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
J.W. ALEXANDER, Note on Riemann spaces, Bull. Amer. Math. Soc. 26 (1920), 370–372.
K. ASANO, Homeomorphisms of prism manifolds, Yokohama Math. J. 26 (1978), 19–25.
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.
J.S. BIRMAN and H. HILDEN, Heegaard splittings of branched coverings of S3, Trans. Amer. Math. Soc. 213 (1975), 315–352.
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.)
F. BONAHON, Diffeotopies des espaces lenticulaires, preprint.
F. BONAHON and J-P OTAL, Scindements de Heegaard des espaces lenticulaires, preprint.
G.E. BREDON, Introduction to compact transformation groups, Academic Press, New York and London, 1972.
E.J. BRODY, The topological classification of the lens spaces, Ann. of Math. 71 (1960), 163–184.
W. BROWDER and G.R. LIVESAY, Fixed point free involutions on homotopy spheres, Bull. Amer. Math. Soc. 73 (1967), 242–245.
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.
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.
A. DURFEE and L. KAUFFMAN, Periodicity of branched cyclic covers, Math. Ann. 218 (1975), 157–174.
R. ENGMANN, Nicht homöomorphe Heegaard Zerlegungen vom Geschlecht 2 der zusammenhängenden Summe zweier Linsenräume, Abh. Math. Sem. Univ. Hamburg 35 (1970), 33–38.
D.B.A. EPSTEIN, Curves on 2-manifolds and isotopies, Acta Math. 115 (1966) 83–107.
R.H. FOX, A note on branched cyclic coverings of spheres, Rev. Mat. Hisp.-Amer. 32 (1972), 158–166.
W. FRANZ, Über die Torsion einer Überdeckung, J.f. reine u. angewandte Math. 173 (1935), 245–253.
M. GOLUBITSKY and V. GUILLEMIN, Stable mappings and their singularities, Springer-Verlag, New York, Heidelberg and Berlin, 1973.
C. McA. GORDON and R.A. LITHERLAND, Incompressible surfaces in branched coverings, to appear.
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.
R.S. HAMILTON, Three manifolds with positive Ricci curvature, J. of Diff. Geom 17 (1982), 255–306.
M.-E. HAMSTROM, Homotopy in homeomorphism spaces, TOP and PL. Bull. Amer. Math. Soc. 80 (1974), 207–230.
R.I. HARTLEY, Knots and involutions, Math. Zeitschr. 171 (1980), 175–185.
A. HATCHER, Homeomorphisms of sufficiently large p2-irreducible 3-manifolds, Topology 15 (1976), 343–347.
A. HATCHER, A proof of the Smale Conjecture, Diff(S3)≃O(4), Ann. of Math. 117 (1983), 553–607.
W. HEIL, On P2-irreducible 3-manifolds, Bull. Amer. Math. Soc. 75 (1969), 772–775.
J. HEMPEL, 3-manifolds, Annals of Math. Studies No. 86, Princeton Univ. Press, Princeton, N.J., 1976.
H. HILDEN, Every closed orientable 3-manifold is a 3-fold branched covering space of S3, Bull. Amer. Soc. 80 (1974), 1243–1244.
M.W. HIRSCH, Differential topology, Springer-Verlag, New York, Heidelberg and Berlin, 1976.
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.)
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.)
W. JACO, Lectures on three-manifold topology, CBMS Regional Conference Series in Math., No. 43, Amer. Math. Soc., Providence R.I., 1980.
B. JAHREN, One-parameter families of spheres in 3-manifolds, Ph.D. thesis, Princeton Univ., 1975.
P.K. KIM, PL involutions on lens spaces and other 3-manifolds, Proc. Amer. Math. Soc. 44 (1974), 467–473.
P.K. KIM, Cyclic actions on lens spaces, Trans. Amer. Math. Soc. 237 (1978), 121–144.
P.K. KIM, Involutions on Klein spaces M(p,q), Notices Amer. Math. Soc. 25 (1978), abstract 752-57-12, A-147.
P.K. KIM and J.L. TOLLEFSON, Splitting the PL involutions of nonprime 3-manifolds, Michigan Math. J. 27 (1980), 259–274.
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.
K.W. KWUN, Scarcity of orientation-reversing PL involutions of lens spaces, Michigan Math. J. 17 (1970), 355–358.
K.W. KWUN, Sense-preserving PL involutions of some lens spaces, Michigan Math. J. 20 (1973), 73–77.
F. LAUDENBACH, Topologie de la dimension trois: homotopie et isotopie, Astérisque No. 12, Soc. Math de France, Paris, 1974.
D. McCULLOUGH, Homotopy groups of the space of self-homotopy-equivalences, Trans. Amer. Math. Soc. 264 (1981), 151–163.
D. McCULLOUGH, The group of homotopy euqivalences for a connected sum of closed aspherical manifolds, Indian Univ. Math. J. 30 (1981), 249–260.
J.W. MILNOR, Morse theory, Annals of Math. Studies No. 51, Princeton Univ. Press, Princeton N.J., 1963.
J.W. MILNOR, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426.
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.
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.
J.M. MONTESINOS, Minimal plat presentations of prime knots and links are not unique, Canad. J. Math. 28 (1976), 161–167.
J.M. MONTESINOS, Revêtements ramifiés de noeuds, espace fibrés de Seifert et scindements de Heegaard, preprint. To appear in Astérisque.
R. MYERS, Free involutions on lens spaces, Topology 20 (1981), 313–318.
M. NEWMAN, Integral matrices, Academic Press, New York and London, 1972.
P. ORLIK, Seifert manifolds, Lecture Notes in Math., vol. 291, Springer-Verlag, Berlin and New York, 1972.
K. REIDEMEISTER, Homotopieringe und Linsenräume, Abh. Math. Sem. Univ. Hamburg 11 (1935), 102–109.
D. ROLFSEN, Knots and links, Publish or Perish, Berkeley, CA, 1976.
J.H. RUBINSTEIN, On 3-manifolds that have finite fundamental group and contain Klein bottles, Trans. Amer. Math. Soc. 251 (1979), 129–137.
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.
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.)
H. SCHUBERT, Knoten mit zwei Brücken, Math. Zeitschr. 65 (1956), 133–170.
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.)
L.C. SIEBENMANN, Exercices sur les noeuds rationnels, notes, Orsay 1975. To appear in Astérisque.
E.H. SPANIER, Algebraic topology, McGraw-Hill, New York, 1966.
N. STEENROD, The topology of fibre bundles, Princeton Univ. Press, Princeton N.J., 1951.
W.P. THURSTON, The geometry and topology of 3-manifolds, preprint, Princeton University.
W.P. THURSTON, Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. AMS 6 (1982), 357–381.
W.P. THURSTON, Three-manifolds with symmetry, preprint.
J.L. TOLLEFSON, Involutions on S1×S2 and other 3-manifolds, Trans. Amer. Math. Soc. 183 (1973), 139–152.
J.L. TOLLEFSON, Involutions on Seifert fiber spaces, Pacific J. Math. 74 (1978), 519–529.
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.)
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.)
F. WALDHAUSEN, Eine Klasse von 3-dimensionalen Mannifaltigkeiten I, II, Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117.
F. WALDHAUSEN, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. 87 (1968), 56–88.
F. WALDHAUSEN, Über Involutionen der 3-Sphäre, Topology 8 (1969), 81–91.
F. WALDHAUSEN, Heegaard Zerlegungen der 3-Sphäre, Topology 7 (1968), 195–203.
A.G. WASSERMAN, Equivariant differential topology, Topology 8 (1969), 127–150.
J.H.C. WHITEHEAD, On C1-complexes, Ann. of Math. 41 (1940), 809–824.
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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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