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         HOMOTOPY UNIQUENESS OF CLASSIFYING SPACES OF COMPACT 
    CONNECTED LIE GROUPS AT PRIMES DIVIDING THE ORDER OF THE WEYL GROUP
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             by       Dietrich Notbohm
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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 fifties: It requires only very weak assumptions on the topology
of a topological group to get a Lie group.

    Trying to distinguish two connected compact 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 
connected compact Lie groups up to the
local isomorphism type are classified by pure combinatorical data, namely 
the Dynkin diagram. Semi simple connected compact Lie groups are distinguished
by their normalizer of the maximal torus.

   Similar phenomena seem to occur, if one considers the classifying space 
BG  of a connected compact 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.
That 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\p.
For a large class of \cclg s, `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 preceeding one. The main theorems will say that, under 
certain conditions, the first two are equivalent and characterize the 
homotopy type of BG\p. That is what we understand by homotopy uniqueness.  

...