Key Takeaway: A closed 4-manifold admitting a finite decomposition into geometric pieces is usually either geometric or aspherical, with the exception of the geometries $S2cross H2$, where the geometric viewpoint is limited in higher dimensions.

Abstract

$\tilde{X}$ 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 1or 2-manifold is geometric. Much current research on 3-manifolds is guided by Thurston’s Geometrization Conjecture, 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 $S^{2}\cross H^{2}$ ,