A new topology for the set \(\text{C}^{\infty,G}(M,N)\) of \(G\)-equivariant smooth maps
type: Item
instance of: scholarly article
title: A new topology for the set \(\text{C}^{\infty,G}(M,N)\) of \(G\)-equivariant smooth maps
zbMATH Open document ID: 0940.57031
author: Marja Kankaanrinta
author: Soeren Illman
published in: Mathematische Annalen
publication date: 2000-07-13T00:00:00Z
review text: Suppose \(M\) and \(N\) are \(C^r\) manifolds, \(1 \leq r \leq \omega\). Let \(C^r(M,N)\) denote the set of \(C^r\) maps from \(M\) to \(N\). The standard choices for a topology on \(C^r(M,N)\) are the weak topology, and the strong topology (also called the Whitney topology). In the paper under review, the authors give a new topology on \(C^r(M,N)\), which is called the strong-weak topology, and is at least as fine as the weak topology and at most as fine as the strong topology. If \(G\) is a Lie group and \(M\), \(N\) are \(C^r\) \(G\)-manifolds, \(C^{r,G} (M,N)\) denotes the set of \(G\)-equivariant \(C^r\) maps from \(M\) to \(N\). The authors also define the strong-weak topology on \(C^{r,G} (M,N)\) to be the relative topology of that of \(C^r (M,N)\). The authors show basic properties of the strong-weak topology. Among others, they show that the strong-weak topology agrees with the strong topology when \(G\) is compact, and that the strong-weak topology agrees with the weak topology when the orbit space \(M/G\) is compact.
Mathematics Subject Classification ID: 57R35
Mathematics Subject Classification ID: 58D10
Mathematics Subject Classification ID: 57S20
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 58D19
Mathematics Subject Classification ID: 57R91
Mathematics Subject Classification ID: 58D15
Mathematics Subject Classification ID: 46E10
zbMATH DE Number: 1406574
zbMATH Keywords: strong-weak topology
zbMATH Keywords: noncompact Lie group actions
zbMATH Keywords: \(G\)-embeddings
zbMATH Keywords: \(G\)-immersions
zbMATH Keywords: weak topology
zbMATH Keywords: strong topology
zbMATH Keywords: Whitney topology
reviewed by: Katsuhiro Komiya
MaRDI profile type: Publication
full work available at URL: s002080050007
OpenAlex ID: W158069766
DOI: 10.1007/S002080050007
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description: scientific article; zbMATH DE number 1406574