Cellularization of classifying spaces and fusion properties of finite groups

Abstract

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.

The cellular structure of the classifying spaces of finite groups

“…In this sense the results here further improve those of [FS07], where this degree of sharpness was only obtained in the description of some concrete examples.…”

“…According to [FS07,3.2], the last statement of our previous Lemma implies that we are actually studying the BZ/p-cellularization of BG ∧ p . More precisely, if G is generated by order p elements, the equivalence…”

“…In particular, it is a consequence of the Puppe sequence and the definition of nullification (see [FS07,Proposition 4.2] for details) that if A 1 (S) = S then P ∧ p is trivial. This leads one to consider the case in which A 1 (S) is strictly contained in S.…”

“…(Section 5 and references [FS07] and [FF08] give illuminating examples of the subgroups just defined. )…”

Postnikov pieces and 𝐵ℤ/𝕡-homotopy theory

“…This is not true in general if we do not assume X to be an H-space. For instance, the BZ/p-cellularization of BΣ 3 is a space with infinitely many non-trivial homotopy groups [11]. Also, it is not true for an arbitrary space A that the A-cellularization of an H-space having a finite number of A-homotopy groups is always a Postnikov piece.…”