Subject: Homotopy equivalence of BG and BG'
From: Jesper Grodal <jg@math.uchicago.edu>
Date: Fri, 2 Dec 2005 16:43:13 -0600 (CST)

Concerning the discussion of homotopy equivalence between BG and BG':

It has been known for a while that two compact (but not necessarily 
connected) Lie groups G and G' are isomorphic if and only if BG is 
homotopy equivalent to BG'. The statement in this generality is in 
Notbohm:  "On the "classifying space" functor for compact Lie groups" 
(JLMS 1995), generalizing Baum-Browder, Moller, Osse, and Scheerer.

Concerning Wolfgang Ziller's question of how different BG is from BG': 
It depends on what you are after, but one question one can ask is at 
which primes p the p-completion BG^_p is homotopy equivalent to BG'^_p? 
Let's for simplicity assume that they are connected (the case where this 
is not the case is also very interesting, but another story, which 
leads to p-local finite groups):

Then BG^_p and BG'^_p are equivalent if and only if the corresponding 
p-adic root data D_G \otimes Z_p and D_{G'} \otimes Z_p are isomorphic, 
which at odd primes just amounts to saying that the reflection groups 
(W_G,L_G \otimes Z_p) and (W_{G'},L_{G'} \otimes Z_p) are isomorphic, 
where L_G is the coweight lattice of G.

This is explained and elaborated on in Thm 11.5 in the appendix of 
"The classification of p-compact groups for p odd" math.AT/0302346 by 
Andersen-Grodal-Moeller-Viruel. The language of root data at the prime 2 
is in math.AT/0510179 and math.AT/0510180.

Beware that compact Lie groups can be equivalent at all primes without 
being isomorphic (the first examples of this kind were pointed out by
Dietrich Notbohm and Larry Smith, I think). Also beware that in general 
whether BG^_p and BG^_p are homotopy equivalent cannot be determined just 
by looking at cohomology together with Steenrod operations, though this 
does happen in certain cases.

In the example G = Sp(n) x_{C_2} S1 and G' = Sp(n) x S1, BG^_2 and 
BG'^_2 are not homotopy equivalent but they are equivalent at all 
other primes. They both have torsion free cohomology ring (eg by a 
spectral sequence argument). They can also be distinguished by mod 2 
cohomology, by observing that there is a Sq2 from degree 4 to 6 in 
the first case (eg by reducing to the case Sp(1) x_{C_2} S1 = U(2), 
by embedding Sp(1) in Sp(n) diagonally).

Trying to understanding the answer to these kind of questions was 
amongst the motivations which led to finite loop spaces and 
p-compact groups.

Jesper