Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

IMPACT FACTOR 2018: 0.726
5-year IMPACT FACTOR: 0.869

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2018: 0.34

ICV 2017: 161.82

Open Access
See all formats and pricing
More options …
Volume 3, Issue 3


Volume 13 (2015)

On the first homology of automorphism groups of manifolds with geometric structures

Kōjun Abe / Kazuhiko Fukui
Published Online: 2005-09-01 | DOI: https://doi.org/10.2478/BF02475921


Hermann and Thurston proved that the group of diffeomorphisms with compact support of a smooth manifold M which are isotopic to the identity is a perfect group. We consider the case where M has a geometric structure. In this paper we shall survey on the recent results of the first homology of the diffeomorphism groups which preserve a smooth G-action or a foliated structure on M. We also work in Lipschitz category.

Keywords: Automorphism group; first homology group; G-manifold; Foliated manifold

Keywords: 47S05; 58D05; 58H10

  • [1] K. Abe and K. Fukui: “On commutators of equivariant diffeomorphisms”, Proc. Japan Acad., Vol. 54, (1978), pp. 52–54. http://dx.doi.org/10.3792/pjaa.54.52CrossrefGoogle Scholar

  • [2] K. Abe and K. Fukui: “On the structure of the group of equivariant diffeomorphisms of G-manifolds with codimension one orbit”, Topology, Vol. 40, (2001), pp. 1325–1337. http://dx.doi.org/10.1016/S0040-9383(00)00014-8CrossrefGoogle Scholar

  • [3] K. Abe and K. Fukui: “On the structure of the group of Lipschitz homeomorphisms and its subgroups”, J. Math. Soc. Japan, Vol. 53(3), (2001), pp. 501–511. CrossrefGoogle Scholar

  • [4] K. Abe and K. Fukui: “On the structure of the group of Lipschitz homeomorphisms and its subgroups II”, J. Math. Soc. Japan, Vol. 55(4), (2003), pp. 947–956. CrossrefGoogle Scholar

  • [5] K. Abe and K. Fukui: “On the first homology of the group of equivariant diffeomorphisms and its applications”, preprint. Google Scholar

  • [6] K. Abe and K. Fukui: “On the structure of the group of diffeomorphisms of manifolds with boundary and its applications”, preprint. Google Scholar

  • [7] K. Abe, K. Fukui and T. Miura: “On the first homology of the group of equivariant Lipschitz homeomorphisms”, preprint. Google Scholar

  • [8] A. Banyaga: “On the structure of the group of equivariant diffeomorphisms”, Topology, Vol. 16, (1977), pp. 279–283. http://dx.doi.org/10.1016/0040-9383(77)90009-XCrossrefGoogle Scholar

  • [9] A. Banyaga; The structure of classical diffeomorphism groups, Mathematics and its Applications, Vol. 400, Kluwer Academic Publishers, Dordrecht, 1997. Google Scholar

  • [10] E. Bierstone: “The Structure of Orbit Spaces and the Singularities of Equivariant Mappings”, Instituto de Mathematica Pura e Aplicada, Rio de Janeiro, 1980. Google Scholar

  • [11] B. Bredon: Introduction to Compact Transformation Groups, Academic Press, New York-London, 1972. Google Scholar

  • [12] D.B.A. Epstein: “The simplicity of certain groups of homeomorphisms”, Compocio Math., Vol. 22, (1970), pp. 165–173. Google Scholar

  • [13] D.B.A. Epstein: “Foliations with all leaves compact”, Ann. Inst. Fourier, Grenoble, Vol. 26, (1976), pp. 265–282. Google Scholar

  • [14] D.B.A. Epstein: “Commutators of C ∞-diffeomorphisms”, Appendix to: John N. Mather: “gA curious remark concerning the geometric transfer map”, Comment. Math. Helv., Vol. 59, (1984), pp. 111–122. CrossrefGoogle Scholar

  • [15] K. Fukui: “Homologies of Dif f ∞(R n , 0) ant its subgroups”, J. Math. Kyoto Univ., Vol. 20, (1980), pp. 457–487. Google Scholar

  • [16] K. Fukui: “Commutators of foliation preserving homeomorphisms for certain compact foliations”, Publ. RIMS Kyoto Univ., Vol. 34(1), (1998), pp. 65–73. http://dx.doi.org/10.2977/prims/1195144828CrossrefGoogle Scholar

  • [17] K. Fukui and H. Imanishi: “On commutators of foliation preserving homeomorphisms”, J. Math. Soc. Japan, Vol. 51(1), (1999), pp. 227–236. CrossrefGoogle Scholar

  • [18] K. Fukui and H. Imanishi: “On commutators of foliation preserving Lipschitz homeomorphisms“; J. Math. Kyoto Univ., Vol. 41 (3), (2001), pp. 507–515. Google Scholar

  • [19] K. Fukui and T. Nakamura: “A topological property of Lipschitz mappings”, Topol. Appl., Vol. 148, (2005), pp. 143–152. http://dx.doi.org/10.1016/j.topol.2004.08.005CrossrefGoogle Scholar

  • [20] K. Fukui and S. Ushiki: “On the homotopy type of F Dif f(S 3, F R )”, J. Math. Kyoto Univ., Vol. 15(1), (1975), pp. 201–210. Google Scholar

  • [21] M. Hermann: “Simplicité du groupe des difféomorphismes de class C ∞, isotopes á l'identité, du tore de dimension n” C. R. Acad. Sci. Pari Sér. A-B, Vol. 273, (1971), pp. 232–234. Google Scholar

  • [22] M. Hermann: “Sur la conjugasion différentiable des difféomorphismes du cercle á des rotations”, Publ. I.H.E.S., Vol. 49, (1979), pp. 5–233. Google Scholar

  • [23] F. Hirzebruch and K.H. Mayer: O(n)-Manigfaltigkeiten, exotische Sphären und Singularitäten, Springer Lecture Notes, Vol. 57, 1968. Google Scholar

  • [24] H. Imanishi: “On the theorem of Denjoy-Sacksteder for codimension one foliations without holonomy”, J. Math. Kyoto Univ., Vol. 14, (1974), pp. 607–634. Google Scholar

  • [25] J. Luukkainen and J. Väisälä: “Elements of Lipschitz topology”, Ann. Acad. Sci. Fennicae, Ser. A.I. Math., Vol. 3, (1977), pp. 85–122. Google Scholar

  • [26] J.N. Mather: “Commutators of diffeomorphisms I and II”, Comment. Math. Helv., Vol. 49, (1992), pp. 512–528; Vol. 50, (1975), pp. 33–40. Google Scholar

  • [27] T. Rybicki: “The identity component of the leaf preserving diffeomorphism group is perfect”, Monatsh. Math., Vol. 120, (1995), pp. 289–305. http://dx.doi.org/10.1007/BF01294862CrossrefGoogle Scholar

  • [28] T. Rybicki: “Commutators of diffeomorphisms of a manifold with boundary”, Ann. Polon. Math., Vol. 68(3), (1998), pp. 199–210. Google Scholar

  • [29] G.W. Schwarz: “Smooth invariant functions under the action of a compact Lie group”, Topology, Vol. 14, (1975), pp. 63–68. http://dx.doi.org/10.1016/0040-9383(75)90036-1CrossrefGoogle Scholar

  • [30] G.W. Schwarz: “Lifting smooth homotopies of orbit spaces”, Inst. Hautes Etudes Sci. Publ. Math., Vol. 51, (1980), pp. 37–135. CrossrefGoogle Scholar

  • [31] R. Strub: “Local classification of quotients of smooth manifolds by discontinuous groups”, Math. Zeitschrift, Vol. 179, (1982), pp. 43–57. http://dx.doi.org/10.1007/BF01173913CrossrefGoogle Scholar

  • [32] S. Sternberg: “Local contractions and a theorem of Poincaré”, Amer. Jour. of Math., Vol. 79, (1957), pp. 809–823. http://dx.doi.org/10.2307/2372437CrossrefGoogle Scholar

  • [33] S. Sternberg: “The structure of local homeomorphisms, II”, Amer. Jour. of Math., Vol. 80, (1958), pp. 623–632. http://dx.doi.org/10.2307/2372774CrossrefGoogle Scholar

  • [34] W. Thurston: “Foliations and group of diffeomorphisms”, Bull. Amer. Math. Soc., Vol. 80, (1974), pp. 304–307. http://dx.doi.org/10.1090/S0002-9904-1974-13475-0CrossrefGoogle Scholar

  • [35] T. Tsuboi: “On the group of foliation preserving diffeomorphisms”, preprint. Google Scholar

  • [36] J.H.C. Whitehead: “Manifolds with transverse fields in euclidean space”, Ann. of Math., Vol. 73(2), (1961), pp. 154–212. http://dx.doi.org/10.2307/1970286CrossrefGoogle Scholar

About the article

Published Online: 2005-09-01

Published in Print: 2005-09-01

Citation Information: Open Mathematics, Volume 3, Issue 3, Pages 516–528, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/BF02475921.

Export Citation

© 2005 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Sergiy Maksymenko
Open Mathematics, 2009, Volume 7, Number 2

Comments (0)

Please log in or register to comment.
Log in