Hopf invariants, Toda brackets and the reduced diagonal map (Q616938)

The decomposition of the reduced diagonal map, \(\overline{\Delta}_{n}: X\to X^{\land n}\), of a space, \(X\), in function of some Hopf invariants was introduced by \textit{J. M. Boardman} and \textit{B. Steer} [Comment. Math. Helv. 42, 180--221 (1967; Zbl 0166.19201)] for \(n=2\) and used by \textit{L. Fernández-Suárez, A. Gómez-Tato, J. Strom} and the reviewer [Proc. Am. Math. Soc. 132, No. 2, 587--595 (2004; Zbl 1038.55004)], in the case \(n=5\), for the determination of the Lusternik-Schnirelmann category of the Lie group \(Sp(3)\). In the paper under review, the authors explore relationships between reduced diagonal maps and Hopf invariants, in terms of conic structures of a space. As applications, using the fact that certain Toda brackets contain crude Hopf invariants as elements, they can detect some crude Hopf invariants. An explicit example is given.