Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ 2$-sided embeddings of projective planes into $ 3$-manifolds

Author: Mitsuyuki Ochiai
Journal: Trans. Amer. Math. Soc. 274 (1982), 641-650
MSC: Primary 57N10; Secondary 57M40, 57Q25
MathSciNet review: 675072
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ be a nonorientable closed $ 3$-manifold which admits a $ 2$-sided embedding of a projective plane. Then we first prove the following theorem: If $ M$ has a Heegaard splitting of genus two, then $ M$ is homeomorphic to $ {P^{2}}\times {S^{1}}$. Next, let $ M$ be a nonorientable $ 3$-manifold whose fundamental group is abelian. We verify that if $ M$ has a Heegaard splitting of genus two, then $ M$ is either the nonorientable $ 2$-sphere bundle over the circle or $ {P^{2}}\times {S^{1}}$.

References [Enhancements On Off] (What's this?)

  • [1] D. B. A. Epstein, Projective planes in 3-manifolds, Proc. London Math. Soc. (3) 11 (1961), 469–484. MR 0152997
  • [2] Wolfgang Haken, Some results on surfaces in 3-manifolds, Studies in Modern Topology, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), 1968, pp. 39–98. MR 0224071
  • [3] John Hempel, 3-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
  • [4] William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450
  • [5] W. B. R. Lickorish, Homeomorphisms of non-orientable two-manifolds, Proc. Cambridge Philos. Soc. 59 (1963), 307–317. MR 0145498
  • [6] Mitsuyuki Ochiai, Homeomorphisms on a three-dimensional handle, J. Math. Soc. Japan 30 (1978), no. 4, 697–702. MR 513078, 10.2969/jmsj/03040697
  • [7] Mitsuyuki Ochiai, Heegaard diagrams and Whitehead graphs, Math. Sem. Notes Kobe Univ. 7 (1979), no. 3, 573–591. MR 567245
  • [8] M. Ochiai and M. Takahashi, Heegaard diagrams of torus bundles over $ S^1$, preprint.
  • [9] Yoko Tao, On fixed point free involutions of 𝑆¹×𝑆², Osaka Math. J. 14 (1962), 145–152. MR 0140092
  • [10] John Stallings, On fibering certain 3-manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 95–100. MR 0158375

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57N10, 57M40, 57Q25

Retrieve articles in all journals with MSC: 57N10, 57M40, 57Q25

Additional Information

Article copyright: © Copyright 1982 American Mathematical Society