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 $$\circ $$ (called the Theriault product) to a sequence $$X_1,\ldots ,X_n$$ of simply connected co-H-spaces to obtain a higher Gray–Whitehead product map $$\begin{aligned} w_n{:}\,\Sigma ^{n-2}(X_1\circ \cdots \circ X_n)\rightarrow T_1(X_1,\ldots ,X_n), \end{aligned}$$ where $$T_1(X_1,\ldots ,X_n)$$ is the fat wedge of $$X_1,\ldots ,X_n$$ .