Triangulation of the map of a \(G\)-manifold to its orbit space
type: Item
instance of: scholarly article
zbMATH Open document ID: 1295.57027
DOI: 10.1215/00277630-2366201
author: Mitsutaka Murayama
author: Masahiro Shiota
published in: Nagoya Mathematical Journal
publication date: 2014-01-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1004.4094
Mathematics Subject Classification ID: 57Q15
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 57S20
Mathematics Subject Classification ID: 58K20
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 6250922
zbMATH Keywords: \(G\)-manifold
zbMATH Keywords: orbit map
zbMATH Keywords: triangulation
zbMATH Keywords: Thom conjecture
zbMATH Keywords: subanalytic set
MaRDI profile type: Publication
OpenAlex ID: W3100607954
arXiv ID: 1004.4094
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title: Triangulation of the map of a \(G\)-manifold to its orbit space
review text: Let \(G\) be a Lie group with a smooth proper action on a manifold \(M\). It is proved that the orbit projection \(M \to M/G\) is topologically equivalent to a PL map from a PL manifold to a polyhedron. The first ingredient of the proof is the proof of the Thom conjecture [\textit{R. Thom}, Enseign. Math. (2) 8, 24--33 (1962; Zbl 0109.40002)] by the second author [Topology 39, No. 2, 383--399 (2000; Zbl 0934.57028)]. This asserts that a locally triangulable map between subsets of Euclidean spaces is triangulable if it is proper and satisfies a condition which roughly says that its fibers do not explode. The orbit projection is shown to possess the latter property; however, it is not proper in general, hence an extra ingredient is needed. This is the second author's theory of subanalytic sets and maps [\textit{M. Shiota}, Geometry of subanalytic and semialgebraic sets. Progress in Mathematics (Boston, Mass.). 150. Boston, MA: Birkhäuser (1997; Zbl 0889.32006)]. A subset of \(\mathbb R^n\) is subanalytic if it can be obtained from finitely many analytic images of real analytic manifolds by taking set differences and unions, and a map is subanalytic if its graph is. To be able to apply this theory, the authors endow \(M\), \(M/G\) and the orbit map with subanalytic structures.
reviewed by: Rainer Löwen
description: scientific article; zbMATH DE number 6250922