Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-03T04:08:28.013Z Has data issue: false hasContentIssue false

Cohomology of Bieberbach groups

Published online by Cambridge University Press:  26 February 2010

Howard Hiller
Affiliation:
Department of Mathematics, Columbia University, New York, N.Y., 10027, USA.
Get access

Extract

Recently, Szczepariski [11] has constructed examples of aspherical manifolds with the ℚ-homology of a sphere. More precisely, if k is a commutative ring of characteristic zero containing , the following theorem holds.

Type
Research Article
Copyright
Copyright © University College London 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bieberbach, L.. Über die Bewegungsgruppen der Euklidischen Raume. Math. Ann., 70 (1910), 297336.Google Scholar
2.Bourbaki, N.. Groupes et algèbres de Lie, Chs. IV-V-VI (Hermann, Paris, 1968).Google Scholar
3.Charlap, L.. Compact flat Riemannian manifolds I. Ann. Math., 81 (1965), 1530.Google Scholar
4.Charlap, L. and Vasquez, A.. Compact flat Riemannian manifolds II: The cohomology of Z/p-manifolds. Amer. J. Math., 87 (1965), 551563.CrossRefGoogle Scholar
5.Chevalley, C.. Invariants of finite groups generated by reflections. Amer. J. Math., 78 (1955), 778782.Google Scholar
6.Hantzsche, W. and Wendt, H.. Dreidimensionale euklidische Raumformen. Math. Ann., 110 (1934/1935), 593611.Google Scholar
7.Hiller, H.. Flat manifolds with Z/p2 holonomy. To appear in L'Enseignement Math.Google Scholar
8.Maxwell, G.. Compact Euclidean space forms. J. Algebra, 44 (1977), 191195.CrossRefGoogle Scholar
9.Maxwell, G.. The crystallography of Coxeter groups. J. Algebra, 35 (1975), 159178.Google Scholar
10.Stanley, R.. Relative invariants of finite groups generated by pseudo-reflections. J. Algebra, 49 (1977), 134148.CrossRefGoogle Scholar
11.Szczepański, A.. Aspherical manifolds with the Q-homology of a sphere. Mathematika, 30 (1983), 291294.CrossRefGoogle Scholar
12.Wolf, J.. Spaces of constant curvature (McGraw-Hill, New York, 1967).Google Scholar
13.Yau, S.-T.. Compact flat Riemannian manifolds. J. Diff. Geom., 6 (1972), 395402.Google Scholar