Actions of \({\mathbb{R}}^ p\) on closed manifolds
type: Item
instance of: scholarly article
title: Actions of \({\mathbb{R}}^ p\) on closed manifolds
zbMATH Open document ID: 0647.57029
DOI: 10.1016/0166-8641(88)90055-7
author: N. M. Dos Santos
author: José L. Arraut
published in: Topology and its Applications
publication date: 1988-01-01T00:00:00Z
review text: The authors associate to a non-singular \(C^ r\) \((r\geq 2)\) action F of \({\mathbb{R}}^ p \)on a closed, orientable, connected m-manifold M a symmetric \((q+1)\)-linear mapping \(\alpha_ F: {\mathbb{R}}^ p\times...\times {\mathbb{R}}^ p\to H_{deR}^{2q+1}(M)\), where \(q=m-p\), called the characteristic mapping of F. They study actions through their characteristic mappings. Denote by \(M_ n\) the manifold obtained from \(T^ 2\times [0,1]\) by identifying (x,1) with (A(x),0), where A is the matrix given by \(a_{11}=a_{22}=1\), \(a_{12}=n\) and \(a_{21}=0.\) The main theorem of the paper is the following: If F is an \({\mathbb{R}}^ 2\)-action on \(M_ n\), \(n\neq 0\), such that \(\alpha_ F\) is non- degenerate, then F has a compact orbit and any \({\mathbb{R}}^ 2\)-action G sufficiently \(C^ 1\)-near to F has a compact orbit, too.
Mathematics Subject Classification ID: 57S20
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 4056600
zbMATH Keywords: foliation
zbMATH Keywords: stability of compact orbit
zbMATH Keywords: characteristic mappings
zbMATH Keywords: \({bbfR}^ 2\)-action
zbMATH Keywords: compact orbit
MaRDI profile type: Publication
full work available at URL: 0166-8641(88)90055-7
OpenAlex ID: W2048679616
cites work: A classification of the topological types of R\(^2\)-actions on closed orientable 3-manifolds
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cites work: Isotopie de formes fermées en dimension trois
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description: scientific article; zbMATH DE number 4056600