Long box bracket operations in homotopy theory
type: Item
instance of: scholarly article
title: Long box bracket operations in homotopy theory
zbMATH Open document ID: 1227.55015
author: Howard J. Marcum
author: Nobuyuki Oda
published in: Applied Categorical Structures
publication date: 2011-03-30T00:00:00Z
review text: 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 T…
Mathematics Subject Classification ID: 55Q40
Mathematics Subject Classification ID: 55Q35
Mathematics Subject Classification ID: 18G55
Mathematics Subject Classification ID: 18D05
zbMATH DE Number: 5871634
zbMATH Keywords: 2-category
zbMATH Keywords: suspension functor
zbMATH Keywords: Toda bracket (quaternary, long)
zbMATH Keywords: box bracket (triple, long)
zbMATH Keywords: homotopy groups of spheres
reviewed by: Sergey A. Melikhov
MaRDI profile type: Publication
full work available at URL: s10485-009-9186-3
OpenAlex ID: W1986790303
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DOI: 10.1007/S10485-009-9186-3
description: scientific article; zbMATH DE number 5871634