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)
    Mathematical Reviews (MathSciNet): MR0413144
    Zentralblatt MATH: 0246.57017
  • M Brown, 0133812
  • M W Hirsch, Differential topology, Graduate Texts in Mathematics 33, Springer, New York (1976)
    Mathematical Reviews (MathSciNet): MR0448362
    Zentralblatt MATH: 0356.57001
  • 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
    Mathematical Reviews (MathSciNet): MR1735082
    Digital Object Identifier: doi:10.1007/s002080050007
  • S Illman, M Kankaanrinta, Three basic results for real analytic proper $G$–manifolds, Math. Ann. 316 (2000) 169–183
    Mathematical Reviews (MathSciNet): MR1735083
    Zentralblatt MATH: 0945.57016
    Digital Object Identifier: doi:10.1007/s002080050008
  • M Kankaanrinta, Proper smooth $G$–manifolds have complete $G$–invariant Riemannian metrics, Topology Appl. 153 (2005) 610–619
    Mathematical Reviews (MathSciNet): MR2193330
    Digital Object Identifier: doi:10.1016/j.topol.2005.01.034
  • 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
    Mathematical Reviews (MathSciNet): MR0059919
    Zentralblatt MATH: 0101.16201
  • S Lang, Differential and Riemannian manifolds, third edition, Graduate Texts in Mathematics 160, Springer, New York (1995)
    Mathematical Reviews (MathSciNet): MR1335233
  • J N Mather, }, Ann. of Math. $(2)$ 89 (1969) 254–291
    Mathematical Reviews (MathSciNet): MR0259953
    Digital Object Identifier: doi:10.2307/1970668
  • G D Mostow, 0087037
  • R S Palais, Imbedding of compact, differentiable transformation groups in orthogonal representations, J. Math. Mech. 6 (1957) 673–678
    Mathematical Reviews (MathSciNet): MR0092927
    Zentralblatt MATH: 0086.02603
  • R S Palais, The classification of $G$–spaces, Mem. Amer. Math. Soc. No. 36 (1960)
    Mathematical Reviews (MathSciNet): MR0177401
  • R S Palais, }, Ann. of Math. $(2)$ 73 (1961) 295–323
    Mathematical Reviews (MathSciNet): MR0126506
    Digital Object Identifier: doi:10.2307/1970335
  • M J Pflaum, Analytic and geometric study of stratified spaces, Lecture Notes in Mathematics 1768, Springer, Berlin (2001)
    Mathematical Reviews (MathSciNet): MR1869601
    Zentralblatt MATH: 0988.58003
  • G W Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math. (1980) 37–135
    Mathematical Reviews (MathSciNet): MR573821
    Zentralblatt MATH: 0449.57009
    Digital Object Identifier: doi:10.1007/BF02684776

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