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Minimal atlases on 3-manifolds

Published online by Cambridge University Press:  24 October 2008

J. C. Gómez-Larrañaga ,
F. González-Acuña  and
Jim Hoste
J. C. Gómez-Larrañaga
Affiliation:
C.I.C.B.–U.A.E.M., Av. Institute Literario 100, Toluca, Edo. México and Instituto de Matematicas, U.N.A.M.
F. González-Acuña
Affiliation:
Instituto de Matemáticas, U.N.A.M., Ciudad Universitaria, 04510 México D.F.
Jim Hoste
Affiliation:
Pitzer College, Claremont, CA 91711, U.S.A.
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Abstract

In this paper we investigate the minimal number of charts (spaces homeomorphic to open subsets of ℝ3) needed to cover a closed 3-manifold M3. We prove that this number is two if the Bockstein βω1(M3)∈H2(M3;ℤ) of the first Stiefel-Whitney class of M3 is zero and three if it is not. We state several properties equivalent to βω1(M3) = 0, for example the condition of being covered by two orientable subspaces, and the condition that all torsion elements of H1(M3;ℤ) be represented by orientation-preserving loops. We also show that the non-orientable 3-manifolds which can be covered with two charts are precisely those manifolds which can be described as sewn-up r-link exteriors. These are manifolds obtained by removing from S3 the interiors of two disjoint handlebodies of genus r and then identifying the two boundary components by a homeomorphism.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 1 , January 1991 , pp. 105 - 115
Copyright
Copyright © Cambridge Philosophical Society 1991

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