Homotopy uniqueness of classifying spaces of compact connected lie groups at primes dividing the order of the weyl group

Abstract

As A truism, Lie groups-in particular compact connected Lie groups-are very rigid objects. The perhaps best known instance of this rigidity was formulated in Hilbert’s fifth problem and proved by Gleason, Montgomery and Zippin in the early 1950s: It requires only very weak assumptions on the topology of a topological group to get a Lie group (for a survey see [19]). Trying to distinguish two compact connected Lie groups, another kind of rigidity occurs. Very often, the rich structure of a Lie group is totally described by little information. For example, simply connected compact Lie groups or compact connected Lie groups up to the local isomorphism type are classified by pure combinatorical data, namely the Dynkin diagram. Semi simple Lie groups are distinguished by their normalizer of the maximal torus [ 121. Similar phenomena seem to occur, if one considers the classifying space BG of a compact connected Lie group G. Surprisingly, pure algebraic data, given by cohomology or complex K-theory, is enough to distinguish BG as a space from other spaces. Which is what most of the paper is about. p-adic completion of spaces makes life a lot easier. Most of the results are about the p-adic completion BG,^. For a large class of compact connected Lie groups, ‘global’ results are also obtained. We are concerned with three concepts: The homotopy type, the p-adic type, and the mod-p type of a classifying space. We will explain these concepts in detail in a moment. The last two notions are purely algebraic. Each concept is weaker than the preceding one. The main theorems will say that, under certain conditions, the first two are equivalent and characterize the homotopy type of BG,^. That is what we understand by homotopy uniqueness. We will use the following notation throughout: T, 4 G denotes a fixed maximal torus of G,N(T,)csG the normalizer of T,, and W, the Weyl group of G. We say a p-complete space X has the mod-p type of BG, if there exists an isomorphism