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

Published electronically:
March 2, 2000

MathSciNet review:
1637102

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

**1.**A. K. Bousfield and D. M. Kan,*Homotopy limits, completions and localizations*, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR**0365573****2.**Albrecht Dold,*Fixed point index and fixed point theorem for Euclidean neighborhood retracts*, Topology**4**(1965), 1–8. MR**0193634****3.**W. G. Dwyer,*Homology decompositions for classifying spaces of finite groups*, Topology**36**(1997), no. 4, 783–804. MR**1432421**, 10.1016/S0040-9383(96)00031-6**4.**J. Hollender and R. M. Vogt,*Modules of topological spaces, applications to homotopy limits and 𝐸_{∞} structures*, Arch. Math. (Basel)**59**(1992), no. 2, 115–129. MR**1170635**, 10.1007/BF01190675**5.**Sören Illman,*Equivariant singular homology and cohomology. I*, Mem. Amer. Math. Soc.**1**(1975), no. issue 2, 156, ii+74. MR**0375286****6.**Stefan Jackowski, James McClure, and Bob Oliver,*Homotopy classification of self-maps of 𝐵𝐺 via 𝐺-actions. I*, Ann. of Math. (2)**135**(1992), no. 1, 183–226. MR**1147962**, 10.2307/2946568

Stefan Jackowski, James McClure, and Bob Oliver,*Homotopy classification of self-maps of 𝐵𝐺 via 𝐺-actions. II*, Ann. of Math. (2)**135**(1992), no. 2, 227–270. MR**1154593**, 10.2307/2946589**7.**Stefan Jackowski and Bob Oliver,*Vector bundles over classifying spaces of compact Lie groups*, Acta Math.**176**(1996), no. 1, 109–143. MR**1395671**, 10.1007/BF02547337**8.**L. G. Lewis Jr., J. P. May, M. Steinberger, and J. E. McClure,*Equivariant stable homotopy theory*, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. MR**866482****9.**Robert Oliver,*A transfer homomorphism for compact Lie group actions*, Math. Ann.**260**(1982), no. 3, 351–374. MR**669300**, 10.1007/BF01461468**10.**J. Slominska, Homology decompositions of Borel constructions, preprint (Torun) 1996.**11.**Tammo tom Dieck,*Transformation groups*, de Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. MR**889050****12.**Stephen J. Willson,*Equivariant homology theories on 𝐺-complexes*, Trans. Amer. Math. Soc.**212**(1975), 155–171. MR**0377859**, 10.1090/S0002-9947-1975-0377859-X

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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:
http://dx.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