Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T18:26:11.003Z Has data issue: false hasContentIssue false
  • English
  • Français

The cohomology ring of a combinatorially aspherical group

Part of: Special aspects of infinite or finite groups Connections with homological algebra and category theory

Published online by Cambridge University Press:  09 April 2009

K. J. Horadam
K. J. Horadam
Affiliation:
Department of Mathematics, Royal Melbourne Institute of TechnologyMelbourne, Victoria 3001, 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.

A presentation is given for the cohomology ring of a finitely presented combinatorially aspherical group with trivial coefficients in an integral domain. Cohomological periodicity is characterized in terms of the cup product.

MSC classification

Secondary: 20J05: Homological methods in group theory 20F05: Generators, relations, and presentations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 3 , December 1989 , pp. 453 - 457
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Brown, K. S., Cohomology of groups, (Springer-Verlag, New York, 1982).Google Scholar
[2]Chiswell, I. M., Collins, D. J. and Huebschmann, J., ‘Aspherical group presentations’, Math. Z. 178 (1981), 1–36.Google Scholar
[3]Collins, D. J. and Huebschmann, J., ‘Spherical diagrams and identities among relations’, Math. Ann. 261 (1982), 155–183.Google Scholar
[4]Diethelm, T., ‘The mod p cohomology rings of the nonabelian split metacyclic p-groups’, Arch. Math. 44 (1985), 29–38.CrossRefGoogle Scholar
[5]Evens, L., ‘The cohomology ring of a finite group’, Trans. Amer. Math. Soc. 101 (1961), 224–239.Google Scholar
[6]Fox, R. H., ‘Free differential calculus I’, Ann. of Math. 57 (1953), 547–560.CrossRefGoogle Scholar
[7]Horadam, K. J., ‘The cup product and coproduct for a combinatorially aspherical group’, J. Pure Appl. Algebra 33 (1984), 41–47.CrossRefGoogle Scholar
[8]Huebschmann, J., ‘Cohomology theory of aspherical groups and of small cancellation groups’, J. Pure Appl. Algebra 14 (1979), 137–143.CrossRefGoogle Scholar
[9]Huebschmann, J., ‘The mod p cohomology rings of metacylic groups’, J. Pure Appl. Algebra, to appear.Google Scholar
[10]Lewis, G., ‘The integral cohomology rings of groups of order p 3’, Trans. Amer. Math. Soc. 132 (1968), 501–529.Google Scholar
[11]Newman, M., Integral matrices, (Academic Press, New York, 1972).Google Scholar
[12]Quillen, D., ‘On the cohomology and K-theory of the general linear groups over a finite field’, Ann. of Math. (2) 96 (1972), 552–586.CrossRefGoogle Scholar
[13]Ratcliffe, J., ‘The cohomology ring of a one-relator group’, in Contributions to Group Theory, edited by Appel, K. I., Ratcliffe, J. G., and Schupp, P. E., (Contemporary Math., 33, Amer. Math. Soc., Providence, R.I., 1984).Google Scholar
[14]Rusin, D., ‘The mod 2 cohomology of metacyclic 2-groups,’ J. Pure Appl. Algebra 44 (1987), 315–327.Google Scholar