Unstable higher Toda brackets III
Hideaki Oshima, Katsumi Oshima
本文在带基点空间范畴中归纳构造了 n>2 的不稳定 n 重 Toda 括号，并进一步定义了相应的稳定 Toda 括号。
我们在带基点的空间范畴中，对每个 n>2 归纳地定义了不稳定的 n 重 Toda 括号，随后定义了稳定的 n 重 Toda 括号。
We define inductively unstable n-fold Toda brackets for every n>2 in the category of spaces with base points, and then define stable ones.
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We define subscripted unstable higher Toda brackets and study their elementary properties. This paper is the continuation of our previous paper in which we defined the non-subscripted unstable higher Toda brackets.
We show that a system of unstable higher Toda brackets can be defined inductively.
This paper investigates the relation between Toda brackets and congruences of modular forms. It determines the $f$-invariant of Toda brackets and thereby generalizes the formulas of J.F.\ Adams for the classical $e$-invariant to the chromatic second filtration.
We define two new unstable n-fold Toda brackets for every composable sequence (f_n, ... ,f_1) of pointed maps f_i : X_i \to X_{i+1} between well-pointed spaces with n > 2. The brackets agree with the classical Toda bracket when n = 3, and they are subsets of both the unstable n-fold Toda brackets of Gershenson and Cohen for every n > 2.
We give two formulas for the generalized Hopf invariant and 4-fold Toda brackets which are useful in computations of homotopy groups of spheres.
We provide a general definition of Toda brackets in a pointed model categories, show how they serve as obstructions to rectification, and explain their relation to the classical stable operations.
On the generalized Jacobi Identity of Toda brackets
本文将 Toda 括号的 Jacobi 恒等式从指标 0 推广到任意 $n \geq 0$，给出了更一般的恒等式形式。
We describe two ways to define higher order Toda brackets in a pointed simplicial model category $\mathcal{D}$: one is a recursive definition using model categorical constructions, and the second uses the associated simplicial enrichment. We show that these two definitions agree, by providing a third, diagrammatic, description of the Toda bracket, and explain how it serves as the obstruction to rectifying a certain homotopy-commutative diagram in $\mathcal{D}$.
In this paper we prove the laws of Toda brackets on the homotopy groups of a connective ring spectrum and the laws of the cup-one square in the homotopy groups of a commutative connective ring spectrum.
On rectangular Toda brackets and Oda's extension problems
本文引入矩形Toda括号工具，解决了Oda关于在2处局部化的三个球面同伦群的扩张问题，这些难题四十多年来未被攻克。
Motivic Toda brackets
本文在不稳定motivic同伦论中构造Toda brackets，证明其基本性质，并给出若干具体例子。
We provide a uniform definition of higher order Toda brackets in a general setting, covering the known cases of long Toda brackets for topological spaces and chain complexes and Massey products for differential graded algebras, among others.
An Unstable Approach to the May-Lawrence Matrix Toda bracket and the \textit{2}nd James-Hopf Invariant
本文提出 May-Lawrence 矩阵 Toda 括号的不稳定方法，推广相关同伦群同构并建立广义 $H$-公式，从而给出 2 局部化元素 $\ct'\inπ_{26}(S^6)$ 的新构造。
A Toda bracket convergence theorem for multiplicative spectral sequences
该研究在对称幺模稳定拓扑模型范畴中推广了联系Massey乘积与Toda括号的Moss定理，使其适用于幺模相容滤过导出的乘性谱序列，可应用于motivic相关谱序列的计算。
The present paper gives a proof of the author's paper in the Proceeding of the International Conference on Homotopy Theory and Related Topics at Korea University (2005), 109--113, on the orders of Whitehead products of iota_n with alpha in pi^n_{n+k}, (n > k+1, k < 25) and improves and extends it. The method is to use composition methods in the homotopy groups of spheres and rotation groups.
Toda brackets in n-angulated categories
本文为n-角范畴引入Toda括号，证明其定义一致性、杂耍公式及Heller定理的推广，并给出计算实例及与三角化情形的关系。
The Adams spectral sequence is available in any triangulated category equipped with a projective or injective class. Higher Toda brackets can also be defined in a triangulated category, as observed by B. Shipley based on J. Cohen's approach for spectra. We provide a family of definitions of higher Toda brackets, show that they are equivalent to Shipley's, and show that they are self-dual. Our main result is that the Adams differential $d_r$ in any Adams spectral sequence can be expressed as an $(r+1)$-fold Toda bracket and as an $r^{\text{th}}$ order cohomology operation. We also show how the …
We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of R-module spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an R-module spectrum. For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex K-theory spectra serve as our main examples.
Relations in the 24-th homotopy groups of spheres
本文证明了Toda括号⟨ν̄,σ,ν̄⟩和⟨ν,η,σ̄⟩非平凡，肯定了Mahowald猜想，并确定了π^6_{30}中ν̄_6ω_{14}和π^7_{31}中ν̄_7ω_{15}的关系。
The main purpose of this note is to give a proof of the fact that the Toda brackets $<\barν,σ,\barν>$ and $<ν,η, \barσ>$ are not trivial. This is an affirmative answer of the second author's Conjecture (Determination of the $P$-image by Toda brackets, Geometry and Topology Monographs 13(2008), 355-383). The second purpose is to show the relation $\barν_7ω_{15}=ν_7σ_{10}κ_{17}$ in $π^7_{31}$.
Using a Toda bracket computation $\langle θ_4, 2, σ^2\rangle$ due to Daniel C. Isaksen [11], we investigate the $45$-stem more thoroughly. We prove that $θ_4^2=0$ using a $4$-fold Toda bracket. By [2], this implies that $θ_5$ exists and there exists a $θ_5$ such that $2θ_5=0$. Based on $θ_4^2=0$, we simplify significantly the $9$-cell complex construction in [1] to a $4$-cell complex, which leads to another proof that $θ_5$ exists.
We construct non-trivial elements of order 2 in the homotopy groups $π_{8j+1+*} Diff(D^6,\partial)$, for * congruent 1 or 2 modulo 8, which are detected by the "assembling homomorphism" (giving rise to the Gromoll filtration), followed by the alpha-invariant in $KO_*=Z/2$. These elements are constructed by means of Morlet's homotopy equivalence between $Diff(D^6,\partial)$ and $Ω^7(PL_6/O_6)$, and Toda brackets in $PL_6/O_6$. We also construct non-trivial elements of order 2 in $π_* PL_m$ for every m greater or equal to 6 and * congruent to 1 or 2 modulo 8, which are detected by the alpha-inva…
We determine the L-S category of Sp(3) by showing that the 5-fold reduced diagonal $\widebarΔ_5$ is given by $ν^2$, using a Toda bracket and a generalised cohomology theory $h^*$ given by $h^*(X,A) = \{X/A,{\mathbb S}[0,2]\}$, where ${\mathbb S}[0,2]$ is the 3-stage Postnikov piece of the sphere spectrum ${\mathbb S}$. This method also yields a general result that $\cat(Sp(n)) \geq n+2$ for $n \geq 3$, which improves the result of Singhof.
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