Homotopy nilpotency of some homogeneous spaces
type: Item
instance of: scholarly article
title: Homotopy nilpotency of some homogeneous spaces
zbMATH Open document ID: 1486.55013
author: Marek Golasiński
published in: Manuscripta Mathematica
publication date: 2022-01-14T00:00:00Z
review text: Let \(\mathbb{K}P^n\) be the projective \(n\)-space for \(\mathbb{K} = \mathbb{R}, \mathbb{C}\), the field of reals or complex numbers and \(\mathbb{H}\), the skew \(\mathbb{R}\)-algebra of quaternions. In [J. Math. Mech. 16, 853--858 (1967; Zbl 0148.17104)], \textit{T. Ganea} developed the homotopy nilpotency of the loop spaces \(\Omega (\mathbb{K}P^n)\), and in [Osaka J. Math. 13, 145-156 (1976; Zbl 0329.55010)], \textit{V. P. Snaith} gave necessary and sufficient conditions for an \(H\)-space to have homotopy nilpotency equal to \(n\). In [Math. Z. 161, 169--183 (1978; Zbl 0367.55009)], \textit{W. Meier} studied relations between the homotopy nilpotency of a loop space and its localizations and profinite completions. Let \(G_{n,m}(\mathbb{K})\), \(F_{n;n_1,\ldots,n_k} (\mathbb{K})\) and \(V_{n,m}(\mathbb{K})\) be the Grassmann, flag and Stiefel manifolds in the \(n\)-dimensional \(\mathbb{K}\)-vector space, respectively. In this paper under review, the author investigates the homotopy nilpotency of the loop spaces on \(G_{n,m}(\mathbb{K})\), \(F_{n;n_1,\ldots,n_k} (\mathbb{K})\) and \(V_{n,m}(\mathbb{K})\), and homotopy nilpotency classes of \(p\)-localized loop spaces \(\Omega(G_{n,m}^+ (\mathbb{K})_{(p)})\) and \(\Omega(V_{n,m} (\mathbb{K})_{(p)})\) at a prime \(p\), where \(G_{n,m}^+ (\mathbb{K})\) is the oriented Grassmann manifold. Moreover, the author calculates the nilpotency of the Cohen group \([J(X), \Omega(Y)]\), where \(J(X)\) is the James reduced product of a space \(X\): For any space \(X\), (1) if \(n \geq 1\), then \(\operatorname{nil}[J(X), \Omega(Y)] < \infty\), where \(Y = SO(2n)/U(n)\), \(Sp(n)/U(n)\), \(SU(2n)/Sp(n)\), \(SO(4n)/Sp(n)\); (2) if \(m<n \leq \infty\), then \(\operatorname{nil}[J(X), \Omega(Y)] < \infty\), where \(Y\) is the (oriented) Grassmann and flag manifolds and their localizations with additional conditions; and (3) if \(1\leq m \leq n\), then \(\operatorname{nil}[J(X), \Omega(Y)] < \infty\), where \(Y = V_{n,m}(\mathbb{C})…
reviewed by: Dae-Woong Lee
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55T20
zbMATH DE Number: 7457114
zbMATH Keywords: homotopy nilpotency
zbMATH Keywords: loop space
zbMATH Keywords: homogeneous space
MaRDI profile type: Publication
full work available at URL: s00229-021-01273-y
OpenAlex ID: W3128385858
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DOI: 10.1007/S00229-021-01273-Y
description: scientific article; zbMATH DE number 7457114