This paper is expository and a record of two talks which I gave at Seoul in Korea, 2007 (Cf.10) and at Halifax in Canada, 2008.Let B be the mapping cone of a map ϕ : A → X.Then, Toda(1956) defined a map T : Ω(B, X) → Ω 2 (ΣA ∧ B) where Ω(B, X) is the homotopy fiber of the inclusion map X → B. Boardman-Steer(1967) and Ganea(1968), independently, constructed a map μ : ΩB → Ω 2 (ΣA ∧ B) using the coaction map of B. Boardman-Steer obtained a very useful result about the functional cup products.These maps, in fact, factor through Ω 2 (ΣA × B, ΣA ∨ B), so they are "delicate Hopf invariants".I describe these construction and show that they are essentially the same and that these invariants have some useful properties related to the following James's old homotopy exact sequence(1954) for some range.here the above exact sequence can be regarded as "relative EHP-sequence".
Kaoru Morisugi (Fri,) studied this question.