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Open Mathematics

formerly Central European Journal of Mathematics

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


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Volume 3, Issue 3

Issues

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

Abstract

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.

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

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