ON 4-MANIFOLDS WITH UNIVERSAL COVERING SPACE A COMPACT GEOMETRIC MANIFOLD

There are 11 closed 4-manifolds which admit geometries of compact type (S4, C P2 or S2 X S2) and two other closely related bundle spaces (S2 x S2 and the total space of the nontrivial R P2-bundle over S2). We s how that the homotopy type of such a manifold is determined up to an a mbiguity of order at most 4 by its quadratic 2-type, and this in turn is (in most cases) determined by the Euler characteristic, fundamental group and Stiefel-Whitney classes. In (at least) seven of the 13 case s, a PL 4-manifold with the same invariants as a geometric manifold or bundle space must be homeomorphic to it.