## Homology decompositions for classifying spaces of compact Lie groups

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**352**(2000), 2643-2657 Request permission

## 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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## Additional Information

**Alexei Strounine**- Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
- Email: alexei.strounine.1@nd.edu
- Received by editor(s): December 18, 1997
- Published electronically: March 2, 2000
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1637102