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Homology decompositions for classifying spaces of compact Lie groups


Author: Alexei Strounine
Journal: Trans. Amer. Math. Soc. 352 (2000), 2643-2657
MSC (1991): Primary 55R35; Secondary 55R40
DOI: https://doi.org/10.1090/S0002-9947-00-02427-2
Published electronically: March 2, 2000
MathSciNet review: 1637102
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Abstract:

Let $p$ be a prime number and $G$ be a compact Lie group. A homology decomposition for the classifying space $BG$ is a way of building $BG$ up to mod $p$ homology as a homotopy colimit of classifying spaces of subgroups of $G$. In this paper we develop techniques for constructing such homology decompositions. Jackowski, McClure and Oliver (Homotopy classification of self-maps of BG via $G$-actions, Ann. of Math. 135 (1992), 183-270) construct a homology decomposition of $BG$ by classifying spaces of $p$-stubborn subgroups of $G$. Their decomposition is based on the existence of a finite-dimensional mod $p$ acyclic $G$-$CW$-complex with restricted set of orbit types. We apply our techniques to give a parallel proof of the $p$-stubborn decomposition of $BG$which does not use this geometric construction.


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  • 1. A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer-Verlag, Berlin, 1972. MR 51:1825
  • 2. A. Dold, Fixed point index and fixed point theorem for Euclidean neighborhood retracts, Topology 4, (1965), 1-8. MR 33:1850
  • 3. W. G. Dwyer, Homology decompositions for classifying spaces of finite groups, Topology 36 (1997), 783-804. MR 97m:55016
  • 4. J. Hollender and R. M. Vogt, Modules of topological spaces, applications to homotopy limits and $E_\infty $ structures, Arch. Math. 59 (1992), 115-129. MR 93e:55015
  • 5. S. Illman, Equivariant singular homology and cohomology, Mem. Amer. Math. Soc. 156, 1975. MR 51:11482
  • 6. S. Jackowski, J. E. McClure and R. Oliver, Homotopy classification of self-maps of $BG$ via $G$-actions. I, II, Annals of Math. 135 (1992), 183-270. MR 93e:55019a; MR 93e:55019b
  • 7. S. Jackowski and R. Oliver, Vector bundles over classifying spaces of compact Lie groups, Acta Math., 176 (1996), 109-143. MR 97h:55005
  • 8. L. G. Lewis, J. P. May, M. Steinberger, Equivariant stable homotopy theory, Lecture Notes in Math. 1213, Springer Verlag (1986). MR 88e:55002
  • 9. R. Oliver, A transfer homomorphism for compact Lie group actions, Mathematische Annalen 260 (1982), 351-374. MR 83m:57034
  • 10. J. Slominska, Homology decompositions of Borel constructions, preprint (Torun) 1996.
  • 11. T. tom Dieck, Transformation groups. de Gruyter Stud. Math., 8. de Gruyter, Berlin-New York, 1987. MR 89c:57048
  • 12. S. J. Wilson, Equivariant homology theories on $G$ -complexes, Trans. Amer. Math. Soc. 212 (1975), 155-171. MR 51:14028

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Additional Information

Alexei Strounine
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: alexei.strounine.1@nd.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02427-2
Received by editor(s): December 18, 1997
Published electronically: March 2, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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