On 4-manifolds which admit geometric decompositions

Abstract

An $n$ -manifold $X$ is geometric in the sense of Thurston if its universal covering space $\tilde{X}$ admits a complete homogeneous Riemannian metric, $\pi_{1}(X)$ acts isometrically on $\tilde{X}$ and $X=\pi_{1}(X)\backslash \tilde{X}$ has finite volume.Every closed 1-or 2-manifold is geometric.Much current research on 3-manifolds is guided by Thurston's Geometrization Con- jecture, that every closed irreducible 3-manifold admits a finite decomposition into geometric pieces [Th82].There are 19 maximal 4-dimensional geometries; one of these is in fact an infinite family of closely related geometries and one is not realized by any closed 4-manifold [F].Our first result (in \S 1) shall illustrate the limitations of geometry in higher dimensions by showing that a closed 4-manifold which admits a finite decomposition into geometric pieces is usually either geometric or aspherical.The geometric viewpoint is nevertheless of considerable interest in connection with complex surfaces [Ue90,91, W185,86].We show also that except for the geometries $S^{2}\cross H^{2}$ , $H^{2}\cross H^{2},$ $H^{2}\cross E^{2}

Locations

• Journal of the Mathematical Society of Japan - View - PDF