# 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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