Homotopical Uniqueness of Classifying Spaces


         W.G.Dwyer, H.R. Miller, and C.W. Wilkerson

                         University of Notre Dame
                  Massachusetts Institute of Technology
                             Purdue University



                             x1. Introduction

   IfG is a connected compact Lie group, then for almost all prime
numbers p the mod p cohomology ring of the classifying space BG is a
finitely generated polynomial algebra. In 1961, N. Steenrod [24] asked
in general for a determination of all spaces X such that H  (X;Fp ) is a
finitely generated polynomial algebra (i.e.,such that X has a polynomial
cohomology ring); at that time, the only examplesknown were spaces
of the form X =B G.
   There has been a lot of subsequent progress on this problem. On one
hand, the topological constructions of Sullivan [25, p. 4.28], as exploited
by Clark-Ewing [8] and Wilkerson [27], have led to the discovery of ex-
otic spaces X with polynomial cohomology rings. On the other hand,
the algebraic arguments of Wilkerson [26] and Adams-Wilkerson [1] have
shown in some generality thatif X  is any space with a polynomial co-
homology ring then H (X; Fp ) must be one of the polynomial algebras
listed in [8].
   However, there is still a gap here between topology and algebra; in this
paper!we!act to narrow the gap andin some cases to close it. Suppose
that!p!is an odd prime. Let K be the category of unstable algebras [18]
over!the mod p Steenrod algebra Ap  and Kpoly thefull subcategory of
K!consisting!of objects which as rings are finitely generated polynomial
algebras.!If X is a p-complete space with H (X;Fp ) 2 Kpoly we extend
the!ideas!of [1] by associating to X a finite p-adic linear group WX
generated!by!pseudoreflections (see1.1); given any such finite group W
such!that p does not divide the order ofW ,we then show (1.2) that there
is!up!to homotopy exactly one p-complete space X  with H  (X;Fp ) 2
Kpoly!and WX = W.  Ina variety of particular situations (1.3, 1.4) this
gives!a!bijective correspondence between finite group data and homotopy
types!of!p-complete spaces with polynomial cohomology rings.
!


2                    W. Dwyer, H. Miller, and C. Wilkerson


   Asa corollary we observe that if G is a connected compact Lie group
and p a prime which does not divide theorder of the Weyl group of G
then the cohomology ring H (BG; Fp ), considered asan ob jectof K,
determines the homotopy type of thep-completion of BG (1.7).  This
generalizes the treatment of G = SU(2) in [12] and is the uniqueness
property referred to in the title of thepap er.
   We will now describe our results in more detail. If R is an object of
Kpoly, let aeR (the rankQof R) be the number of polynomial generators in
R and R  the pro duct   i(jxij=2) indexedby a set fxig of polynomial
generators for R; recall that the prime p is odd, so that the degree
jxij of each polynomial generator xi is even.  The integer R does not
depend upon a choice of polynomial generators for R. If X is a space
with H (X; Fp ) 2 Kpoly, wewill write aeX  and X  for aeR  and R  with
R = H  (X;Fp ).  If H  (X; Fp ) 2Kp olyand X  = BG for a connected
compact Lie group G,then aeX  is the Lie-theoretic rank of G and X  the
order of the Weyl groupof G.
   LetZp  denote the ring of p-adic integers.  The Eilenberg-MacLane
space K((Zp)r; 2) is the p-completion of the classifying space of the r-
torus Tr, and we will denote it ^BTr or ^BT if r is understood; it is clear
that the general linear group GL(r;Zp) acts on ^BTr in a natural way.
   Given a commutative domain D, an element g 2 GL(r;D) of finite
order is said to be a pseudoreflection if ther  r matrix (g  Ir) has
rank at most one (here Ir 2GL(r; D) is the identity matrix). If X is a
space, say that a subgroup W ae GL(r; Zp) is adapted to X if there is
a map | : B^Tr ! X , equivariant up to homotopy with respect to the
natural action of W on ^BT and the trivial action of W on X ,such that
| induces an isomorphism


                        H  (X;Fp  )=  H  (^BT; Fp )W  :


1.1 Theorem. Let p be an odd prime and let X be a p-complete space
with H (X; Fp ) in Kpoly. Then there exists up to conjugacy a unique
finite subgroup WX  ae GL(aeX ;Zp) adapted to X.  The group WX  is
generated by pseudoreflections andjWX j = X .

Remark: Theorem 1.1 does not require the assumption p - X and for
this reason leads to non-realizability results for many objects of Kpoly.
The image in GL(aeX ;Fp ) of the group WX  aeGL(aeX ; Zp) is the "Galois
group" of H  (X;Fp ) constructed in [1] (see x2). If H  (X; Fp ) 2 Kpoly
and X is the p-completion of BG for a connected compact Lie group G,
then WX  is the image in GL(aeX ;Zp) of the Lie-theoretic Weyl group