Approximating the homotopy sequence of a pair of spaces (Q1177317)

Let \((X,A)\) be a pair of spaces and \(h: X\rightarrow B\) be a map such that \(h(A)=*\). Replacing \(\pi_ n(X,A)\) by \(\pi_ n(B)\) via \(h_*\) in the exact sequence of homotopy groups of the pair \((X,A)\), some other sequence \[ \longrightarrow \pi_nA \overset{i_*}{\longrightarrow} \pi_ nX \overset{H}{\longrightarrow} \pi_ nB \overset{\Delta}{\longrightarrow} \pi_{n-1}A \overset{i_ *}{\longrightarrow} \pi_{n-1}X \longrightarrow \] is obtained, which is no longer exact. Also, \(\Delta\) is in general not a map, but something similar. Toda brackets are used to define \(\Delta^\leftarrow\): \[ \Delta^ \leftarrow(\mu)=-\{^ \circ\{h\},\{i\},\mu\}\subset\pi_ nB, \] for every \(\mu\in\pi_{n-1}A\) with \(i_*(\mu)=0\), where the little circle indicates that the trivial nullhomotopy for \(hi=*\) should be taken for the brackets. It is shown that the indeterminacy of \(\Delta^\leftarrow\) is the image of \(H\). Further investigations are done in the special case where \(i\) factors as \(i=i''\circ i'\): \[ A \overset{i'}{\longrightarrow} A\cup_ \lambda CM \overset{i''}{\longrightarrow} X\] with some \(\lambda\in\pi(M,A)\). The following main Proposition is proved: 0.6. Proposition. Let \(i=i''\circ i'\) as above, \(j: \Sigma M\rightarrow B\) be the map induced by \(h\circ i''\). If \(\gamma\in\pi_ n M\) is such that \(\lambda\circ\gamma=0\), then there exists an element \(\xi\in\pi_{n+1}X\) such that \(H\xi=j_*E_\gamma\). Moreover if \(m\gamma=0\) for some \(m\in{\mathbb{N}}\) then \(m\xi\in-i_*\{\{\lambda\}, \{\gamma\}, m\iota_ n\}\). The examples in mind are the canonical maps \(h_t: (S^n_ \infty,S^n_{t-1}) \rightarrow (S^{tn}_\infty,*)\), (\(t\in{\mathbb{N}}, t>1\)), leading to the famous \(EH\Delta\) sequences considered by \textit{I. M. James} [Ann. Math. (2) 62, 170-197 (1955; Zbl 0064.41505)]\ and \textit{H. Toda} [Composition methods in homotopy groups of spheres. Princeton, N. J.: Princeton University Press (1962; Zbl 0101.40703)]. The map \(h_t\) induces a generalized higher Hopf invariant \(H_t\). For \(t=2\) the map \(\lambda\) above is usually the Whitehead product \([\iota_n,\iota_n]\). As applications for homotopy groups of spheres the image \(H_t\) is related to certain Toda brackets and other elements. Concerning these examples the author remarks: ``Proposition 0.6. specializes to a result that appears new''.