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Generalized Eilenberg–Moore spectral sequences for elementary abelian groups and tori

Published online by Cambridge University Press:  24 October 2008

J. P. C. Greenlees
J. P. C. Greenlees
Affiliation:
Department of Mathematics, University of Sheffield, The Hicks Building, Sheffield, S3 7RH
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Abstract

In this note we prove universal coefficient theorems for Borel cohomology and related theories. Whatever other merit this may have the comment of Borel [5] applies ‘ …elle a au moms l'utilité de bien mettre en évidence le rôle fondamental joué dans cette question par la cohomologie des groupes’.

Indeed the purpose of the enterprise is to use homological properties of the group cohomology ring H*(BG+) to study properties of G-spaces. Because of the relative simplicity of ordinary cohomology much attention in the proofs and applications is concentrated on change of groups, and on changes in the way the group action is exploited. Nonetheless we are able to adapt the non-equivariant approach of Adams ([1, 2]; see also [3]). Thus the existence of universal coefficient theorems automatically gives Kiinneth theorems as special cases.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 1 , July 1992 , pp. 77 - 89
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

[1]Adams, J. F.. Lectures on Generalised Cohomology. Lecture Notes in Math. vol. 99 (Springer-Verlag, 1969), PP. 1–138.Google Scholar
[2]Adams, J. F.. Stable Homotopy and Generalised Homology, part III. Chicago Lectures in Math. (University of Chicago Press. 1974).Google Scholar
[3]Atiyah, M. F.. Vector bundles and the Künneth formula. Topology 1 (1962), 245–248.CrossRefGoogle Scholar
[4]Boardman, J. M.. Conditionally convergent spectral sequences. Preprint (The Johns Hopkins University, 1981).Google Scholar
[5]Borel, A.. Nouvelle démonstration d'un théorème de P. A. Smith. Comment. Math. Helv. 29 (1955), 27–39.CrossRefGoogle Scholar
[6]Borel, A.. Seminar on Transformation Groups. Annals of Math. Studies vol. 46 (Princeton University Press, 1960).Google Scholar
[7]Bredon, G. E.. Cohomological aspects of transformation groups. In Proceedings of the Tulane Conference on Transformation Groups (editor Mostert, P. S.), (Springer-Verlag, 1968), PP. 247–280.Google Scholar
[8]Bredon, G. E.. An Introduction to Compact Transformation Groups. Pure and Applied Math. vol. 46 (Academic Press, 1972).Google Scholar
[9]Brown, K. S.. Cohomology of Groups. Graduate texts in Math. vol. 87 (Springer-Verlag, 1982).CrossRefGoogle Scholar
[10]Dwyer, W. G.. Strong convergence of the Eilenberg–Moore spectral sequence. Topology 13 (1974), 255–265.CrossRefGoogle Scholar
[11]Dwyer, W. G.. Exotic convergence of the Eilenberg–Moore spectral sequence. Illinois J. Math. 19 (1975), 607–617.CrossRefGoogle Scholar
[12]Friedlander, E. M. and Mislin, G.. Locally finite approximation of Lie groups, I. Invent. Math. 83 (1986), 425–436.CrossRefGoogle Scholar
[13]Friedlander, E. M. and Mislin, G.. Locally finite approximation of Lie groups, II. Math. Proc. Cambridge Philos. Soc. 100 (1986), 505–517.CrossRefGoogle Scholar
[14]Greenlees, J. P. C.. Adams spectral sequences in equivariant topology. Ph.D. thesis, Cambridge University (1985).Google Scholar
[15]Greenlees, J. P. C.. Representing Tate cohomology of G-spaces. Proc. Edinburgh Math. Soc. (2) 30 (1987), 435–443.CrossRefGoogle Scholar
[16]Greenlees, J. P. C.. Stable maps into free G-spaces. Proc. Amer. Math. Soc. 310 (1988), 199–215.Google Scholar
[17]Greenlees, J. P. C.. Topological methods in equivariant cohomology. In Proceedings of the 1987 Singapore Group Theory Conference(W. de Gruyter, 1989), pp. 373–389.CrossRefGoogle Scholar
[18]Greenlees, J. P. C.. The power of mod p Borel homology. In Homotopy Theory and Related Topics (editor Mimura, M.), Lecture Notes in Math. vol. 1418 (Springer-Verlag, 1990), pp. 140–151.CrossRefGoogle Scholar
[19]Lewis, L. G., May, J. P. and McClure, J. E.. Ordinary RO(G)-graded cohomology. Bull. Amer. Math. Soc. (N.S.) 4 (1981), 208–212.CrossRefGoogle Scholar
[20]Lewis, L. G., May, J. P., McClure, J. E. and Waner, S.. Equivariant cohomology theory. In preparation.Google Scholar
[21]Lewis, L. G., May, J. P. and Steinberger, M. (with contributions by McClure, J. E.). Equivariant Stable Homotopy Theory. Lecture Notes in Math. vol. 1213 (Springer-Verlag, 1986).CrossRefGoogle Scholar
[22]McClure, J. E. M.. Restriction maps in equivariant K-theory. Topology 25 (1986), 399–409.CrossRefGoogle Scholar
[23]Swan, R. G.. A new method in fixed point theory. Comment. Math. Helv. 34 (1960), 1–16.CrossRefGoogle Scholar
[24]Waner, S.. Equivariant, RO(G)-graded singular cohomology. Preprint (1979).Google Scholar