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Locally finite approximation of Lie groups. II

Published online by Cambridge University Press:  24 October 2008

Eric M. Friedlander  and
Guido Mislin
Eric M. Friedlander
Affiliation:
Northwestern University, Evanston, IL 60201, U.S.A.
Guido Mislin
Affiliation:
Eidgenössische Technische Hochschule, 8092 Zürich, Switzerland
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Extract

In an earlier paper [10], we constructed a ‘locally finite approximation away from a given prime p’ of the classifying space BG of a Lie group with finite component group. Such an approximation consists of a locally finite group g and a homotopy class of maps which in particular induces an isomorphism in cohomology with finite coefficients of order prime to p. The usefulness of such a construction is that it reduces various homotopy-theoretic questions concerning the space BG to the corresponding questions concerning Bπ for finite subgroups π. For example, we demonstrated in [10] how H. Miller's proof of the Sullivan conjecture concerning maps from , where π is a finite group and X is a finite-dimensional complex, can be extended to maps BG→X for G a Lie group with finite component group.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 100 , Issue 3 , November 1986 , pp. 505 - 517
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

REFERENCES

[1]Adams, J. F.. Stable Homotopy and Generalised Homology. Chicago Lectures in Math. (University of Chicago Press, 1974).Google Scholar
[2]Atiyah, M. F. and Segal, G. B.. Equivariant K-theory and completion. J. Differential Geom. 3 (1967), 1–18.Google Scholar
[3]Benson, D.. Modular Representation Theory: New Trends and Methods. Lecture Notes in Math. 1081 (Springer-Verlag, 1984).Google Scholar
[4]Borel, A.. On proper actions and maximal compact subgroups of locally compact groups (to appear).Google Scholar
[5]Cartan, H. and Eilenberg, S.. Homological Algebra (Princeton University Press, 1956).Google Scholar
[6]Carlsson, G.. Equivariant stable homotopy and Segal's Burnside ring conjecture. Ann. of Math 120 (1984), 189–224.CrossRefGoogle Scholar
[7]Curtis, C. W. and Reiner, I.. Methods of Representation Theory with Applications to Finite Groups and Orders. Volume 1 (Wiley-Interscience, 1981).Google Scholar
[8]Demazure, M. and Gabriel, P.. Introduction to Algebraic Geometry and Algebraic Groups. North-Holland Mathematical Studies 39 (North-Holland, 1980).Google Scholar
[9]Fesbach, M.. The Segal Conjecture for compact Lie groups (to appear).Google Scholar
[10]Friedlander, E. M. and Mislin, G.. Locally finite approximation of Lie groups, I. Inventiones Math. 83 (1986), 425–436.Google Scholar
[11]Madsen, I.. Smooth spherical space forms. In Geometric Applications of Homotopy Theory, Lecture Notes in Math. vol. 657 (Springer-Verlag, 1978), 303–352.Google Scholar
[12]Meier, W.. Localisation, complétion, et applications fantômes. C.R. Acad. Sci. Paris 281, Serie A (1975), 787–789.Google Scholar
[13]Meier, W.. Détermination de certaines groupes d'applications fantômes. C.R. Acad. Sci. Paris 283, Serie A (1976), 971–974.Google Scholar
[14]Miller, H.. The Sullivan Conjecture on maps from classifying spaces. Ann. of Math 120 (1984), 39–87.CrossRefGoogle Scholar
[15]Milne, J.. Etale Cohomology (Princeton University Press, 1980).Google Scholar
[16]Quillen, D.. The spectrum of an equivariant cohomology ring, I, II. Ann. of Math 94 (1971), 549–602.Google Scholar
[17] Séminaire de Géométric Algébrique (SGA3). Schémas en Groupes I, III. Lecture Notes in Math. vol. 151, 153 (Springer-Verlag, 1970).Google Scholar
[18]Serre, J.-P.. Représentations Linéaires des Groupes Finis (Hermann, Paris, 1967).Google Scholar
[19]Springer, T. A. and Steinberg, R.. Conjugacy classes. In Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Math. vol. 131 (Springer-Verlag, 1970), 167–266.Google Scholar
[20]Zabrodsky, A.. Maps between classifying spaces (to appear).Google Scholar