Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-24T12:41:12.520Z Has data issue: false hasContentIssue false
  • English
  • Français

The Seifert Fiber Space Conjecture and Torus Theorem for Nonorientable 3-Manifolds

Published online by Cambridge University Press:  20 November 2018

Wolfgang Heil  and
Wilbur Whitten
Wolfgang Heil
Affiliation:
Department of Mathematics Florida State University Tallahassee, Florida 32306-3027 U.S. A.
Wilbur Whitten
Affiliation:
Department of Mathematics, University of Southwestern Louisiana Lafayette, Louisiana 70504 U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Seifert-fiber-space conjecture for nonorientable 3-manifolds states that if M denotes a compact, irreducible, nonorientable 3-manifold that is not a fake P2 x S1, if π1M is infinite and does not contain Z2 * Z2 as a subgroup, and if π1M does however contain a nontrivial, cyclic, normal subgroup, then M is a Seifert bundle. In this paper, we construct all compact, irreducible, nonorientable 3-manifolds (that do not contain a fake P2 × I) each of whose fundamental group contains Z2 * Z2 and an infinité cyclic, normal subgroup; none of these manifolds admits a Seifert fibration, but they satisfy Thurston's Geometrization Conjecture. We then reformulate the statement of the (nonorientable) SFS-conjecture and obtain a torus theorem for nonorientable manifolds.

Keywords

Primary: 57N10secondary: 57M50Seifert bundleSeifert bundle mod ℙ
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 4 , 01 December 1994 , pp. 482 - 489
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Casson, A. and Jungreis, D., Convergence groups and Seifert fibered 3-manifolds, preprint.Google Scholar
2. Epstein, D. B. A., Projective planes in 3-manifolds, Proc. London Math. Soc. (3) 11(1961), 469–484. 3 , Periodic flows on 3-manifolds, Ann. of Math. 95(1972), 66–82.Google Scholar
4. Feustel, C. D., On the torus theorem for closed 3-manifolds, Trans. Amer. Math. Soc. 217(1976), 45–47.Google Scholar
5. Feustel, C. D., On the torus theorem and its applications, Trans. Amer. Math. Soc. 217(1976), 1–43.Google Scholar
6. Gabai, D., Convergence groups are Fuchsian groups, Bull. Amer. Math. Soc. (2) 25(1991), 395–402.Google Scholar
7. Gabai, D., Convergence groups are Fuchsian groups, preprint.Google Scholar
8. Holzman, W. H., An equivariant torus theorem for involutions, Trans. Amer. Math. Soc. 326(1991), 887– 906.Google Scholar
9. Jaco, W. and Shalen, P., Seifert fiber spaces in 3-manifolds, Mem. Amer. Math. Soc. (220) 21(1979).Google Scholar
10. Johannson, K., Homotopy Equivalences of 3-Manifolds with Boundary, Lecture Notes in Math. 761, Springer, 1979.Google Scholar
11. Luft, E., Equivariant surgery on incompressible tori and Klein bottles in 3-manifolds with respect to involutions, Math. Ann. 272(1985), 519–544.Google Scholar
12. Meeks, W. H. and Scott, P., Finite group actions on 3-manifolds, Invent. Math. 86(1986), 287–346.Google Scholar
13. Orlik, P., Seifert Manifolds, Lecture Notes in Math. 291, Springer, 1972.Google Scholar
14. Scott, G. P., The geometries of 3-manifolds, Bull. London Math. Soc. 15(1983), 401–487.Google Scholar
15. Scott, G. P., Strong annulus and torus theorems and the enclosing property of characteristic submanifolds of 3-manifolds, Quart. J. Math. Oxford 35(1984), 485–506.Google Scholar
16. Tollefson, J., Free involutions on non-prime 3-manifolds, Osaka J. Math. 7(1970), 161–164.Google Scholar
17. Tollefson, J., Involution of Seifert fiber spaces, Pacific J. Math. 74(1978), 519–529.Google Scholar
18. Waldhausen, F., On the determination of some bounded 3-manifolds by their fundamental group alone, Proc. Internat. Symp. Topology, Herce-Novi, Yugoslavia 1968; Beograd, 1969, 331-332.Google Scholar
19. Whitten, W., Recognizing nonorientable Seifert bundles, J. Knot Theory Ramifications 1(1992), 471–475.Google Scholar