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Triangulation of the map of a G-manifold to its orbit space

Published online by Cambridge University Press:  11 January 2016

Mitsutaka Murayama  and
Masahiro Shiota
Mitsutaka Murayama
Affiliation:
Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan, murayama@math.titech.ac.jp
Masahiro Shiota
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa, Nagoya 464-8602, Japan, shiota@math.nagoya-u.ac.jp
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Abstract

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Let G be a Lie group, and let M be a smooth proper G-manifold. Let M/G denote the orbit space, and let π : M → M/G be the natural map. It is known that M/G is homeomorphic to a polyhedron. In the present paper we show that there exist a piecewise linear (PL) manifold P, a polyhedron L, and homeomorphisms τ : P → M and σ : M/G → L such that σ o π o τ is PL. This is an application of the theory of subanalytic sets and subanalytic maps of Shiota. If M and the G-action are, moreover, subanalytic, then we can choose τ and σ subanalytic and P and L unique up to PL homeomorphisms.

Keywords

57S1557S2058K20
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 212 , December 2013 , pp. 159 - 195
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2013

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