Algebraic & Geometric Topology

Equivariant collaring, tubular neighbourhood and gluing theorems for proper Lie group actions

Marja Kankaanrinta

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Abstract

The purpose of this paper is to prove equivariant versions of some basic theorems in differential topology for proper Lie group actions. In particular, we study how to extend equivariant isotopies and then apply these results to obtain equivariant smoothing and gluing theorems. We also study equivariant collars and tubular neighbourhoods. When possible, we follow the ideas in the well-known book of M W Hirsch. When necessary, we use results from the differential topology of Hilbert spaces.

Article information

Source
Algebr. Geom. Topol., Volume 7, Number 1 (2007), 1-27.

Dates
Received: 19 January 2006
Revised: 30 October 2006
Accepted: 4 December 2006
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796653

Digital Object Identifier
doi:10.2140/agt.2007.7.1

Mathematical Reviews number (MathSciNet)
MR2289802

Zentralblatt MATH identifier
1181.57037

Subjects
Primary: 57S20: Noncompact Lie groups of transformations

Keywords
smooth proper action Lie group collar gluing

Citation

Kankaanrinta, Marja. Equivariant collaring, tubular neighbourhood and gluing theorems for proper Lie group actions. Algebr. Geom. Topol. 7 (2007), no. 1, 1--27. doi:10.2140/agt.2007.7.1. https://projecteuclid.org/euclid.agt/1513796653


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References

  • G E Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics 46, Academic Press, New York (1972)
  • M Brown, 0133812
  • M W Hirsch, Differential topology, Graduate Texts in Mathematics 33, Springer, New York (1976)
  • S Illman, M Kankaanrinta, A new topology for the set $C^{\infty,G}(M,N)$ of $G$–equivariant smooth maps, Math. Ann. 316 (2000) 139–168
  • S Illman, M Kankaanrinta, Three basic results for real analytic proper $G$–manifolds, Math. Ann. 316 (2000) 169–183
  • M Kankaanrinta, Proper smooth $G$–manifolds have complete $G$–invariant Riemannian metrics, Topology Appl. 153 (2005) 610–619
  • J L Koszul, Sur certains groupes de transformations de Lie, from: “Géométrie différentielle. Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1953”, Centre National de la Recherche Scientifique, Paris (1953) 137–141
  • S Lang, Differential and Riemannian manifolds, third edition, Graduate Texts in Mathematics 160, Springer, New York (1995)
  • J N Mather, }, Ann. of Math. $(2)$ 89 (1969) 254–291
  • G D Mostow, 0087037
  • R S Palais, Imbedding of compact, differentiable transformation groups in orthogonal representations, J. Math. Mech. 6 (1957) 673–678
  • R S Palais, The classification of $G$–spaces, Mem. Amer. Math. Soc. No. 36 (1960)
  • R S Palais, }, Ann. of Math. $(2)$ 73 (1961) 295–323
  • M J Pflaum, Analytic and geometric study of stratified spaces, Lecture Notes in Mathematics 1768, Springer, Berlin (2001)
  • G W Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math. (1980) 37–135