Long box bracket operations in homotopy theory
The triple bracket of H. Toda is a secondary composition operation which has played a fundamental role in computations of homotopy groups of spheres. It is defined for a sequence of maps \(X_1 \overset {f_1}\longrightarrow X_2 \overset {f_2}\longrightarrow X_3 \overset {f_3}\longrightarrow X_4\) such that both compositions \(f_2f_1\) and \(f_3f_2\) are null-homotopic, and consists of all homotopy classes of maps \(\Sigma X_1\to X_4\) of the form \(T(h_1,h_2):=h_2f_1\underset{f_3f_2f_1}\cup f_3h_1\), for some null-homotopies \(h_i:f_{i+1}f_i\sim *\). (All spaces, maps and homotopies are assumed to be pointed in this review.) On interpreting the data of the Toda bracket as a homotopy commutative diagram \[ \begin{tikzcd} X_1 \rar["f_1"]\dar & X_2 \rar\dar["f_2" '] & \ast\dar\\ \ast \rar & X_3 \rar["f_3" '] & X_4 \end{tikzcd} \] the Toda bracket is a special case of the double box bracket operation studied by the authors and \textit{K. A. Hardie} [Rend. Ist. Mat. Univ. Trieste 33, No.~1--2, 19--70 (2001; Zbl 1005.18007)]. The latter is defined for a homotopy-commutative diagram \[ \begin{tikzcd} A_1 \rar["a_1"]\dar["f_1" '] & A_2 \rar["a_2"]\dar["f_2" '] & A_3\dar["f_3"]\\ B_1 \rar["b_1" '] & B_2 \rar["b_2" '] & B_3 \end{tikzcd} \] with null-homotopic row compositions, and consists of all homotopy classes of maps \(\Sigma A_1\to B_3\) of the form \(BB(\beta,\varphi_1,\varphi_2,\alpha):=\beta f_1\underset{b_2b_1f_1}\cup b_2\varphi_1\underset{b_2f_2a_1}\cup\varphi_2a_1\underset{f_3a_2a_1}\cup f_3\alpha\), for some homotopies \(\varphi_i:b_if_i\sim f_{i+1}a_i\), \(i=1,2\), and some null-homotopies \(\alpha:a_2a_1\sim *\) and \(\beta:*\sim b_2b_1\). In the present paper, the authors introduce higher order box brackets and study in detail the triple box bracket, which they relate to the quaternary Toda bracket of K. Ôguchi. As noted in the introduction, different authors including E. Spanier, B. Gray, J. Cohen and G. Walker ``have employed different methods to define long Toda brackets, but all definitions are quite complicated because coherence conditions for higher homotopies arise''. In fact, it is shown in the paper under review (see Remark 4.4) that Gray's \(4\)-fold bracket is generally a larger operation than Ôguchi's quaternary Toda bracket, although no example confirming that the inclusion is strict seems to be provided. The triple box bracket is defined for a homotopy-commutative diagram \[ \begin{tikzcd} A_1 \rar["a_1"]\dar["f_1" '] & A_2 \rar["a_2"]\dar["f_2" '] & A_3 \rar["a_3"]\dar["f_3"] & A_4 \dar["f_4"]\\B_1 \rar["b_1" '] & B_2 \rar["b_2" '] & B_3 \rar["b_3" '] & B_4 \end{tikzcd} \] with null-homotopic double compositions in rows such that there exists a choice of homotopies \(\varphi_i:b_if_i\sim f_{i+1}a_i\), \(i=1,2,3\) and null-homotopies \(\alpha_i:a_{i+1}a_i\sim *\) and \(\beta_i:*\sim b_{i+1}b_i\), \(i=1,2\), that is ``coherent'' in the sense that the maps \(T(\alpha_1,\alpha_2):\Sigma A_1\to A_4\) and \(T(\beta_1,\beta_2):\Sigma B_1\to B_4\), as well as \(BB(\beta_i,\varphi_i,\varphi_{i+1},\alpha_i):\Sigma A_i\to B_{i+2}\), \(i=1,2\), are null-homotopic. A coherent choice of homotopies \(\alpha_1,\alpha_2,\beta_1,\beta_2,\varphi_1,\varphi_2,\varphi_3\) yields a double-square diagram \[ \begin{tikzcd}\Sigma A_1 \rar\dar["\Sigma f_1" '] & C(a_2) \rar\dar & A_4\dar["f_4"]\\ \Sigma B_1 \rar & C(b_2) \rar & B_4, \end{tikzcd} \] where \(C(f)\) denotes the mapping cone of \(f\), and all horizontal arrows are of the form \(h_*:C(g_1)\to C(g_2)\), corresponding to a homotopy \(h\) between two compositions \(X_1\longrightarrow X_2\overset {g_2}\longrightarrow Y_2\) and \(X_1\overset {g_1}\longrightarrow Y_1\longrightarrow Y_2\). Due to the null-homotopies of \(BB(\beta_i,\varphi_i,\varphi_{i+1},\alpha_i)\), \(i=1,2\), the double-square diagram is homotopy commutative, and due to the null-homotopies of \(T(\alpha_1,\alpha_2)\) and \(T(\beta_1,\beta_2)\), it has null-homotopic row compositions. Hence its double box bracket is defined, as a subset of the pointed homotopy set \([\Sigma^2 A_1,B_4]\). Now the triple box bracket of the original triple-square diagram is defined to be the union of all such double box brackets, for all coherent choices of homotopies \(\alpha_1,\alpha_2,\beta_1,\beta_2,\varphi_1,\varphi_2,\varphi_3\). The authors use the triple box bracket (i) to elaborate on Oda's 1979 computation of \(\pi_{32}(S^7)\) up to isomorphism, by providing an explicit presentation for this group (one of the generators is given by a triple box bracket), and (ii) to show that the quaternary Toda bracket of Ôguchi is defined and nontrivial for the sequence \(\begin{tikzcd} S^{30}\rar["\sigma_{23}"] & S^{23}\rar["\eta_{15}\circ\sigma_{16}"] & S^{15} \rar["2\iota_{15}"] & S^{15} \rar["\bar\nu_7"] & S^7\end{tikzcd}\) (in Toda's notation). The second half of the paper is devoted to a version of Toda's formula for triple box brackets, which is in turn applied to show triviality of the triple box brackets for some specific triple-square diagrams of maps between spheres, and to show that the quaternary box bracket is defined for a certain specific quaternary-square diagram of maps between spheres. The authors warn that the quaternary box bracket operation is different from the box quartet operation which they studied elsewhere.
scientific article; zbMATH DE number 5871634