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Involutions and isotopies of lens spaces

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Knot Theory and Manifolds

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1144))

Abstract

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.

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References

  1. J.W. ALEXANDER, Note on Riemann spaces, Bull. Amer. Math. Soc. 26 (1920), 370–372.

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. 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.

    Article  MathSciNet  MATH  Google Scholar 

  4. J.S. BIRMAN and H. HILDEN, Heegaard splittings of branched coverings of S3, Trans. Amer. Math. Soc. 213 (1975), 315–352.

    MathSciNet  MATH  Google Scholar 

  5. 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. F. BONAHON, Diffeotopies des espaces lenticulaires, preprint.

    Google Scholar 

  7. F. BONAHON and J-P OTAL, Scindements de Heegaard des espaces lenticulaires, preprint.

    Google Scholar 

  8. G.E. BREDON, Introduction to compact transformation groups, Academic Press, New York and London, 1972.

    MATH  Google Scholar 

  9. E.J. BRODY, The topological classification of the lens spaces, Ann. of Math. 71 (1960), 163–184.

    Article  MathSciNet  MATH  Google Scholar 

  10. W. BROWDER and G.R. LIVESAY, Fixed point free involutions on homotopy spheres, Bull. Amer. Math. Soc. 73 (1967), 242–245.

    Article  MathSciNet  MATH  Google Scholar 

  11. 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.

    Book  MATH  Google Scholar 

  12. 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. A. DURFEE and L. KAUFFMAN, Periodicity of branched cyclic covers, Math. Ann. 218 (1975), 157–174.

    Article  MathSciNet  MATH  Google Scholar 

  14. R. ENGMANN, Nicht homöomorphe Heegaard Zerlegungen vom Geschlecht 2 der zusammenhängenden Summe zweier Linsenräume, Abh. Math. Sem. Univ. Hamburg 35 (1970), 33–38.

    Article  MathSciNet  MATH  Google Scholar 

  15. D.B.A. EPSTEIN, Curves on 2-manifolds and isotopies, Acta Math. 115 (1966) 83–107.

    Article  MathSciNet  MATH  Google Scholar 

  16. R.H. FOX, A note on branched cyclic coverings of spheres, Rev. Mat. Hisp.-Amer. 32 (1972), 158–166.

    MathSciNet  MATH  Google Scholar 

  17. W. FRANZ, Über die Torsion einer Überdeckung, J.f. reine u. angewandte Math. 173 (1935), 245–253.

    MathSciNet  MATH  Google Scholar 

  18. M. GOLUBITSKY and V. GUILLEMIN, Stable mappings and their singularities, Springer-Verlag, New York, Heidelberg and Berlin, 1973.

    Book  MATH  Google Scholar 

  19. C. McA. GORDON and R.A. LITHERLAND, Incompressible surfaces in branched coverings, to appear.

    Google Scholar 

  20. 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. R.S. HAMILTON, Three manifolds with positive Ricci curvature, J. of Diff. Geom 17 (1982), 255–306.

    MathSciNet  MATH  Google Scholar 

  22. M.-E. HAMSTROM, Homotopy in homeomorphism spaces, TOP and PL. Bull. Amer. Math. Soc. 80 (1974), 207–230.

    Article  MathSciNet  MATH  Google Scholar 

  23. R.I. HARTLEY, Knots and involutions, Math. Zeitschr. 171 (1980), 175–185.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. HATCHER, Homeomorphisms of sufficiently large p2-irreducible 3-manifolds, Topology 15 (1976), 343–347.

    Article  MathSciNet  MATH  Google Scholar 

  25. A. HATCHER, A proof of the Smale Conjecture, Diff(S3)≃O(4), Ann. of Math. 117 (1983), 553–607.

    Article  MathSciNet  MATH  Google Scholar 

  26. W. HEIL, On P2-irreducible 3-manifolds, Bull. Amer. Math. Soc. 75 (1969), 772–775.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. HEMPEL, 3-manifolds, Annals of Math. Studies No. 86, Princeton Univ. Press, Princeton, N.J., 1976.

    MATH  Google Scholar 

  28. H. HILDEN, Every closed orientable 3-manifold is a 3-fold branched covering space of S3, Bull. Amer. Soc. 80 (1974), 1243–1244.

    Article  MathSciNet  MATH  Google Scholar 

  29. M.W. HIRSCH, Differential topology, Springer-Verlag, New York, Heidelberg and Berlin, 1976.

    Book  MATH  Google Scholar 

  30. 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.)

    Article  MATH  Google Scholar 

  31. 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.)

    MATH  Google Scholar 

  32. W. JACO, Lectures on three-manifold topology, CBMS Regional Conference Series in Math., No. 43, Amer. Math. Soc., Providence R.I., 1980.

    MATH  Google Scholar 

  33. B. JAHREN, One-parameter families of spheres in 3-manifolds, Ph.D. thesis, Princeton Univ., 1975.

    Google Scholar 

  34. P.K. KIM, PL involutions on lens spaces and other 3-manifolds, Proc. Amer. Math. Soc. 44 (1974), 467–473.

    MathSciNet  MATH  Google Scholar 

  35. P.K. KIM, Cyclic actions on lens spaces, Trans. Amer. Math. Soc. 237 (1978), 121–144.

    Article  MathSciNet  MATH  Google Scholar 

  36. 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. P.K. KIM and J.L. TOLLEFSON, Splitting the PL involutions of nonprime 3-manifolds, Michigan Math. J. 27 (1980), 259–274.

    Article  MathSciNet  MATH  Google Scholar 

  38. 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. K.W. KWUN, Scarcity of orientation-reversing PL involutions of lens spaces, Michigan Math. J. 17 (1970), 355–358.

    Article  MathSciNet  MATH  Google Scholar 

  40. K.W. KWUN, Sense-preserving PL involutions of some lens spaces, Michigan Math. J. 20 (1973), 73–77.

    Article  MathSciNet  MATH  Google Scholar 

  41. F. LAUDENBACH, Topologie de la dimension trois: homotopie et isotopie, Astérisque No. 12, Soc. Math de France, Paris, 1974.

    MATH  Google Scholar 

  42. D. McCULLOUGH, Homotopy groups of the space of self-homotopy-equivalences, Trans. Amer. Math. Soc. 264 (1981), 151–163.

    Article  MathSciNet  MATH  Google Scholar 

  43. D. McCULLOUGH, The group of homotopy euqivalences for a connected sum of closed aspherical manifolds, Indian Univ. Math. J. 30 (1981), 249–260.

    Article  MathSciNet  MATH  Google Scholar 

  44. J.W. MILNOR, Morse theory, Annals of Math. Studies No. 51, Princeton Univ. Press, Princeton N.J., 1963.

    Book  MATH  Google Scholar 

  45. J.W. MILNOR, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426.

    Article  MathSciNet  MATH  Google Scholar 

  46. 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.

    Article  MathSciNet  MATH  Google Scholar 

  47. 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.

    Article  MathSciNet  MATH  Google Scholar 

  48. J.M. MONTESINOS, Minimal plat presentations of prime knots and links are not unique, Canad. J. Math. 28 (1976), 161–167.

    Article  MathSciNet  MATH  Google Scholar 

  49. J.M. MONTESINOS, Revêtements ramifiés de noeuds, espace fibrés de Seifert et scindements de Heegaard, preprint. To appear in Astérisque.

    Google Scholar 

  50. R. MYERS, Free involutions on lens spaces, Topology 20 (1981), 313–318.

    Article  MathSciNet  MATH  Google Scholar 

  51. M. NEWMAN, Integral matrices, Academic Press, New York and London, 1972.

    MATH  Google Scholar 

  52. P. ORLIK, Seifert manifolds, Lecture Notes in Math., vol. 291, Springer-Verlag, Berlin and New York, 1972.

    MATH  Google Scholar 

  53. K. REIDEMEISTER, Homotopieringe und Linsenräume, Abh. Math. Sem. Univ. Hamburg 11 (1935), 102–109.

    Article  MathSciNet  MATH  Google Scholar 

  54. D. ROLFSEN, Knots and links, Publish or Perish, Berkeley, CA, 1976.

    MATH  Google Scholar 

  55. J.H. RUBINSTEIN, On 3-manifolds that have finite fundamental group and contain Klein bottles, Trans. Amer. Math. Soc. 251 (1979), 129–137.

    MathSciNet  MATH  Google Scholar 

  56. 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. 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. H. SCHUBERT, Knoten mit zwei Brücken, Math. Zeitschr. 65 (1956), 133–170.

    Article  MathSciNet  MATH  Google Scholar 

  59. 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. L.C. SIEBENMANN, Exercices sur les noeuds rationnels, notes, Orsay 1975. To appear in Astérisque.

    Google Scholar 

  61. E.H. SPANIER, Algebraic topology, McGraw-Hill, New York, 1966.

    MATH  Google Scholar 

  62. N. STEENROD, The topology of fibre bundles, Princeton Univ. Press, Princeton N.J., 1951.

    MATH  Google Scholar 

  63. W.P. THURSTON, The geometry and topology of 3-manifolds, preprint, Princeton University.

    Google Scholar 

  64. W.P. THURSTON, Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. AMS 6 (1982), 357–381.

    Article  MathSciNet  MATH  Google Scholar 

  65. W.P. THURSTON, Three-manifolds with symmetry, preprint.

    Google Scholar 

  66. J.L. TOLLEFSON, Involutions on S1×S2 and other 3-manifolds, Trans. Amer. Math. Soc. 183 (1973), 139–152.

    MathSciNet  MATH  Google Scholar 

  67. J.L. TOLLEFSON, Involutions on Seifert fiber spaces, Pacific J. Math. 74 (1978), 519–529.

    Article  MathSciNet  MATH  Google Scholar 

  68. 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.)

    Article  MathSciNet  MATH  Google Scholar 

  69. 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.)

    Article  MATH  Google Scholar 

  70. F. WALDHAUSEN, Eine Klasse von 3-dimensionalen Mannifaltigkeiten I, II, Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  72. F. WALDHAUSEN, Über Involutionen der 3-Sphäre, Topology 8 (1969), 81–91.

    Article  MathSciNet  MATH  Google Scholar 

  73. F. WALDHAUSEN, Heegaard Zerlegungen der 3-Sphäre, Topology 7 (1968), 195–203.

    Article  MathSciNet  MATH  Google Scholar 

  74. A.G. WASSERMAN, Equivariant differential topology, Topology 8 (1969), 127–150.

    Article  MathSciNet  MATH  Google Scholar 

  75. J.H.C. WHITEHEAD, On C1-complexes, Ann. of Math. 41 (1940), 809–824.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Princeton University, 08544, Princeton, NJ, USA

    Craig Hodgson

  2. University of Melbourne, Parkville, Australia

    J. H. Rubinstein

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  1. Craig Hodgson
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  2. J. H. Rubinstein
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Dale Rolfsen

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© 1985 Springer-Verlag

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Hodgson, C., Rubinstein, J.H. (1985). Involutions and isotopies of lens spaces. In: Rolfsen, D. (eds) Knot Theory and Manifolds. Lecture Notes in Mathematics, vol 1144. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075012

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  • DOI: https://doi.org/10.1007/BFb0075012

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  • Online ISBN: 978-3-540-39616-1

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