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On the homotopy of \(p\)-completed classifying spaces. (English) Zbl 0888.55009
Adem, Alejandro (ed.) et al., Group representations: cohomology, group actions and topology. Summer Research Institute on cohomology, representations, and actions of finite groups, Seattle, WA, USA, July 7–27, 1996, Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 63, 157-182 (1998).
One of the persisting challenges to homotopy theory is to describe the effect of Quillen’s plus-construction on the homotopy theoretic invariants and properties of a given space \(X\). This paper surveys aspects of this problem in the special case where \(X=BG\) is the classifying space of a finite perfect group \(G\). Here are some of the key words associated to topics discussed in this paper: exponents for homotopy groups; resolvability of \(BG^+\) as the total space of an iterated fibration of spheres and their loop spaces; structural properties of \(H_*(\Omega BG^{\wedge}_{p};\mathbb F_p)\); (un-)stable homotopy theory of \(\Omega BG^+\); relation with finite complexes.
The paper contains a wealth of information tying \(BG^+\) into its homotopy theoretic environment. It closes with a section on problems and conjectures regarding \(BG^+\).
For the entire collection see [Zbl 0882.00036].