J Austral Math Soc Ser A 55 pp137--148, 1993.

On 4-Manifolds with Universal Covering Space a Compact Geometric Manifold

Jonathan A. Hillman

Abstract

There are 11 closed 4-manifolds which admit geometries of compact type (S⁴, C P² or S² ×S²) and two other closely related bundle spaces (S² ~× S² and the total space of the nontrivial RP²-bundle over S²). We show that the homotopy type of such a manifold is determined up to an ambiguity 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 cases, a PL 4-manifold with the same invariants as a geometric manifold or bundle space must be homeomorphic to it.

1991 AMS Subject Classification: 57N13

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Authors

Johnathan A. Hillman

School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, AUSTRALIA.