Finite approximations of $p$-local compact groups

Abstract

In this paper we show how every p-local compact group can be described as a telescope of p-local finite groups. As a consequence, we deduce several corollaries, such as a Stable Elements Theorem for the mod p cohomology of their classifying spaces, and a generalized Dwyer-Zabrodsky description of certain related mapping spaces. 55R35; 55R40, 20D20For a fixed prime p, the concept of p-local compact group was introduced in the last decade by C. Broto, R. Levi and B. Oliver in [7]. It provides a unifying, categorical language to study several classes of groups from a p-local point of view, including finite groups, compact Lie groups, p-compact groups and algebraic groups over algebraic closures of fields of characteristic other than p.In view of the many classes of groups and other mathematical objects that p-local compact groups model, it is no surprise that this theory is not very well understood yet. Nevertheless several structural results have already been conjectured since its introduction in [7], of which two are of special relevance in this work. The first conjecture asks about the existence of a version of the Stable Elements Theorem by H. Cartan and S. Eilenberg [10, Theorem 10.1] for p-local compact groups, while the second conjecture suggests the extension of the classic result of W. Dwyer and A. Zabrodsky [17, Theorem 1.1] on mapping spaces to p-local compact groups.A reader familiar with the theory of p-local finite groups will know that the proofs of these results in the finite case, as well as many others, depend ultimately on a certain biset, which is at the core this theory. And it is there where the main obstacle to study p-local compact groups resides, since there is no such biset in the context of p-local compact groups.An alternative to the biset is necessary, and evidence that such alternative may exist is found in a result by E. M. Friedlander and G. Mislin,[19,20], that relates compact Lie groups to towers of finite groups. In this paper we prove a version of the result of Friedlander and Mislin for p-local compact groups.Many statements for p-local finite groups extend to p-local compact groups as a consequence of this approximation result that replaces the use of the biset by arguments on Mapping spaces 54Bibliography 69Acknowledgements. The author is specially grateful to C. Broto, R. Levi and B. Oliver for numerous inspirational conversations and their support. The author would like to thank also N. Castellana, A. Libman and R. Stancu for their help in various stages of this work, and A. Chermak and R. Molinier for their support during the final stages of this work. Finally, the author is truly indebted to the referee, without whose assistance and thorough job the reader's experience with this paper would be greatly diminished. Background on p-local compact groupsLet p a prime, to remain fixed for the rest of the paper unless otherwise stated. In this section we review all the definitions and results about p-local compact groups that we Finally, we say that P is fully F -centralized, resp...

Automorphisms of p-local compact groups

“…Although our main theorem also holds with respect to the original Robinson amalgam, a certain modification given in [LS] behaves better with respect to a variety of example. Most of the discussion here is based on part of the first author's thesis [Gon0, Section 2] and on [Gon3].…”

“…The construction of Φ in the general case uses the finite approximations of p-local finite groups from [Gon3]. In order to facilitate the reading, we recall here some of the definitions and results from [Gon3] that will be used here.…”

“…By [Gon3,Proposition 1.11], L is a transporter system which contains L as a full subcategory, and |L| ≃ | L|. Furthermore, the functor (−)…”

“…We recall some of the main properties of the triples (S i , F i , L i ) from [Gon3] in the following.…”

“…Parts (i) and (ii) follow from [Gon3, Theorem 2.13] and its proof, by taking the p-adic valuation of ζ − 1 to be sufficiently large. Part (iii) is [Gon3,Lemma 2.3].…”