A note about \(G\)-equivariant homotopies and isotopies, where \(G\) is a Lie group
type: Item
instance of: scholarly article
title: A note about \(G\)-equivariant homotopies and isotopies, where \(G\) is a Lie group
zbMATH Open document ID: 0930.57032
author: Marja Kankaanrinta
published in: Archiv der Mathematik
publication date: 2000-02-07T00:00:00Z
review text: Let \(G\) be a good Lie group, i.e., a closed subgroup of a Lie group having only finitely many connected components. For example, every linear Lie group is good. Let \(M\) be a proper \(C^r\) \(G\)-manifold \((1 \leq r \leq \omega)\), and \(N\) a \(C^r\) \(G\)-manifold. Assume that \(N\) is also proper if \(G\) has infinitely many connected components. The author proves that (1) a continuous \(G\)-map \(M \to N\) is \(G\)-homotopic to a \(C^r\) \(G\)-map by a homotopy which approximates the constant homotopy, (2) two \(G\)-homotopic \(C^r\) \(G\)-maps \(M \to N\) are \(C^r\) \(G\)-homotopic. He also proves that if \(M\) is a proper \(C^{\omega}\) \(G\)-manifold, then (1) a \(C^{\infty}\) \(G\)-diffeomorphism \(M \to M\) is \(C^{\infty}\) \(G\)-isotopic to a \(C^{\omega}\) \(G\)-diffeomorphism, and (2) two \(C^{\infty}\) \(G\)-isotopic \(C^{\omega}\) \(G\)-diffeomorphisms \(M \to M\) are \(C^{\omega}\) \(G\)-isotopic. The proofs of the case where \(G\) has only finitely many connected components are based on the existence of global slices of proper \(G\)-manifolds.
Mathematics Subject Classification ID: 57S20
Mathematics Subject Classification ID: 57R52
Mathematics Subject Classification ID: 57R91
zbMATH DE Number: 1346259
zbMATH Keywords: proper \(G\)-manifolds
zbMATH Keywords: real analytic \(G\)-manifolds
zbMATH Keywords: \(G\)-homotopies
zbMATH Keywords: \(G\)-isotopies
zbMATH Keywords: global slices
reviewed by: Katsuhiro Komiya
MaRDI profile type: Publication
full work available at URL: s000130050381
OpenAlex ID: W2030898756
Wikidata QID: Q115390130
DOI: 10.1007/S000130050381
description: scientific article; zbMATH DE number 1346259