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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$S^{2}$-bundles over aspherical surfaces
and 4-dimensional geometries


Authors: Robin J. Cobb and Jonathan A. Hillman
Journal: Proc. Amer. Math. Soc. 125 (1997), 3415-3422
MSC (1991): Primary 57N50; Secondary 57N13, 55R25
MathSciNet review: 1422856
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Abstract: Melvin has shown that closed 4-manifolds that arise as $S^{2}$-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 $S^{2}\times E^{2}$ or $S^{2}\times \mathbb {H}^{2}$ [depending on whether $\chi (M)=0$ or $\chi (M)<0$]. Conversely a geometric closed, connected 4-manifold $M$ of type $S^{2}\times E^{2}$ or $S^{2}\times \mathbb {H}^{2}$ is the total space of an $S^{2}$-bundle over a closed, connected aspherical surface precisely when its fundamental group $\Pi _{1}(M)$ is torsion free. Furthermore the total spaces of $\mathbb {RP}^{2}$-bundles over closed, connected aspherical surfaces are all geometric. Conversely a geometric closed, connected 4-manifold $M'$ is the total space of an $\mathbb {RP}^{2}$-bundle if and only if $\Pi _{1}(M')\cong \mathbb {Z}/2\mathbb {Z}\times K$ where $K$ is torsion free.


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Additional Information

Robin J. Cobb
Affiliation: School of Mathematics and Statistics, The University of Sydney, Sydney, New South Wales 2006, Australia
Email: robinc@maths.usyd.edu.au

Jonathan A. Hillman
Affiliation: School of Mathematics and Statistics, The University of Sydney, Sydney, New South Wales 2006, Australia
Email: jonh@maths.usyd.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9939-97-04099-9
PII: S 0002-9939(97)04099-9
Keywords: Aspherical surface, $S^{2}$-bundle, 4-dimensional geometry, Stiefel-Whitney class
Received by editor(s): May 10, 1996
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 1997 American Mathematical Society