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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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$S^2$-bundles over aspherical surfaces and 4-dimensional geometries
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by Robin J. Cobb and Jonathan A. Hillman PDF
Proc. Amer. Math. Soc. 125 (1997), 3415-3422 Request permission

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
  • Received by editor(s): May 10, 1996
  • Communicated by: Ronald A. Fintushel
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 3415-3422
  • MSC (1991): Primary 57N50; Secondary 57N13, 55R25
  • DOI: https://doi.org/10.1090/S0002-9939-97-04099-9
  • MathSciNet review: 1422856