Principal bundle maps via rational homotopy theory
type: Item
instance of: scholarly article
title: Principal bundle maps via rational homotopy theory
zbMATH Open document ID: 1032.55005
author: H. Shiga
author: Toshihiro Yamaguchi
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2004-03-08T00:00:00Z
review text: The authors study the space \(\text{aut }P\) of self equivalences of a principal \(S^1\)-bundle \(P\) over \(B\), and the group \(\Aut P\) of homotopy classes of elements. For \(S^1\)-equivariant self-equivalences, they write \(\text{aut}_{S^1}P\) and \(\Aut_{S^1}P\). There is a natural forgetful map \[ \Aut_{S^1} P\to \Aut P. \] Finally, they write \(\text{aut}_{\text{Id}}B\) for the identity component of \(\text{aut}_B\). The main result is that when \(\pi_2(\text{aut}_{\text{Id}}B)\otimes \mathbb{Q}= 0\), then the kernel of the forgetful map is a finite group. The methods use the ideas of \textit{D. H. Gottlieb} [Trans. Am. Math. Soc. 171, 23-50 (1972; Zbl 0251.55018)] and the theory of minimal models. A corollary treats the case when the base space is a homogeneous space \(G/U\), with \(G\) and \(U\) of equal rank.
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 55R10
zbMATH DE Number: 1917049
zbMATH Keywords: homotopy self-equivalences
reviewed by: Donald W. Kahn
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1145476148
OpenAlex ID: W2041204388
DOI: 10.2977/PRIMS/1145476148
description: scientific article; zbMATH DE number 1917049