Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T20:58:15.017Z Has data issue: false hasContentIssue false
  • English
  • Français

Groups which satisfy a weak form of Poincaré duality

Published online by Cambridge University Press:  20 January 2009

J. E. Roberts
J. E. Roberts
Affiliation:
School of Mathematical SciencesQueen Mary and Westfield CollegeMile End RoadLondon E1 4NS
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.

Our main result is that a “restricted Poincaré duality” property with respect to finite dimensional coefficient modules over a field holds for a certain class of groups which includes all soluble groups of finite Hirsch length. This relies on a generalisation to the given class of a module construction by Stammbach; an extension of his result on homological dimension to these groups is given. We also generalise the well-known result that torsion-free soluble groups of finite rank are countable.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 3 , October 1991 , pp. 463 - 486
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

REFERENCES

1.Atiyah, M. F., Characters and cohomology of finite groups, in: Algebraic Topology—a Student's Guide (ed. Adams, J. F.) London Math. Soc. Lecture Note Series 4, Cambridge University Press 1972.Google Scholar
2.Bieri, R., Homological Dimension of Discrete Groups (Queen Mary College Maths. Notes, 2nd. edition 1981).Google Scholar
3.Brown, K. S., Cohomology of Groups (Graduate Texts in Maths. 87, Springer, Berlin 1982).Google Scholar
4.Gildenhuys, D. and Strebel, R., On the cohomological dimension of soluble groups, Canad. Math. Bull. 24 (1981), 385–392.Google Scholar
5.Hilton, P. J. and Stammbach, U., A Course in Homological Algebra (Graduate Texts in Maths. 4, Springer, Berlin 1970).Google Scholar
6.Kropholler, P. H., On finitely generated soluble groups with no large wreath product sections. Proc. London Math. Soc. 49 (1984), 155–169.Google Scholar
7.Kropholler, P. H., The cohomology of soluble groups of finite rank, Proc. London Math. Soc. 53 (1986), 453–473.CrossRefGoogle Scholar
8.Kropholler, P. H., Cohomological dimension of soluble groups, J. Pure Appl. Algebra 43 (1986), 281–287.CrossRefGoogle Scholar
9.Maclane, S., Homology (Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, 114, Springer, Berlin 1963).CrossRefGoogle Scholar
10.Milnor, J., Hyperbolic Geometry: the first 150 years, Bull. Amer. Math. Soc. 6 (1982), 9–24.CrossRefGoogle Scholar
11.Stammbach, U., On the weak (homological) dimension of the group algebra of solvable groups, J. London Math. Soc. 2 (1970), 567–570.CrossRefGoogle Scholar
12.Wehrfritz, B. A. F., Infinite Linear Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete 76, Springer, Berlin 1973).CrossRefGoogle Scholar