arXiv subjects

Toshiyuki Miyauchi

Publications and source records attributed to Toshiyuki Miyauchi.

9 recordsLinked to original sources

arXiv2026

Relations in the 24-th homotopy groups of spheres

The main purpose of this note is to give a proof of the fact that the Toda brackets \ $\langle\barν,σ,\barν\rangle$ and $\langleν,η, \barσ\rangle$ are not trivial. This is an affirmative answer of M.~Mahowald's Conjecture (J. Mukai, Determination of the $P$-image by Toda brackets, Geometry and Topology Monographs \textbf{13}(2008), 355--383). The second purpose is to determine the relations including $\barν_6ω_{14}$ in $π^6_{30}$ and $\barν_7ω_{15}$ in $π^7_{31}$. To this end, we provide relations between the Toda bracket and the $J$-homomorphism, and between the Toda bracket and the generalized $P$-homomorphism.

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arXiv2024

An Unstable Approach to the May-Lawrence Matrix Toda bracket and the \textit{2}nd James-Hopf Invariant

In this paper, we give an unstable approach of the May-Lawrence matrix Toda bracket, which becomes a useful tool for the theory of determinations of unstable homotopy groups. Then, we give a generalization of the classical isomorphisms between homotopy groups of $(JS^{m},S^{m}) $ and $(JS^{2m},*)$ localized at 2. After that we provide a generalized $H$-formula for matrix Toda brackets. As an application, we show a new construction of $\ct'\inπ_{26}(S^{6})$ localized at 2 which improves the construction of $\ct'$ given by \cite{20STEM}.

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arXiv2023

Homotopy commutativity in symmetric spaces

We extend the former results of Ganea and the two of the authors with Takeda on the homotopy commutativity of the loop spaces of Hermitian symmetric spaces such that the loop spaces of all irreducible symmetric spaces but $\mathbb{C}P^3$ are not homotopy commutative.

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arXiv2021

The 23-rd and 24-th homotopy groups of the n-th rotation group

We denote by $π_k(R_n)$ the $k$-th homotopy group of the $n$-th rotation group $R_n$ and $π_k(R_n:2)$ the 2-primary components of it. We determine the group structures of $π_k(R_n:2)$ for $k = 23$ and $24$ by use of the fibration $R_{n+1}\overset{R_n}{\longrightarrow}S^n$. The method is based on Toda's composition methods.

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arXiv2019

Higher homotopy associativity in the Harris decomposition of Lie groups

Let $(G,H)=(SU(2n+1),SO(2n+1)),\,(SU(2n),Sp(n)),\,(SO(2n),SO(2n-1)),\,(E_6,F_4),\,(Spin(8),G_2)$, and let $p$ be any prime $\ge 5$ for $(G,H)=(E_6,F_4)$, any prime $p\ne 3$ for $(G,H)=(Spin(8),G_2)$, and any odd prime otherwise. The classical result of Harris on the relation between the homotopy groups of $G$ and $H$ is reinterpreted as a $p$-local homotopy equivalence $G\simeq_{(p)}H\times G/H$, which yields a projection $G_{(p)}\to H_{(p)}$. We show how much this projection preserves the higher homotopy associativity.

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arXiv2017

Samelson products in quasi-$p$-regular exceptional Lie groups

There is a product decomposition of a compact connected Lie group $G$ at the prime $p$, called the mod $p$ decomposition, when $G$ has no $p$-torsion in homology. Then in studying the multiplicative structure of the $p$-localization of $G$, the Samelson products of the factor space inclusions of the mod $p$ decomposition are fundamental. This paper determines (non-)triviality of these fundamental Samelson products in the $p$-localized exceptional Lie groups when the factor spaces are of rank $\le 2$, that is, $G$ is quasi-$p$-regular.

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arXiv2017

On Mimura's extension problem

We determine the group strucure of the $23$-rd homotopy group $π_{23}(G_2 : 2)$, where $G_2$ is the Lie group of exceptional type, which hasn't been determined for $50$ years.

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arXiv2013

On Lusternik-Schnirelmann category of SO(10)

Let $G$ be a compact connected Lie group and $p : E\to ΣA$ be a principal G-bundle with a characteristic map $α: A\to G$, where $A=ΣA_{0}$ for some $A_{0}$. Let $\{K_{i}{\to} F_{i-1}{\hookrightarrow} F_{i} \,|\, 1{\le} i {\le} n,\, F_{0}{=} \{\ast\} \; F_{1}{=} Σ{K_{1}} \; \text{and}\; F_{n}{\simeq} G \}$ be a cone-decomposition of $G$ of length $m$ and $F'_{1}=Σ{K'_{1}} \subset F_{1}$ with $K'_{1} \subset K_{1}$ which satisfy $F_{i}F'_{1} \subset F_{i+1}$ up to homotopy for any $i$. Our main result is as follows: we have $\operatorname{cat}(X) \le m{+}1$, if firstly the characteristic map $α$ is compressible into $F'_{1}$, secondly the Berstein-Hilton Hopf invariant $H_{1}(α)$ vanishes in $[A, ΩF'_1{\ast}ΩF'_1]$ and thirdly $K_{m}$ is a sphere. We apply this to the principal bundle $\mathrm{SO}(9)\hookrightarrow\mathrm{SO}(10)\to S^{9}$ to determine L-S category of $\mathrm{SO}(10)$.

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arXiv2012

Relations in the 24-th homotopy groups of spheres

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 $π^7_{31}$.

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