Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-06-02T23:17:29.677Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Baues, H. J., ‘Combinatorial homotopy and 4-dimensional complexes’, in: De Gruyter Expositions in Mathematics 2 (de Gruyter, Berlin-New York, 1991).Google Scholar
[2]Cochran, T. D. and Habegger, N., ‘On the homotopy theory of simply-connected fourmanifolds’, Topology 29 (1990), 419–440.CrossRefGoogle Scholar
[3]Frank, D., ‘The signature mod 8’, Comment. Math. Helv. 48 (1973), 520–524.CrossRefGoogle Scholar
[4]Freedman, M. H. and Quinn, F. R., Topology of Four-Manifolds (Princeton University Press, Princeton N.J., 1990).CrossRefGoogle Scholar
[5]Habegger, N., ‘Une variété de dimension 4 avec forme d'intersection paire et signature -8’, Comment. Math. Helv. 57 (1982), 22–24.CrossRefGoogle Scholar
[6]Hambleton, I. and Kreck, M., ‘On the homotopy classification of topological 4-manifolds with finite fundamental group’, Math. Ann. 280 (1988), 85–104.CrossRefGoogle Scholar
[7]Higman, G., ‘The units of group rings’, Proc. London Math. Soc. 46 (1940), 231–248.CrossRefGoogle Scholar
[8]Hillman, J. A., ‘On 4-manifolds homotopy equivalent to surface bundles over surfaces’, Top. Appl. 40 (1991), 275–286.CrossRefGoogle Scholar
[9]Hillman, J. A., ‘Geometries on 4-manifolds, Euler characteristic and elementary amenable groups’, in: Proceedings of the International Conference on Knots, Osaka 1990 (de Gruyter, Berlin, 1992) pp. 25–46.Google Scholar
[10]Hillman, J. A., ‘On 4-manifolds with universal covering space S 2 × R 2 or S 3 × R’, Top. Appl. (to appear).Google Scholar
[11]Kim, M. H., Kojima, S. and Raymond, F., ‘Homotopy invariants of nonorientable 4-manifolds’, Trans. Amer. Math. Soc. 333 (1992), 71–83.CrossRefGoogle Scholar
[12]Matsumoto, T., ‘On homotopy equivalences of S2 × R P2 to itself’, J. Math. Kyoto University 19 (1979), 1–17.Google Scholar
[13]Olum, P., ‘Mappings of manifolds and the notion of degree’, Ann. Math. 58 (1953), 458–480.CrossRefGoogle Scholar
[14]Rubermann, D., ‘Invariant knots of free involutions of S4’, Top. Appl. 18 (1984), 217–224.CrossRefGoogle Scholar
[15]Siebenmann, L. C., ‘Topological manifolds’, in: Proceedings of the International Congress of Mathematicians, Nice, 1970 (Gauthier-Villars, Paris, 1971), vol. 2 pp. 133–163.Google Scholar
[16]Wall, C. T. C., Surgery on Compact Manifolds (Academic Press, London, 1970).Google Scholar
[17]Wall, C. T. C., ‘Classification of hermitian forms: VI group rings’, Ann. Math. 103 (1976), 1–80.CrossRefGoogle Scholar