Involutions and isotopies of lens spaces

  • Craig Hodgson
  • J. H. Rubinstein
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1144)


In this paper we study the topology of the three-dimensional lens spaces by regarding them as two-fold branched coverings. The main result obtained is a classification of the smooth involutions on lens spaces having one-dimensional fixed point sets. We show that each such involution is conjugate, by a diffeomorphism isotopic to the identity, to an isometry of the lens space (given the standard spherical metric).

Using this classification of involutions, we deduce that genus one Heegaard splittings of lens spaces are unique up to isotopy. We apply this result to give a new proof of the classification of lens spaces up to diffeomorphism. We also compute the group of isotopy classes of diffeomorphisms of each lens space.


Homology Class Orbit Space Morse Function Lens Space Isotopy Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A]
    J.W. ALEXANDER, Note on Riemann spaces, Bull. Amer. Math. Soc. 26 (1920), 370–372.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [As]
    K. ASANO, Homeomorphisms of prism manifolds, Yokohama Math. J. 26 (1978), 19–25.MathSciNetzbMATHGoogle Scholar
  3. [BGM]
    J.S. BIRMAN, F. GONZALEZ-ACUÑA and J.M. MONTESINOS, Minimal Heegaard splittings of 3-manifolds are not unique, Michigan Math. J. 23 (1976), 97–103.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [BH]
    J.S. BIRMAN and H. HILDEN, Heegaard splittings of branched coverings of S3, Trans. Amer. Math. Soc. 213 (1975), 315–352.MathSciNetzbMATHGoogle Scholar
  5. [BR]
    J.S. BIRMAN and J.H. RUBINSTEIN, Homeotopy groups of some non-Haken 3-manifolds, to appear. (See also: Abstracts Amer. Math. Soc. 1 (1980), 773-57-13, 136.)Google Scholar
  6. [Bo]
    F. BONAHON, Diffeotopies des espaces lenticulaires, preprint.Google Scholar
  7. [BO]
    F. BONAHON and J-P OTAL, Scindements de Heegaard des espaces lenticulaires, preprint.Google Scholar
  8. [Br]
    G.E. BREDON, Introduction to compact transformation groups, Academic Press, New York and London, 1972.zbMATHGoogle Scholar
  9. [Bro]
    E.J. BRODY, The topological classification of the lens spaces, Ann. of Math. 71 (1960), 163–184.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [BL]
    W. BROWDER and G.R. LIVESAY, Fixed point free involutions on homotopy spheres, Bull. Amer. Math. Soc. 73 (1967), 242–245.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Ce 1]
    J. CERF, Sur les difféomorphismes de la sphere de dimension trois (Γ4=0), Lecture Notes in Math., Vol. 53, Springer-Verlag, Berlin and New York, 1968.CrossRefzbMATHGoogle Scholar
  12. [Ce 2]
    J. CERF, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. No. 39 (1970), 5–173.Google Scholar
  13. [DK]
    A. DURFEE and L. KAUFFMAN, Periodicity of branched cyclic covers, Math. Ann. 218 (1975), 157–174.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [Eng]
    R. ENGMANN, Nicht homöomorphe Heegaard Zerlegungen vom Geschlecht 2 der zusammenhängenden Summe zweier Linsenräume, Abh. Math. Sem. Univ. Hamburg 35 (1970), 33–38.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [Ep]
    D.B.A. EPSTEIN, Curves on 2-manifolds and isotopies, Acta Math. 115 (1966) 83–107.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [F]
    R.H. FOX, A note on branched cyclic coverings of spheres, Rev. Mat. Hisp.-Amer. 32 (1972), 158–166.MathSciNetzbMATHGoogle Scholar
  17. [Fr]
    W. FRANZ, Über die Torsion einer Überdeckung, J.f. reine u. angewandte Math. 173 (1935), 245–253.MathSciNetzbMATHGoogle Scholar
  18. [GG]
    M. GOLUBITSKY and V. GUILLEMIN, Stable mappings and their singularities, Springer-Verlag, New York, Heidelberg and Berlin, 1973.CrossRefzbMATHGoogle Scholar
  19. [GL]
    C. McA. GORDON and R.A. LITHERLAND, Incompressible surfaces in branched coverings, to appear.Google Scholar
  20. [H]
    W. HAKEN, Some results on surfaces in 3-manifolds, in: Studies in modern topology, M.A.A. Studies in Math. Vol. 5, M.A.A., Englewood Cliffs, N.J. (1968), 39–98.Google Scholar
  21. [Ha]
    R.S. HAMILTON, Three manifolds with positive Ricci curvature, J. of Diff. Geom 17 (1982), 255–306.MathSciNetzbMATHGoogle Scholar
  22. [Ham]
    M.-E. HAMSTROM, Homotopy in homeomorphism spaces, TOP and PL. Bull. Amer. Math. Soc. 80 (1974), 207–230.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [Har]
    R.I. HARTLEY, Knots and involutions, Math. Zeitschr. 171 (1980), 175–185.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [Hat 1]
    A. HATCHER, Homeomorphisms of sufficiently large p2-irreducible 3-manifolds, Topology 15 (1976), 343–347.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [Hat 2]
    A. HATCHER, A proof of the Smale Conjecture, Diff(S3)≃O(4), Ann. of Math. 117 (1983), 553–607.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [He]
    W. HEIL, On P2-irreducible 3-manifolds, Bull. Amer. Math. Soc. 75 (1969), 772–775.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [Hem]
    J. HEMPEL, 3-manifolds, Annals of Math. Studies No. 86, Princeton Univ. Press, Princeton, N.J., 1976.zbMATHGoogle Scholar
  28. [Hi]
    H. HILDEN, Every closed orientable 3-manifold is a 3-fold branched covering space of S3, Bull. Amer. Soc. 80 (1974), 1243–1244.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [Hir]
    M.W. HIRSCH, Differential topology, Springer-Verlag, New York, Heidelberg and Berlin, 1976.CrossRefzbMATHGoogle Scholar
  30. [Iv 1]
    N.V. IVANOV, Diffeomorphism groups of Waldhausen manifolds, J. Soviet Math. 12 (1979), 115–118. (English translation of: Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov 66 (1976), 172–176.)CrossRefzbMATHGoogle Scholar
  31. [Iv 2]
    N.V. IVANOV, Homotopy of spaces of automorphisms of some three-dimensional manifolds, Soviet Math. Dokl. 20 (1979), 47–50. (English translation of: Dokl. Akad. Nauk. SSSR 244 (1979), 274–277.)zbMATHGoogle Scholar
  32. [J]
    W. JACO, Lectures on three-manifold topology, CBMS Regional Conference Series in Math., No. 43, Amer. Math. Soc., Providence R.I., 1980.zbMATHGoogle Scholar
  33. [Jah]
    B. JAHREN, One-parameter families of spheres in 3-manifolds, Ph.D. thesis, Princeton Univ., 1975.Google Scholar
  34. [K 1]
    P.K. KIM, PL involutions on lens spaces and other 3-manifolds, Proc. Amer. Math. Soc. 44 (1974), 467–473.MathSciNetzbMATHGoogle Scholar
  35. [K 2]
    P.K. KIM, Cyclic actions on lens spaces, Trans. Amer. Math. Soc. 237 (1978), 121–144.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [K 3]
    P.K. KIM, Involutions on Klein spaces M(p,q), Notices Amer. Math. Soc. 25 (1978), abstract 752-57-12, A-147.Google Scholar
  37. [KT]
    P.K. KIM and J.L. TOLLEFSON, Splitting the PL involutions of nonprime 3-manifolds, Michigan Math. J. 27 (1980), 259–274.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [Ki]
    R. KIRBY, Problems in low dimensional manifold theory, Proc. Sympos. Pure Math., vol. 32, part 2, Amer. Math. Soc., Providence R.I., 1978, 273–312.Google Scholar
  39. [Kw 1]
    K.W. KWUN, Scarcity of orientation-reversing PL involutions of lens spaces, Michigan Math. J. 17 (1970), 355–358.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [Kw 2]
    K.W. KWUN, Sense-preserving PL involutions of some lens spaces, Michigan Math. J. 20 (1973), 73–77.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [La]
    F. LAUDENBACH, Topologie de la dimension trois: homotopie et isotopie, Astérisque No. 12, Soc. Math de France, Paris, 1974.zbMATHGoogle Scholar
  42. [M 1]
    D. McCULLOUGH, Homotopy groups of the space of self-homotopy-equivalences, Trans. Amer. Math. Soc. 264 (1981), 151–163.MathSciNetCrossRefzbMATHGoogle Scholar
  43. [M 2]
    D. McCULLOUGH, The group of homotopy euqivalences for a connected sum of closed aspherical manifolds, Indian Univ. Math. J. 30 (1981), 249–260.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [Mi 1]
    J.W. MILNOR, Morse theory, Annals of Math. Studies No. 51, Princeton Univ. Press, Princeton N.J., 1963.CrossRefzbMATHGoogle Scholar
  45. [Mi 2]
    J.W. MILNOR, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [Moi]
    E.E. MOISE, Affine structures in 3-manifolds IV, V. Piecewise linear approximations of homeomorphisms. The triangulation theorem and Hauptvermutung, Ann. of Math. 55 (1952), 215–222; ibid. 56 (1952), 96–114.MathSciNetCrossRefzbMATHGoogle Scholar
  47. [Mon 1]
    J.M. MONTESINOS, A representation of closed, orientable 3-manifolds as 3-fold branched coverings of S3, Bull. Amer. Math. Soc. 80 (1974) 845–846.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [Mon 2]
    J.M. MONTESINOS, Minimal plat presentations of prime knots and links are not unique, Canad. J. Math. 28 (1976), 161–167.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [Mon 3]
    J.M. MONTESINOS, Revêtements ramifiés de noeuds, espace fibrés de Seifert et scindements de Heegaard, preprint. To appear in Astérisque.Google Scholar
  50. [My]
    R. MYERS, Free involutions on lens spaces, Topology 20 (1981), 313–318.MathSciNetCrossRefzbMATHGoogle Scholar
  51. [Ne]
    M. NEWMAN, Integral matrices, Academic Press, New York and London, 1972.zbMATHGoogle Scholar
  52. [O]
    P. ORLIK, Seifert manifolds, Lecture Notes in Math., vol. 291, Springer-Verlag, Berlin and New York, 1972.zbMATHGoogle Scholar
  53. [Re]
    K. REIDEMEISTER, Homotopieringe und Linsenräume, Abh. Math. Sem. Univ. Hamburg 11 (1935), 102–109.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [Ro]
    D. ROLFSEN, Knots and links, Publish or Perish, Berkeley, CA, 1976.zbMATHGoogle Scholar
  55. [Ru 1]
    J.H. RUBINSTEIN, On 3-manifolds that have finite fundamental group and contain Klein bottles, Trans. Amer. Math. Soc. 251 (1979), 129–137.MathSciNetzbMATHGoogle Scholar
  56. [Ru 2]
    J.H. RUBINSTEIN, Representations of some 3-manifolds as 2-fold cyclic branched covers of S3, Notices Amer. Math. Soc. 25 (1978), abstract 78T-G7, A-18.Google Scholar
  57. [Ru 3]
    J.H. RUBINSTEIN, Involutions and the homeotopy groups of lens spaces, unpublished manuscript. (See also: Abstracts Amer. Math. Soc. 1 (1980), 773-57-12, 135.)Google Scholar
  58. [Sch]
    H. SCHUBERT, Knoten mit zwei Brücken, Math. Zeitschr. 65 (1956), 133–170.MathSciNetCrossRefzbMATHGoogle Scholar
  59. [Sei]
    H. SEIFERT, Topology of 3-dimensional fibred spaces, in: H. Seifert and W. Threlfall, A textbook of topology, Academic Press, New York, 1980. (English translation of: Acta Math. 60 (1933), 147–238.)Google Scholar
  60. [Sie]
    L.C. SIEBENMANN, Exercices sur les noeuds rationnels, notes, Orsay 1975. To appear in Astérisque.Google Scholar
  61. [Sp]
    E.H. SPANIER, Algebraic topology, McGraw-Hill, New York, 1966.zbMATHGoogle Scholar
  62. [St]
    N. STEENROD, The topology of fibre bundles, Princeton Univ. Press, Princeton N.J., 1951.zbMATHGoogle Scholar
  63. [Th 1]
    W.P. THURSTON, The geometry and topology of 3-manifolds, preprint, Princeton University.Google Scholar
  64. [Th 2]
    W.P. THURSTON, Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. AMS 6 (1982), 357–381.MathSciNetCrossRefzbMATHGoogle Scholar
  65. [Th 3]
    W.P. THURSTON, Three-manifolds with symmetry, preprint.Google Scholar
  66. [To 1]
    J.L. TOLLEFSON, Involutions on S1×S2 and other 3-manifolds, Trans. Amer. Math. Soc. 183 (1973), 139–152.MathSciNetzbMATHGoogle Scholar
  67. [To 2]
    J.L. TOLLEFSON, Involutions on Seifert fiber spaces, Pacific J. Math. 74 (1978), 519–529.MathSciNetCrossRefzbMATHGoogle Scholar
  68. [V 1]
    O. Ja. VIRO, Linkings, Two sheeted branched coverings and braids, Math. U.S.S.R. Sbornik 16 (1972), 222–236. (English translation of: Mat. Sb. 87 (1972), 216–228.)MathSciNetCrossRefzbMATHGoogle Scholar
  69. [V 2]
    O. Ja. VIRO, Nonprojecting isotopies and knots with homeomorphic coverings, J. Soviet Math. 12 (1979), 86–96. (English translation of: Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov 66 (1976), 133–147.)CrossRefzbMATHGoogle Scholar
  70. [W 1]
    F. WALDHAUSEN, Eine Klasse von 3-dimensionalen Mannifaltigkeiten I, II, Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117.MathSciNetCrossRefzbMATHGoogle Scholar
  71. [W 2]
    F. WALDHAUSEN, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. 87 (1968), 56–88.MathSciNetCrossRefzbMATHGoogle Scholar
  72. [W 3]
    F. WALDHAUSEN, Über Involutionen der 3-Sphäre, Topology 8 (1969), 81–91.MathSciNetCrossRefzbMATHGoogle Scholar
  73. [W 4]
    F. WALDHAUSEN, Heegaard Zerlegungen der 3-Sphäre, Topology 7 (1968), 195–203.MathSciNetCrossRefzbMATHGoogle Scholar
  74. [Wa]
    A.G. WASSERMAN, Equivariant differential topology, Topology 8 (1969), 127–150.MathSciNetCrossRefzbMATHGoogle Scholar
  75. [Wh]
    J.H.C. WHITEHEAD, On C1-complexes, Ann. of Math. 41 (1940), 809–824.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Craig Hodgson
    • 1
  • J. H. Rubinstein
    • 2
  1. 1.Princeton UniversityPrincetonUSA
  2. 2.University of MelbourneParkvilleAustralia

Personalised recommendations