S2-bundles over aspherical surfaces and 4-dimensional geometries

Author

Robin J. Cobb and Jonathan A. Hillman

Status

Research Report 96-18
Date: 17 April 1996

Abstract

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 x E2 or S2 x H2 [depending on whether chi(M) = 0 or chi(M) < 0]. Conversely a geometric closed, connected 4-manifold M of type S2 x E2 or S2 x H2 is the total space of an S2-bundle over a closed, connected aspherical surface precisely when its fundamental group Pi_1(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 Pi_1(M') is congruent to Z/2 Z x K where K is torsion free.

Key phrases

aspherical surface. sphere bundle. 4-dimensional geometry. Stiefel-Whitney class.

AMS Subject Classification (1991)

Primary: 57N50
Secondary: 57N13, 55R25

Content

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