Skip to main content
SpringerLink
Log in

Elliptic Three-Manifolds and the Smale Conjecture

  • Chapter
  • First Online:
Diffeomorphisms of Elliptic 3-Manifolds

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2055))

  • 909 Accesses

Abstract

After a discussion of the (Generalized) Smale Conjecture, the main results of the monograph are summarized. The extent to which the Smale Conjecture extends to larger classes of three-manifolds—usually in a limited form called the Weak Smale Conjecture, if at all—is detailed. The chapter closes with a brief discussion of why Perelman’s methods appear not to give progress on the Smale Conjecture. As noted in the Preface, theSmale Conjecture is the assertion that the inclusion \(\mathrm{Isom}(M) \rightarrow \mathrm{Diff}(M)\) is a homotopy equivalence whenever M is an elliptic three-manifold, that is, a three-manifold with a Riemannian metric of constant positive curvature (which may be assumed to be 1). TheGeometrization Conjecture, now proven byPerelman, shows that all closed three-manifolds with finite fundamental group are elliptic.In this chapter, we will first review elliptic three-manifolds and their isometry groups. In the second section, we will state our main results on the Smale Conjecture, and provide some historical context. In the final two sections, we discuss isometries of nonelliptic three-manifolds, and address the possibility of applying Perelman’s methods to the Smale Conjecture.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aneziris, C., Balachandran, A.P., Bourdeau, M., Jo, S., Sorkin, R.D., Ramadas, T.R.: Aspects of spin and statistics in generally covariant theories. Int. J. Mod. Phys. A 4(20), 5459–5510 (1989)

    Google Scholar 

  2. Asano, K.: Homeomorphisms of prism manifolds. Yokohama Math. J. 26(1), 19–25 (1978)

    Google Scholar 

  3. Banyaga, A.: The structure of classical diffeomorphism groups. In: Mathematics and Its Applications, vol. 400. Kluwer Academic, Dordrecht (1997)

    Google Scholar 

  4. Bessaga, C., Pelczynski, A.: Selected topics in infinite-dimensional topology. In: Monografie Matematyczne, Tom 58 [Mathematical Monographs, vol. 58]. PWN—Polish Scientific Publishers, Warsaw (1975)

    Google Scholar 

  5. Boileau, M., Otal, J.P.: Scindements de Heegaard et groupe des homeotopies des petites varietes de Seifert. Invent. Math. 106(1), 85–107 (1991)

    Google Scholar 

  6. Bonahon, F.: Difféotopies des espaces lenticulaires. Topology 22(3), 305–314 (1983)

    Google Scholar 

  7. Bredon, G.E., Wood, J.W.: Non-orientable surfaces in orientable 3-manifolds. Invent. Math. 7, 83–110 (1969)

    Google Scholar 

  8. Bruce, J.W.: On transversality. Proc. Edinb. Math. Soc. (2) 29(1), 115–123 (1986)

    Google Scholar 

  9. Casson, A.J., Gordon, C.McA.: Reducing Heegaard splittings. Topology Appl. 27(3), 275–283 (1987)

    Google Scholar 

  10. Cerf, J.: Topologie de certains espaces de plongements. Bull. Soc. Math. Fr. 89, 227–380 (1961)

    Google Scholar 

  11. Cerf, J.: Sur les difféomorphismes de la sphere de dimension trois \(({\Gamma }_{4} = 0)\). In: Lecture Notes in Mathematics, vol. 53. Springer, Berlin (1968)

    Google Scholar 

  12. Charlap, L.S., Vasquez, A.T.: Compact flat riemannian manifolds. III. The group of affinities. Am. J. Math. 95, 471–494 (1973)

    Google Scholar 

  13. Filipkiewicz, R.P.: Isomorphisms between diffeomorphism groups. Ergod. Theor. Dyn. Syst. 2(2), 159–171 (1982)

    Google Scholar 

  14. Friedman, J., Sorkin, R.: Spin 1/2 from gravity. Phys. Rev. Lett. 44, 1100–1103 (1980)

    Google Scholar 

  15. Gabai, D.: The Smale conjecture for hyperbolic 3-manifolds: \(\mathrm{Isom}(\mathrm{{M}}^{3}) \simeq \mathrm{ Diff}(\mathrm{{M}}^{3})\). J. Differ. Geom. 58(1), 113–149 (2001)

    Google Scholar 

  16. Gibson, C.G., Wirthmuller, K., du Plessis, A.A., Looijenga, E.J.N.: Topological stability of smooth mappings. In: Lecture Notes in Mathematics, vol. 552. Springer, Berlin (1976)

    Google Scholar 

  17. Giulini, D.: On the configuration space topology in general relativity. Helv. Phys. Acta 68(1), 86–111 (1995)

    Google Scholar 

  18. Giulini, D., Louko, J.: No-boundary θ sectors in spatially flat quantum cosmology. Phys. Rev. D (3) 46(10), 4355–4364 (1992)

    Google Scholar 

  19. Gramain, A.: Le type d’homotopie du groupe des difféomorphismes d’une surface compacte. Ann. Sci. École Norm. Sup. (4) 6, 53–66 (1973)

    Google Scholar 

  20. Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. (N.S.) 7(1), 65–222 (1982)

    Google Scholar 

  21. Hantsche, W., Wendt, W.: Drei dimensionali Euklidische Raumformen. Math. Ann. 110, 593–611 (1934)

    Google Scholar 

  22. Hatcher, A.E.: Homeomorphisms of sufficiently large P 2-irreducible 3-manifolds. Topology 15(4), 343–347 (1976)

    Google Scholar 

  23. Hatcher, A.E.: On the diffeomorphism group of \({S}^{1} \times {S}^{2}\). Proc. Am. Math. Soc. 83(2), 427–430 (1981)

    Google Scholar 

  24. Hatcher, A.E.: A proof of the Smale conjecture, \(\mathrm{Diff}({S}^{3}) \simeq \mathrm{ O}(4)\). Ann. Math. (2) 117(3), 553–607 (1983)

    Google Scholar 

  25. Hatcher, A.: On the diffeomorphism group of \({S}^{1} \times {S}^{2}\). Revised version posted at http://www.math.cornell.edu/~hatcher/#papers (2003)

  26. Hempel, J.: 3-Manifolds. In: Annals of Mathematics Studies, vol. 86. Princeton University Press, Princeton (1976)

    Google Scholar 

  27. Henderson, D.W.: Corrections and extensions of two papers about infinite-dimensional manifolds. Gen. Topology Appl. 1, 321–327 (1971)

    Google Scholar 

  28. Henderson, D.W., Schori, R.: Topological classification of infinite dimensional manifolds by homotopy type. Bull. Am. Math. Soc. 76, 121–124 (1970)

    Google Scholar 

  29. Hendriks, H.: La stratification naturelle de l’espace des fonctions différentiables reelles n’est pas la bonne. C. R. Acad. Sci. Paris Ser. A-B 274, A618–A620 (1972)

    Google Scholar 

  30. Isham, C.J.: Topological θ-sectors in canonically quantized gravity. Phys. Lett. B 106(3), 188–192 (1981)

    Google Scholar 

  31. Ivanov, N.V.: Groups of diffeomorphisms of Waldhausen manifolds, Studies in topology, II. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 66, 172–176, 209 (1976)

    Google Scholar 

  32. Ivanov, N.V.: Diffeomorphism groups of Waldhausen manifolds. J. Sov. Math. 12, 115–118 (1979)

    Google Scholar 

  33. Ivanov, N.V.: Homotopies of automorphism spaces of some three-dimensional manifolds. Dokl. Akad. Nauk SSSR 244(2), 274–277 (1979)

    Google Scholar 

  34. Ivanov, N.V.: Corrections: Homotopies of automorphism spaces of some three-dimensional manifolds. Dokl. Akad. Nauk SSSR 249(6), 1288 (1979)

    Google Scholar 

  35. Ivanov, N.V.: Homotopy of spaces of diffeomorphisms of some three-dimensional manifolds. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 122, 72–103, 164–165 (1982)

    Google Scholar 

  36. Ivanov, N.V.: Homotopy of spaces of diffeomorphisms of some three-dimensional manifolds. J. Soviet Math. 26, 1646–1664 (1984)

    Google Scholar 

  37. Jaco, W.: Lectures on three-manifold topology. In: CBMS Regional Conference Series in Mathematics, vol. 43. American Mathematical Society, Providence (1980)

    Google Scholar 

  38. Jaco, W., Shalen, P.B.: Seifert fibered spaces in 3-manifolds. Memoir. Am. Math. Soc. 21(220), viii+192 (1979)

    Google Scholar 

  39. Kalliongis, J., McCullough, D.: Isotopies of 3-manifolds. Topology Appl. 71(3), 227–263 (1996)

    Google Scholar 

  40. Karcher, H.: Riemannian center of mass and mollifier smoothing. Comm. Pure Appl. Math. 30(5), 509–541 (1977)

    Google Scholar 

  41. Kobayashi, T., Saeki, O.: The Rubinstein-Scharlemann graphic of a 3-manifold as the discriminant set of a stable map. Pac. J. Math. 195(1), 101–156 (2000)

    Google Scholar 

  42. Kriegl, A., Michor, P.W.: The convenient setting of global analysis. In: Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence (1997)

    Google Scholar 

  43. Friedman, J.L., Witt, D.M.: Homotopy is not isotopy for homeomorphisms of 3-manifolds. Topology 25(1), 35–44 (1986)

    Google Scholar 

  44. Lundell, A., Weingram, S.: The Topology of CW Complexes. Van Nostrand Reinhold, Princeton (1969)

    Google Scholar 

  45. Mather, J.N.: Stability of \(\mathrm{{C}}^{\infty }\) mappings. III. Finitely determined mapgerms. Inst. Hautes Etudes Sci. Publ. Math. 35, 279–308 (1968)

    Google Scholar 

  46. McCullough, D.: Isometries of elliptic 3-manifolds. J. Lond. Math. Soc. (2) 65(1), 167–182 (2002)

    Google Scholar 

  47. McCullough, D., Soma, T.: The Smale conjecture for Seifert fibered spaces with hyperbolic base orbifold (2010) [ArXiv:1005.5061]

    Google Scholar 

  48. Neumann, W., Raymond, F.: Automorphisms of Seifert manifolds (1979). Preprint

    Google Scholar 

  49. Orlik, P.: Seifert manifolds. In: Lecture Notes in Mathematics, vol. 291. Springer, Berlin (1972)

    Google Scholar 

  50. Orlik, P., Vogt, E., Zieschang, H.: Zur Topologie gefaserter dreidimensionaler Mannigfaltigkeiten. Topology 6, 49–64 (1967)

    Google Scholar 

  51. Palais, R.S.: Local triviality of the restriction map for embeddings. Comment. Math. Helv. 34, 305–312 (1960)

    Google Scholar 

  52. Palais, R.S.: Homotopy theory of infinite dimensional manifolds. Topology 5, 1–16 (1966)

    Google Scholar 

  53. Park, C.Y.: Homotopy groups of automorphism groups of some Seifert fiber spaces. Dissertation at the University of Michigan (1989)

    Google Scholar 

  54. Park, C.Y.: On the weak automorphism group of a principal bundle, product case. Kyungpook Math. J. 31(1), 25–34 (1991)

    Google Scholar 

  55. Pitts, J.T., Rubinstein, J.H.: Applications of minimax to minimal surfaces and the topology of 3-manifolds. Miniconference on geometry and partial differential equations, 2 (Canberra, 1986), pp. 137–170. In: Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 12. Austral. Nat. Univ., Canberra (1987)

    Google Scholar 

  56. Rubinstein, J.H.: On 3-manifolds that have finite fundamental group and contain Klein bottles. Trans. Am. Math. Soc. 251, 129–137 (1979)

    Google Scholar 

  57. Rubinstein, J.H., Birman, J.S.: One-sided Heegaard splittings and homeotopy groups of some 3-manifolds. Proc. Lond. Math. Soc. (3) 49(3), 517–536 (1984)

    Google Scholar 

  58. Rubinstein, J.H., Scharlemann, M.: Comparing Heegaard splittings of non-Haken 3-manifolds. Topology 35(4), 1005–1026 (1996)

    Google Scholar 

  59. Sakuma, M.: The geometries of spherical Montesinos links. Kobe J. Math. 7(2), 167–190 (1990)

    Google Scholar 

  60. Scott, P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15(5), 401–487 (1983)

    Google Scholar 

  61. Seeley, R.T.: Extension of C∞ functions defined in a half space. Proc. Am. Math. Soc. 15, 625–626 (1964)

    Google Scholar 

  62. Seifert, H.: Topologie dreidimensionaler gefaserter raume. Acta Math. 60(1), 147–238 (1933)

    Google Scholar 

  63. Sergeraert, F.: Un théoréme de fonctions implicites sur certains espaces de Fréchet et quelques applications. Ann. Sci. École Norm. Sup. (4) 5, 599–660 (1972)

    Google Scholar 

  64. Smale, S.: Diffeomorphisms of the 2-sphere. Proc. Am. Math. Soc. 10, 621–626 (1959)

    Google Scholar 

  65. Sorkin, R.D.: Classical topology and quantum phases: Quantum geons. In: Geometrical and Algebraic Aspects of Nonlinear Field Theory. North-Holland Delta Series, pp. 201–218. North-Holland, Amsterdam (1989)

    Google Scholar 

  66. Takens, F.: Characterization of a differentiable structure by its group of diffeomorphisms. Bol. Soc. Brasil. Mat. 10(1), 17–25 (1979)

    Google Scholar 

  67. Tougeron, J.C.: Une généralisation du théoréme des fonctions implicites. C. R. Acad. Sci. Paris Ser. A-B 262, A487–A489 (1966)

    Google Scholar 

  68. Tougeron, J.C.: Idéaux de fonctions différentiables. I. Ann. Inst. Fourier (Grenoble) 18(fasc. 1), 177–240 (1968)

    Google Scholar 

  69. Waldhausen, F.: Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II. (German). Invent. Math. 3, 308–333 (1967). Ibid: 4, 87–117 (1967)

    Google Scholar 

  70. Waldhausen, F.: Gruppen mit Zentrum und 3-dimensionale Mannigfaltigkeiten. Topology 6, 505–517 (1967)

    Google Scholar 

  71. Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math. (2) 87, 56–88 (1968)

    Google Scholar 

  72. Wall, C.T.C.: Finite determinacy of smooth map-germs. Bull. Lond. Math. Soc. 13(6), 481–539 (1981)

    Google Scholar 

  73. Witt, D.M.: Symmetry groups of state vectors in canonical quantum gravity. J. Math. Phys. 27(2), 573–592 (1986)

    Google Scholar 

  74. Wolf, J.A.: Spaces of Constant Curvature, 3rd edn. Publish or Perish Inc., Boston (1974)

    Google Scholar 

Download references

Author information

Authors and Affiliations

    Authors
    1. Sungbok Hong
      View author publications

      You can also search for this author in PubMed Google Scholar

    2. John Kalliongis
      View author publications

      You can also search for this author in PubMed Google Scholar

    3. Darryl McCullough
      View author publications

      You can also search for this author in PubMed Google Scholar

    4. J. Hyam Rubinstein
      View author publications

      You can also search for this author in PubMed Google Scholar

    Rights and permissions

    Reprints and permissions

    Copyright information

    © 2012 Springer-Verlag Berlin Heidelberg

    About this chapter

    Cite this chapter

    Hong, S., Kalliongis, J., McCullough, D., Rubinstein, J.H. (2012). Elliptic Three-Manifolds and the Smale Conjecture. In: Diffeomorphisms of Elliptic 3-Manifolds. Lecture Notes in Mathematics, vol 2055. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31564-0_1

    Download citation

    Publish with us

    Policies and ethics

    Search

    Navigation