Abstract. One way to understand the mod p homotopy theory of classifying spaces of finite groups is to compute their BZ/p-cellularization. In the easiest cases this is a classifying space of a finite group (always a finite p-group). If not, weshow that it has infinitely many non-trivial homotopy groups. Moreover they are either p-torsion free or else infinitely many of them contain p-torsion. By means of techniques related to fusion systems we exhibit concrete examples where p-torsion appears.
Abstract. In this note we discuss the effect of the BZ/p-nullification P BZ/p and the BZ/p-cellularization CW BZ/p over classifying spaces of finite groups, and we relate them with the corresponding functors with respect to Moore spaces that have been intensively studied in the last years. We describe P BZ/p BG by means of a covering fibration, and we classify all finite groups G for which BG is BZ/p-cellular. We also carefully study the analogous functors in the category of groups, and their relationship with the fundamental groups of P BZ/p BG and CW BZ/p BG.
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