Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-14T09:32:20.627Z Has data issue: false hasContentIssue false
  • English
  • Français

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, [email protected]
Masahiro Shiota
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa, Nagoya 464-8602, Japan, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1] Bredon, G. E., Introduction to Compact Transformation Groups, Pure Appl. Math. 46, Academic Press, New York, 1972. MR 0413144.Google Scholar
[2] Gibson, C. G., Wirthmüller, K., du Plessis, A. A., and Looijenga, E. J. N., Topological Stability of Smooth Mappings, Lecture Notes in Math. 552, Springer, Berlin, 1976. MR 0436203.Google Scholar
[3] Grauert, H., On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460–472. MR 0098847.Google Scholar
[4] Hironaka, H., “Subanalytic sets” in Number Theory, Algebraic Geometry and Commutative Algebra, in Honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, 453–493. MR 0377101.Google Scholar
[5] Illman, S., “Every proper smooth action of a Lie group is equivalent to a real analytic action” in Prospects in Topology (Princeton, 1994), Ann. of Math. Stud. 138, Princeton University Press, Princeton, 1995, 189–220. MR 1368660.Google Scholar
[6] Koszul, J. L., “Sur certains groupes de transformations de Lie” in Gèomètrie diffèrentielle (Strasbourg, 1953), Centre National de la Recherche Scientifique, Paris, 1953, 137–141. MR 0059919.Google Scholar
[7] Lojasiewicz, S., Triangulation of semi-analytic sets, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3) 18 (1964), 449–474. MR 0173265.Google Scholar
[8] Matumoto, T. and Shiota, M., “Proper subanalytic transformation groups and unique triangulation of the orbit spaces” in Transformation Groups (Poznan, 1985), Lecture Notes in Math. 1217, Springer, Berlin, 1986, 290–302. MR 0874184. DOI 10.1007/ BFb0072829.Google Scholar
[9] Matumoto, T. and Shiota, M., “Unique triangulation of the orbit space of a differentiable transformation group and its applications” in Homotopy Theory and Related Topics (Kyoto, 1984), Adv. Stud. Pure Math. 9, North-Holland, Amsterdam, 1987, 41–55. MR 0896944.Google Scholar
[10] Milnor, J., Two complexes which are homeomorphic but combinatorially distinct, Ann. of Math. (2) 74 (1961), 575–590. MR 0133127.Google Scholar
[11] Munkres, J. R., Elementary Differential Topology, Lectures Given at Massachusetts Institute of Technology, Fall, 1961, rev. ed., Ann. of Math. Stud. 54, Princeton University Press, Princeton, 1966. MR 0198479.Google Scholar
[12] Palais, R. S., On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323. MR 0126506.Google Scholar
[13] Palais, R. S., C1 actions of compact Lie groups on compact manifolds are C1-equivalent to C°° actions, Amer. J. Math. 92 (1970), 748–760. MR 0268912.Google Scholar
[14] Shiota, M., Geometry of Subanalytic and Semialgebraic Sets, Progr. Math. 150, Birkhäuser, Boston, 1997. MR 1463945. DOI 10.1007/978-1-4612-2008-4.Google Scholar
[15] Shiota, M., Thom’s conjecture on triangulations of maps, Topology 39 (2000), 383-399. MR 1722004. DOI 10.1016/S0040-9383(99)00022-1.Google Scholar
[16] Thom, R., La stabilitè topologique des applications polynomiales, Enseign. Math. (2) 8 (1962), 24–33. MR 0148079.Google Scholar
[17] Verona, A., Stratified Mappings—Structure and Triangulability, Lecture Notes in Math. 1102, Springer, Berlin, 1984. MR 0771120.Google Scholar
[18] Whitney, H., Differentiable manifolds, Ann. of Math. (2) 37 (1936), 645–680. MR 1503303.Google Scholar
[19] Yang, C. T., The triangulability of the orbit space of a differentiable transformation group, Bull. Amer. Math. Soc. (N.S.) 69 (1963), 405–408. MR 0146291.Google Scholar