Journal of the Mathematical Society of Japan

Commutators of C-diffeomorphisms preserving a submanifold

Kōjun ABE and Kazuhiko FUKUI

Full-text: Open access

Abstract

We consider the group of C-diffeomorphisms of M which is isotopic to the identity through C-diffeomorphisms preserving N for a compact manifold pair (M,N) and prove that the group is perfect. Also we prove that it is uniformly perfect for a certain compact manifold with boundary.

Article information

Source
J. Math. Soc. Japan, Volume 61, Number 2 (2009), 427-436.

Dates
First available in Project Euclid: 13 May 2009

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1242220717

Digital Object Identifier
doi:10.2969/jmsj/06120427

Mathematical Reviews number (MathSciNet)
MR2532896

Zentralblatt MATH identifier
1169.57029

Subjects
Primary: 57R50: Diffeomorphisms
Secondary: 58D05: Groups of diffeomorphisms and homeomorphisms as manifolds [See also 22E65, 57S05]

Keywords
diffeomorphism group commutator subgroup compact manifold with boundary uniformly perfect

Citation

ABE, Kōjun; FUKUI, Kazuhiko. Commutators of $C^{\infty}$ -diffeomorphisms preserving a submanifold. J. Math. Soc. Japan 61 (2009), no. 2, 427--436. doi:10.2969/jmsj/06120427. https://projecteuclid.org/euclid.jmsj/1242220717


Export citation

References

  • K. Abe and K. Fukui, On commutators of equivariant diffeomorphisms, Proc. Japan Acad. Ser. A Math. Sci., 54 (1978), 52–54.
  • K. Abe and K. Fukui, The first homology of the group equivariant diffeomorphisms and its applications, J. of Topology, 1 (2008), 461–476.
  • D. Burago, S. Ivanov and L. Polterovich, Conjugation-invariant norms on groups of geometric origin, Groups of Diffeomorphisms, Adv. Stud. Pure Math., 52, Math. Soc. Japan, 2008, pp. 221–250.
  • M. Entov, Commutator length of symplectomorphisms, Comment. Math. Helv., 79 (2004), 58–104.
  • D. B. A. Epstein, Commutators of $C^{\infty}$-diffeomorphisms, Comment. Math. Helv., 59 (1984), 111–122.
  • K. Fukui, Homologies of the group $\mathrm{Diff}^{\infty}(\mbi{R}^{n},0)$ and its subgroups, J. Math. Kyoto Univ., 20 (1980), 475–487.
  • J. M. Gambaudo and É. Ghys, Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Systems, 24 (1980), 1591–1617.
  • M. Herman, Simplicité du groupe des difféomorphismes de classe $C^{\infty}$, isotopes à l'identité, du tore de dimension $n$, C. R. Acad. Sci. Paris, 273 (1971), 232–234.
  • M. Herman, Sur la conjugation différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5–233.
  • J. Mather, Commutators of diffeomorphisms, Comment. Math. Helv., 49 (1974), 512–518; II 50 (1975), 33–40; III 60 (1985), 122–124.
  • T. Rybicki, The identity component of the leaf preserving diffeomorphism group is perfect, Monatsh. Math., 120 (1995), 289–305.
  • T. Rybicki, Commutators of diffeomorphisms of a manifold with boundary, Ann. Polon. Math., 68 (1998), 199–210.
  • T. Rybicki, On the group of diffeomorphisms preserving a submanifold, Demonstratio Math., 31 (1998), 103–110.
  • S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math., 79 (1957), 809–823.
  • S. Sternberg, The structure of local homeomorphisms, II, Amer. J. Math., 80 (1958), 623–632.
  • W. Thurston, Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc., 80 (1974), 304–307.
  • T. Tsuboi, On the group of foliation preserving diffeomorphisms, (ed. P. Walczak et al.) Foliations 2005, World scientific, Singapore, 2006, pp. 411–430.
  • T. Tsuboi, On the uniform perfectness of diffeomorphism groups, Groups of Diffeomorphisms, Adv. Stud. Pure Math., 52, Math. Soc. Japan, 2008, pp. 505–524.