S2-Bundles Over Aspherical Surfaces and 4-Dimensional Geometries

Melvin has shown that closed 4-manifolds that arise as S2-bundles over closed, connected aspherical surfaces are classified up to diffeomorphism by the Stiefel-Whitney classes of the associated bundles. We show that each such 4-manifold admits one of the geometries S2× E2or S2× H2[depending on wheth...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 1997-11, Vol.125 (11), p.3415-3422
Main Authors: Cobb, Robin J., Hillman, Jonathan A.
Format: Article
Language:English
Subjects:
Geometry
Klein bottles
Mathematical functions
Mathematical manifolds
Mathematical surfaces
Mathematical theorems
Orientable surfaces
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Summary:Melvin has shown that closed 4-manifolds that arise as S2-bundles over closed, connected aspherical surfaces are classified up to diffeomorphism by the Stiefel-Whitney classes of the associated bundles. We show that each such 4-manifold admits one of the geometries S2× E2or S2× H2[depending on whether χ(M) = 0 or$\chi(M) < 0$]. Conversely a geometric closed, connected 4-manifold M of type S2× E2or S2× H2is the total space of an S2-bundle over a closed, connected aspherical surface precisely when its fundamental group Π1(M) is torsion free. Furthermore the total spaces of RP2-bundles over closed, connected aspherical surfaces are all geometric. Conversely a geometric closed, connected 4-manifold M' is the total space of an RP2-bundle if and only if Π1(M') ≅ Z/2Z × K where K is torsion free.
ISSN:0002-9939
1088-6826