On the higher order exterior and interior Whitehead products

TL;DR

This paper generalizes Whitehead products to higher orders, extending classical constructions and applying Gray's product to co-H spaces to produce new higher Whitehead product maps.

Contribution

It introduces a higher order exterior and interior Whitehead product framework and applies Gray's product to construct higher Whitehead maps for co-H spaces.

Findings

Generalization of Whitehead products to higher orders

Extension of the Hopf invariant in this context

Construction of higher Whitehead maps using Gray's product

Abstract

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 $X_1,\dots,X_n$.