A system of unstable higher Toda brackets
One of the important tools in classical homotopy theory is the Toda bracket [\textit{H. Toda}, Composition methods in homotopy groups of spheres. Princeton, NJ: Princeton University Press (1962; Zbl 0101.40703)]. A number of definitions of higher Toda brackets (\(n\)-fold brackets for \(n \geq 3\)) have been investigated as a generalization of the classical Toda bracket. The followings are extracts from this paper based on the pointed homotopy category. \begin{itemize} \item \(n\) is a positive integer which is bigger than or equal to \(3\). \item \((X_{n+1}, \ldots, X_1)\) is a sequence of well-pointed spaces. \item \(\overrightarrow{\boldsymbol m} = (m_n,\ldots , m_1)\) is a sequence of non-negative integers. \item \(\overrightarrow{\boldsymbol f} = (f_n,\ldots , f_1)\) is a sequence of morphisms \(f_k : \Sigma^{m_k} X_k \rightarrow X_{k+1}\). \item \(A : CX \rightarrow Z\) is a null homotopy of \(g \circ f : X \overset{-f}\longrightarrow Y \overset{-g}\longrightarrow Z\); that is, \(A : g \circ f \simeq *\). \item \([g,A,f] : Y \cup_f CX \rightarrow Z\) is a morphism given by \(g\) and \(A\) on \(Y\) and \(CX\), respectively. \item \((g,A,f) : \Sigma X \rightarrow Z\cup_g CY\) is a morphism given by \[ (g,A,f)(x \wedge t) = \begin{cases} f(x) \wedge (1 - 2t) &\text{for} \,\, t \in [0,\frac{1}{2}], \\ A(x \wedge (2t-1)) &\text{for} \,\, t \in [\frac{1}{ 2}, 1]. \end{cases} \] \item \(\tilde \Sigma^m H : \Sigma^m X \times I \rightarrow \Sigma^m Y\) is a morphism given by \(\tilde \Sigma^m H (x \wedge s,t) = H(x,t) \wedge s\) for a homotopy \(H : X \times I \rightarrow Y\). \end{itemize} Under the above data, the authors defined the \textit{unstable \(n\)-fold Toda bracket} \(\{\overrightarrow{\boldsymbol f}\}^{(\text{\"{s}}_t )}_{\overrightarrow{\boldsymbol m}}\) inductively depending only on the homotopy classes of \(f_k\) for \(1\leq k \leq n\) as a subset of \([\Sigma^{n-2} \Sigma^{m_n} \cdots \Sigma^{m_1} X_1 , X_{n+1}]\) in [\textit{H. Oshima} and \textit{K. Oshima}, Math. J. Okayama Univ. 62, 27--86 (2020; Zbl 1437.55018)]. In this paper under review, the authors show that \[ \begin{array}{lll} \{\overrightarrow{\boldsymbol f}\}^{(\text{\"{s}}_t )}_{\overrightarrow{\boldsymbol m}} &= \quad \bigcup \,\, \{f_n, \ldots, f_4, [f_3,A_2,\Sigma^{m_3} f_2], (\Sigma^{m_3}f_2, \tilde \Sigma^{m_3}A_1, \Sigma^{m_3}\Sigma^{m_2}f_1 )\}^{(\text{\"{s}}_t )}_{(m_n, \ldots, m_4, 0,0)} \\ & \qquad \qquad \circ (1_{\Sigma^{m_3}\Sigma^{m_2} \Sigma^{m_1} X_1} \wedge \tau(\text{S}^{m_4} \wedge\cdots \wedge \text{S}^{m_n}, \text{S}^{1}) \wedge 1_{(\text{S}^{1})^{\wedge (n-3)}} ) \end{array} \] for \(n \geq 4\), where \(\tau\) is the switching map, and the union is taken over (1) all pairs \((A_2, A_1)\) of null homotopies \(A_2 : f_3 \circ \Sigma^{m_3}f_2 \simeq *\) and \(A_1 : f_2 \circ \Sigma^{m_2}f_1 \simeq *\), or (2) all (admissible) sequences \((A_{n-1}, \ldots, A_1)\) of null homotopies \(A_k : f_{k+1} \circ \Sigma^{k+1}f_k \simeq *\). The authors prove that the generalized Hopf invariant preserves the unstable \(n\)-fold Toda bracket for a sequence of connected CW-complexes \((\text{S}^{m+1}, X_{n}, \ldots, X_1)\), and make some corrections of errors in [loc. cit.].
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