2-sided embeddings of projective planes into 3-manifolds

Ochiai, Mitsuyuki

doi:10.1090/S0002-9947-1982-0675072-3

$2$-sided embeddings of projective planes into $3$-manifolds
HTML articles powered by AMS MathViewer

by Mitsuyuki Ochiai PDF

Trans. Amer. Math. Soc. 274 (1982), 641-650 Request permission

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

D. B. A. Epstein, Projective planes in $3$-manifolds, Proc. London Math. Soc. (3) 11 (1961), 469–484. MR 152997, DOI 10.1112/plms/s3-11.1.469
Wolfgang Haken, Some results on surfaces in $3$-manifolds, Studies in Modern Topology, Math. Assoc. America, Buffalo, N.Y.; distributed by Prentice-Hall, Englewood Cliffs, N.J., 1968, pp. 39–98. MR 0224071
John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450
W. B. R. Lickorish, Homeomorphisms of non-orientable two-manifolds, Proc. Cambridge Philos. Soc. 59 (1963), 307–317. MR 145498, DOI 10.1017/s0305004100036926
Mitsuyuki Ochiai, Homeomorphisms on a three-dimensional handle, J. Math. Soc. Japan 30 (1978), no. 4, 697–702. MR 513078, DOI 10.2969/jmsj/03040697
Mitsuyuki Ochiai, Heegaard diagrams and Whitehead graphs, Math. Sem. Notes Kobe Univ. 7 (1979), no. 3, 573–591. MR 567245

Heegaard diagrams of torus bundles over

Yoko Tao, On fixed point free involutions of $S^{1}\times S^{2}$, Osaka Math. J. 14 (1962), 145–152. MR 140092
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