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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.
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Abstract

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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

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