A system of unstable higher Toda brackets
type: Item
instance of: scholarly article
title: A system of unstable higher Toda brackets
zbMATH Open document ID: 1505.55024
author: Katsumi Ōshima
author: Hideaki Ōshima
published in: Journal of the Mathematical Society of Japan
publication date: 2022-10-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/2004.11047
review text: 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. O…
reviewed by: Dae-Woong Lee
Mathematics Subject Classification ID: 55P99
Mathematics Subject Classification ID: 55Q05
zbMATH DE Number: 7608348
zbMATH Keywords: unstable higher Toda brackets
zbMATH Keywords: generalized Hopf invariants
MaRDI profile type: Publication
OpenAlex ID: W4205204969
arXiv ID: 2004.11047
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DOI: 10.2969/JMSJ/86558655
description: scientific article