On 4-manifolds with universal covering space \(S^ 2\times R^ 2\) or \(S^ 3\times R\)
The author sets out to characterise algebraically those closed connected 4-manifolds covered by either \(S^ 2 \times \mathbb{R}^ 2\) or \(S^ 3 \times \mathbb{R}^ 1\), and to bring the resulting manifolds into the geometric framework established by W. Thurston and C. T. C. Wall. In the first case one's model is an \(S^ 2\)-bundle over an aspherical surface, in the second a fibration over \(S^ 1\) with fibre an elliptic 3- manifold. These models do indeed tell a great deal of the story, which (subject to certain technical assumptions) is developed in the sequence: fundamental group \(\to\) homotopy type \(\to\) homeomorphism type using surgical methods. This works better in the first case than in the second, partly because of our continuing lack of a complete theory for 3- manifolds with finite fundamental group.
closed connected 4-manifolds covered by \(S^ 2 \times \mathbb{R}^ 2\) or \(S^ 3 \times \mathbb{R}^ 1\)
\(S^ 2\)-bundle over an aspherical surface
fibration over \(S^ 1\) with fibre an elliptic 3-manifold
scientific article; zbMATH DE number 447264
scientific article