Homotopy idempotent functors on classifying spaces

Abstract

Fix a prime p. Since their definition in the context of localization theory, the homotopy functors P BZ/p and CW BZ/p have shown to be powerful tools used to understand and describe the mod p structure of a space. In this paper, we study the effect of these functors on a wide class of spaces which includes classifying spaces of compact Lie groups and their homotopical analogues. Moreover, we investigate their relationship in this context with other relevant functors in the analysis of the mod p homotopy, such as Bousfield-Kan completion and Bousfield homological localization.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1218 NATÀLIA CASTELLANA AND RAMÓN FLORES book ([Far96, page 175]), where he asked about the cellularity of the p-completion of BG. This seemed a natural extension of the problems considered by the second author concerning BZ/p-homotopy of finite groups (see [Flo07], [FS07], and [FF]), so it was natural to investigate this structure with similar methods.We show that under certain hypotheses, we are able to characterize the effect of the nullification functor P BZ/p by means of a fibration.Theorem 4.1. Let X be a connected space with finite fundamental group and such that (P BZ/p (X 1 )) ∧ p * . Then there is a fibration, where X p is the covering of X whose fundamental group is T p (π 1 (X)), and L Z[ 1 p ] (X p ) denotes Bousfield homological localization of X p with respect to H * (−; Z[ 1 p ]). In particular, one can compute the homotopy groups of P BZ/p (X) in terms of those of X if X is good enough. This result is quite general and in fact describes in a single statement a phenomenon which was previously known for finite groups, p-compact groups and some compact Lie groups, but not for p-local compact or Kač-Moody groups (Corollary 4.13); so, it can then be read as a common property of a big family of homotopy meaningful spaces. Moreover, finite loop spaces also satisfy this property (Corollary 4.12).Another source of examples is the theory of infinite loop spaces. McGibbon [McG] shows that infinite loop spaces satisfy the hypothesis of Theorem 4.1 (see Corollary 4.14).We also obtain some interesting consequences of these results, including a detailed analysis of the relationship between the BZ/p-nullification and Z[1/p]-localization of these spaces -which is very much in the spirit of [Dwy96, Section 6]-and the commutation of nullity functors on them, a situation that was discussed in [RS00] in a general framework.The second part of the paper deals with the effect of the cellularization functor CW BZ/p on classifying spaces of compact Lie groups. We show a Serre-type dichotomy theorem. Theorem 6.9. Let G be a compact connected Lie group. If there exists a non-pcohomologically central element of order p, then the BZ/p-cellullarization of BG has infinitely many non-zero homotopy groups. Otherwise, it has the homotopy typeThis statement is in fact a consequence of a more general statement which extends Proposition 2.3 in [FS07].Theo...

Cellular approximations of p‐local compact groups

“…Applying the nullification functor with respect to , , to the previous equivalence, we get Finally, we proceed to ‐complete the previous equivalence. One can apply [, Lemma 3.9] and then . On the right, by the same result we have , since is 1‐connected, and therefore, and is an equivalence.…”

“…Proof Since the classifying space is nilpotent, the observation after Theorem implies that is so, and we can use Sullivan's arithmetic square. By [, Lemma 2.8], we have and for , so the statement follows from Theorem .…”

“…A necessary step to understand cellular approximations of ‐local compact groups is then to describe cellular approximations of its Sylow ‐subgroups, maximal discrete ‐toral subgroups. For the case , this task was undertaken in [, Example 6.16] and it is not difficult to extend this result to ‐cellular approximation for . Proposition Let be a discrete ‐toral group with a maximal torus of rank .…”

“…In , the second author computed explicitly the ‐cellularization of classifying spaces of finite ‐groups. Then, while obtaining more information in the finite case , the case of connected compact Lie groups was undertaken in ; there more qualitative results were described, as a dichotomy result on the homotopy groups of , and also concrete examples for where , and . The case of is especially interesting since the explicit computation strongly suggested a more general description: The homotopy fibre of the coaugmentation is homotopy equivalent, up to ‐completion, to the loop space of the rationalization of .…”

“…When is a compact Lie group, results of Notbohm allow to describe more explicitly the ‐cellular approximation of for every in terms of strongly closed subgroups of . Theorem in Section 6 is the main result which gives an answer to the final question of . When is a simple compact connected Lie group, Notbohm uses the classification of compact 1‐connected simple Lie groups to classify strongly closed subgroups.…”