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Minimal 4-manifolds for groups of cohomological dimension 2

Published online by Cambridge University Press:  20 January 2009

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

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We show that if π is a group with a finite 2-dimensional Eilenberg-Mac Lane complex then the minimum of the Euler characteristics of closed 4-manifolds with fundamental group π is 2χ(K(π, 1)). If moreover M is such a manifold realizing this minimum then π2(M) ≅ Similarly, if π is a PD3-group and w1(M) is the canonical orientation character of π then χ(M)≧l and π2(M) is stably isomorphic to the augmentation ideal of Z[π].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 37 , Issue 3 , October 1994 , pp. 455 - 461
Copyright
Copyright © Edinburgh Mathematical Society 1994

References

REFERENCES

1.Dicks, W. and Dunwoody, M. J., Groups acting on Graphs (Cambridge studies in advanced mathematics 17, Cambridge University Press, Cambridge—New York—New Rochelle—Melbourne—Sydney, 1989).Google Scholar
2.Freedman, M. H. and Quinn, F., Topology of 4-Manifolds (Princeton University Press, Princeton, New Jersey, 1990).Google Scholar
3.Hambleton, I. and Kreck, M., Smooth structures on algebraic surfaces with finite fundamental group, Invent. Math. 102 (1990), 109–114.CrossRefGoogle Scholar
4.Hausmann, J.-C. and Weinberger, S., Caractérisques d'Euler et groupes fondamentaux des variétés de dimension 4, Comment. Math. Helv. 60 (1985), 139–144.CrossRefGoogle Scholar
5.Hillman, J. A., 2-Knots and their Groups (Australian Mathematical Society Lecture Series 5, Cambridge University Press, Cambridge—New York—New Rochelle—Melbourne—Sydney, 1989).Google Scholar
6.Hillman, J. A., A homotopy fibration theorem in dimension four, Topology Appl. 33 (1989), 151–161.CrossRefGoogle Scholar
7.Hillman, J. A., On 4-manifolds homotopy equivalent to surface bundles over surfaces, Topology Appl. 40 (1991), 275–286.CrossRefGoogle Scholar
8.Hillman, J. A., Elementary amenable groups and 4-manifolds with Euler characteristic 0, J. Austral. Math. Soc. 50 (1991), 160–170.CrossRefGoogle Scholar
9.Kaplansky, I. R., Fields and Rings (Chicago University Press, Chicago—London, 1969).Google Scholar
10.Kreck, M., Surgery and Duality, preprint, Johannes-Gutenberg-Universitát Mainz (1985).Google Scholar
11.Melvin, P., 2-sphere bundles over compact surfaces, Proc. Amer. Math. Soc. 92 (1984), 567–572.CrossRefGoogle Scholar