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On 4-manifolds with universal covering space a compact geometric manifold

Part of: Topological manifolds

Published online by Cambridge University Press:  09 April 2009

Jonathan A. Hillman
Jonathan A. Hillman
Affiliation:
School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, AUSTRALIA
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Abstract

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There are 11 closed 4-manifolds which admit geometries of compact type (S4, CP2 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.

Keywords

4-manifoldgeometryk-invariantquadratic 2-typeStiefel-Whitney class.

MSC classification

Secondary: 57N13: Topology of $E^4$, $4$-manifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 55 , Issue 2 , October 1993 , pp. 137 - 148
Copyright
Copyright © Australian Mathematical Society 1993

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