An n-manifold X is geometric in the sense of Thurston if its universal covering space X admits a complete homogeneous Riemannian metric, π_1(X) acts isometrically on X and X = π_1 (X)X has finite volume. Every closed 1- or 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 §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.
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