A Cohomology Decomposition Theorem


                 W. G. Dwyer and C. W. Wilkerson

                         University of Notre Dame
                             Purdue University


                             x1. Introduction

   In[9] Jackowski and McClure gave a homotopy decomp osition theo-
rem for the classifying space of a compact Lie group G; their theorem
states that for any prime p the space B G canb e constructed at p as
the homotopy direct limit of a specificdiagram involving the classifying
spaces of centralizers of elementary abelian p-subgroups of G.  In this
paper we will prove a parallel algebraic decomposition theorem for cer-
tain kinds of unstable algebras over the mod p Steenrod algebra.  This
algebraic result gives a new proof of the theorem of Jackowski and Mc-
Clure and has the potential to lead to homotopy decompositon theorems
for many spaces which are notof the form BG (see x6).
   Before stating our results we will recall some material from [9]. Choose
a prime p.  Let G be a compact Lie group, and let AG b e the cate-
gory whose objects are the non-trivial elementary abelian p-subgroups
of G; a morphism A ! A0 in AG  isa homomorphism f  : A ! A0of
abelian groups with the property that there exists an element g 2  G
such that f(x) = gxg1   for all x 2 A. There is a functor from AopGto
the category of topological spaces whichsends A to the Borel construc-
tion EG G (G=C(A)), where C(A) denotes the centralizer of A in G.
(Notice!that this Borel construction hasthe homotopy type of the clas-
sifying!space BC(A).)  Jackowski and McClure prove that the natural
map!from!the homotopy direct limitof this functor to EG G  = BG is
an!isomorphism on mod p cohomology.  They derive this from a spectral
sequence!argument![2, XII, 5.8] and the following calculation with the
inverse!limit!functor lim  and its right derived functors lim i.  Let H
denote!mod p cohomology and ffG  the functor on AG  which sends A to
H!(EG!G (G=C(A)).
!
Theorem!1.1 [9, Prop. 3-4]. The natural map H BG ! lim ffG  is
an!isomorphism!and the groups limiffG  vanish for i> 0.

!! Thepro ofof Theorem 1.1 in [9] uses the Feshbach double coset formula
and!so!depends heavily on the presence of a genuine compact Lie group.
!


2                            Dwyer and Wilkerson


What we do is much morealgebraic. Let K be the category of unstable
algebras over the mod p Steenrod algebraAp . Given an object R of Kwe
will build an index category AR together with a functor ffR  : AR  ! K
and natural map R ! lim ffR ; if R = H BG then AR  is equivalent
to AG  in such a way that ffR  corresponds to ffG .  This construction
depends heavily on work of Lannes [11].  Using [11] again, we will define
what it means for an object R of K to "have a non-trivial center"; if
R = H BG this condition holds if G has a non-trivial central element
of order p. Our main result is the following one.

Theorem 1.2. Suppose that i : R ! S is a map of Ksuch that:

   (1)  Both Rand S are finitely generated as algebras, and the map i
        makes S into a finitely generated module over R.
   (2)  The mapi has an additive left inverse S !R which is both a map
        of Rmo dules and a map of unstable modules over the Steenrod
        algebra.
   (3)  The algebra S has a non-trivial center.

Then the natural map R ! lim ffR is an isomorphism and the groups
lim iffR vanish for i > 0.

Remark: For a functor such as ffR which takes values in the category
K, we write lim iffR  (i > 0) for the ordinary higher limits [2, p. 305] of
the composite of ffR with the forgetful functor from K to the category
of graded Fp  vector spaces.

   Theconnection between Theorem 1.2 and Theorem 1.1 is provided
by the following proposition, which lists some standard properties of
compact Lie groups.

Proposition 1.3. Let G be a compact Lie group,T a maximal torus in
G, N(T ) the normalizer of T , Np(T ) the inverseimage in N (T ) of a p-
Sylow subgroup of N(T )=T , and i  the natural restriction map H BG !
H BNp (T). Then the following assertions hold.

   (1)  Both H B G and H B Np(T) are finitely generated algebras [17]
        [15,2.2]. The map i  makes H B Np(T) into a finitely generated
        module over H  BG [15, 2.4].
   (2)  The mapi   has an additiveleft inverse H  BNp(T ) ! H  BG
        which is both a H  BG module map and a map of unstablemo d-
        ules over the mod p Steenrod algebra [1].
   (3)  The group Np (T) has a non-trivial central element of order p.


Remark 1.4: Note that 1.3(3) follows from the fact that the conjuga-
tion action of Np(T)=T on the elements of order p in T must pointwise