Hostname: page-component-cb9f654ff-c75p9 Total loading time: 0 Render date: 2025-08-20T04:54:01.029Z Has data issue: false hasContentIssue false
  • English
  • Français

Normal subgroups of diffeomorphism and homeomorphism groups of ℝn and other open manifolds

Published online by Cambridge University Press:  12 January 2011

PAUL A. SCHWEITZER, S. J.
PAUL A. SCHWEITZER, S. J.*
Affiliation:
Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, RJ 22453-900, Brazil (email: paul37sj@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

We determine all the normal subgroups of the group of Cr diffeomorphisms of ℝn, 1≤r≤∞, except when r=n+1 or n=4, and also of the group of homeomorphisms of ℝn ( r=0). We also study the group A0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compact manifold with non-empty boundary, then the quotient of A0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 6 , December 2011 , pp. 1835 - 1847
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Anderson, R. D.. The algebraic simplicity of certain groups of homeomorphisms. Amer. J. Math. 80 (1958), 955–963.CrossRefGoogle Scholar
[2]Anderson, R. D.. On homeomorphisms as products of a given homeomorphism and its inverse. Topology of 3-Manifolds. Ed. Fort, M.. Prentice-Hall, Englewood Cliffs, NJ, 1961, pp. 231–237.Google Scholar
[3]Banyaga, A.. The Structure of Classical Diffeomorphism Groups (Mathematics and its Applications, 400). Kluwer Academic Publications Group, Dordrecht, 1997, ISBN 0-7923-4475-9.CrossRefGoogle Scholar
[4]Cerf, J.. The pseudo-isotopy theorem for simply connected differentiable manifolds. Manifolds—Amsterdam 1970 (Springer Lecture Notes in Mathematics, 197). 1971, pp. 76–82.CrossRefGoogle Scholar
[5]Cernavskii, A. V.. Local contractibility of the homeomorphism group of a manifold. Math. USSR Sb. 8 (1969), 287–333.CrossRefGoogle Scholar
[6]Edwards, R. D. and Kirby, R. C.. Deformations of spaces of embeddings. Ann. of Math. (2) 93 (1971), 63–88.CrossRefGoogle Scholar
[7]Epstein, D. B. A.. The simplicity of certain groups of homeomorphisms. Compositio Math. 2 (1970), 165–173.Google Scholar
[8]Herman, M.. Sur le groupe des difféomorphismes du tore. Ann. Inst. Fourier 23 (1973), 75–86.CrossRefGoogle Scholar
[9]Kervaire, M. and Milnor, J.. Groups of homotopy spheres, I. Ann. of Math. (2) 77 (1963), 504–537.CrossRefGoogle Scholar
[10]Kirby, R. C.. Stable homeomorphisms and the annulus conjecture. Ann. of Math. (2) 89 (1969), 575–582.CrossRefGoogle Scholar
[11]Ling, W.. Factorizable groups of homeomorphisms. Compositio Math. 51 (1984), 41–50.Google Scholar
[12]Ling, W.. Normal subgroups of the group of automorphisms of an open manifold that has boundary. Preprint, 1977.Google Scholar
[13]Ling, W.. Translations on M×R. Amer. Math. Soc. Proc. Symp. Pure Math. 32(Part 2) (1978), 167–180.CrossRefGoogle Scholar
[14]Mather, J. N.. Commutators of diffeomorphisms. Comment. Math. Helv. 49 (1974), 512–528.CrossRefGoogle Scholar
[15]Mather, J. N.. Commutators of diffeomorphisms II. Comment. Math. Helv. 50 (1975), 33–40.CrossRefGoogle Scholar
[16]McDuff, D.. The lattice of normal subgroups of the group of diffeomorphisms or homeomorphisms of an open manifold. J. London Math. Soc. (2) 18 (1978), 353–364.CrossRefGoogle Scholar
[17]Milnor, J.. On manifolds homeomorphic to the 7-sphere. Ann. of Math. (2) 64 (1956), 399–405.CrossRefGoogle Scholar
[18]Palis, J. and Smale, S.. Structural stability theorems. Amer. Math. Soc. Proc. Symp. Pure Math. 14 (1970), 223–231.CrossRefGoogle Scholar
[19]Penner, R.et al. Groups of Diffeomorphisms: In Honor of Shigeyuki Morita on the Occasion of his 60th Birthday (Advanced Studies in Pure Mathematics, 52). Eds. Penner, R.et al. Kinokuniya, Mathematical Society of Japan, Tokyo, Japan, 2008, ISBN 978-4-931469-48-8.CrossRefGoogle Scholar
[20]Thurston, W.. Foliations and groups of diffeomorphisms. Bull. Amer. Math. Soc. 80 (1974), 304–307.CrossRefGoogle Scholar