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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.
We will use the following notation throughout:
T_G denotes a fixed maximal torus of G,
N(T_G) the normalizer of T_G, and W_G the
Weyl group of G.
We say a p-complete space X has the mod-p type of BG, if there exists an
isomorphism \phi:H^*(X;Z/p) ---> H^*(BG;Z/p) between the mod-p cohomology
as algebras over the Steenrod algebra. Under this condition, with an extra
assumption for p=2, Dwyer, Miller, and Wilkerson constructed a `maximal
torus' f_T:BT_X ---> X of X, T_X a torus, and an action of
W_G on BT_X. With the trivial action on X the map
f_T is W_G--equivariant up to homotopy.
Moreover, they proved that the two associated p-adic W_G representations
induced by the action on the 2-dimensional cohomology of BT_G and BT_X are isomorphic mod-p. This isomorphism can be realized by a map BT_G ---> BT_X.
For a space X with the mod--p type of BG, we say that X has the
p--adic type of BG, if, roughly speaking, the
W_G--actions on BT_G and BT_X are conjugate over the p-adic integers Z\p.
For every \cclg\ G, there exists a finite
covering K --> G_s x T --> G, where K is finite abelian, G_s is simply
connected, and T is a torus. Such coverings we call finite universal. The factor G_s is called the simply connected part of G.
DEFINITION;
Let G be a connected compact Lie group.
i) BG is called p-torsion free, if
H^*(BG;Z) or, equivalently H^*(G;Z), contains no p--torsion.
ii) G is called p-convenient if BG is p--torsion
free and if
H^*(BG_s;Z/p)\cong H^*(BT_{G_s};Z/p)^{W_G}.
iii) G is called pseudo simply connected, if G is a product of
unitary groups and simply connected compact Lie groups which are
not isomorphic to SU(n) (i.e. we replace SU(n) by U(n)).
Now we are prepared to state our main theorems.
THEOREM A:
Let G be a connected compact Lie group and
let X be a p--complete space with the mod--p type of BG.
i) If G is p-convenient, there exists a connected compact Lie group H
such that BH has the same mod-p type as BG and X and such that X has
the p--adic type of BH.
ii) If G is p-convenient and simply connected or pseudo simply connected,
or if G is a product of unitary groups, then X has the
p--adic type of BG.
iii) If p does not divide the order of the Weyl group W_G,
then X has also the p--adic type of BG.
THEOREM B:
Let G be a p-convenient connected compact Lie group or a product
of unitary groups. If X has the p--adic type of BG, then X and BG\p
are homotopy equivalent.
Theorem 1.1 and 1.2 together imply the following corollary:
CORALLARY C:
Let G satisfy one of the following conditions:
i) G is p-convenient and simply connected or pseudo simply connected.
ii) G is a product of unitary groups.
iii) p does not divide the order of W_G.
If X has the mod--p type of BG, then X and BG\p are homotopy equivalent.
The condition of being p-convenient is essential for our proofs.
A more natural condition for theorems of this type would be given
by conditions like BG is mod--p polynomial, i.e.
H^*(BG;Z/p) is a polynomial algebra, or BG is p--torsion free.
For odd primes we can weaken our technical assumptions.
PROPOSITION D:
Let G be a compact connected Lie group. For p an odd prime,
the following conditions are equivalent:
i) G is p-convenient.
ii) BG is p--torsionfree.
iii) BG is mod--p polynomial.
For p=2, the spaces B\SO(n), n > 2, and BG_2 are
mod--2 polynomial, but not 2--torsion free. BSp(n) is 2--torsion
free, but Sp(n) not 2-convenient. Our methods do not cover the case
G=Sp(n) for p=2.
But we can handle the case of U(2), respectively the case of
products of unitary groups, which are not 2-convenient in general.
For odd primes the results cover all classical matrix groups and for some
primes even some of the exeptional Lie groups.
After having studied the local problem, it is natural to ask for
the analogous global question.
In some cases, we can offer a characterization of the
integral homotopy type of BG.
THEOREM E:
Let G be a 2-convenient connected compact Lie group, such that BG is
torsion free, or let G be a product of
unitary groups. Let X be a simply connected CW-complex of finite type,
such that H^*(X;Z) is torsion free. If there exists
an isomorphism K(Y) ---> K(BG) between the complex toplogical K-theory
as \lambda-rings, then Y and BG are homotopy equivalent.
Questions of this type were first studied by Dwyer, Miller,
and Wilkerson. They proved the analogous statement to Corollary C
for SU(2) and SO(3) at all primes. They also proved Corollary C under the condition (iii). McClure and Smith got the same result for U(2)
at p=2. For U(2), Theorem E was already proved by Smith and the author.
The problem when proving Theorems A and B is to pass from
more or less pure algebraic data to geometric information.
Let X be a space of the mod--p type of BG. Then, as mentioned above,
there exists a maximal torus f_T:BT_X ---> X and Weyl group action of
W_G on (BT_X)\p. The associated representation (L^*T_X)\p:= H^2(BT_X;Z\p)
of W_G is mod-p isomorphic to (L^*T_G)\p:=H^2(BT_G;Z\p).
To prove Theorem A we have to study the p--adic liftings of the W_G--module L^*T_G/p:=H^2(BT_G;Z/p).
The construction of a homotopy equivalence BG\p ---> X (i.e the proof
of Theorem B), is based on the mod--p decomposition of BG
of Jackowski, McClure, and Oliver and on two
ways to calculate the mapping space map(BV,BG), V an elementary
abelian p--group, by Lannes--theory and by using the centralizer of
V in G. The structure of BG which has to be taken into account is too
rich for constructing a map BG --> X directly. The mod--p decomposition of BG
breaks BG\p into simpler spaces, namely the classifying spaces
of particular p--toral subgroups. We set up a program which reduces
the construction of a homotopy equivalence BG\p ---> X to a series of
propositions and lemmas.
This program goes through for pseudo simply connected Lie groups and products of unitary groups. The results for general connected compact Lie groups follow by classical Lie group theory, namely the classification
of connected compact Lie groups.