SPACES WITH POLYNOMIAL MOD-p COHOMOLOGY
D. Notbohm
Abstract. In the early seventieth, Steenrod posed the question which poly*
*nomial
algebras over the Steenrod algebra appear as the cohomology ring of a top*
*ological
space. For odd primes, work of Adams and Wilkerson and Dwyer, Miller and *
*Wilker-
son showed that all such algebras are given as the mod-p reduction of the*
* invariants
of a pseudo reflection group acting on a polynomial algebra over the p-ad*
*ic integers.
We show that this necessary condition is also sufficient for finding a re*
*alization of
such a polynomial algebra.
1. Introduction.
In 1970, Steenrod [St] posed the question, which polynomial algebras over the
field Fp of p elements appear as the mod-p cohomology of a topological space. W*
*ork
of Adams and Wilkerson [A-W] and Dwyer, Miller and Wilkerson [D-M-W] showed
that, at least for odd primes, such algebras are given as a ring of invariants *
*of a
pseudo reflection group acting on a polynomial algebra with generators of degree
2. More precisely, for every space X, for which H*(X; Fp) is a polynomial algeb*
*ra,
there exits a Fp-vector space V and a monomorphism __ae: W -! Gl(V ) representi*
*ng
W as pseudo reflection group such that H*(X; Fp) ~= Fp[V ]W . Actually, this *
*is
an isomorphism of algebras over the Steenrod algebra. Here, Fp[V ] denotes the
ring of polynomial functions on V with values in Fp. Following the topological
conventions, the elements of V get the degree 2. And a (mod-p) pseudo reflecti*
*on
group is a faithful representation __ae: W -! Gl(V ) of a finite group such th*
*at the
image is generated by pseudo reflections, that are elements of finite order fix*
*ing a
hyperlane of codimension 1. This definition works for vector spaces over any fi*
*eld
and for lattices over the p-adic integers. In fact, Dwyer, Miller and Wilkerson*
* also
showed, still for odd primes, that the representation __aehas a p-adic integral*
* lift,
i.e. there exists a p-adic lattice L, such that L=p := L Fp ~=V and such that*
* __ae
lifts to a homomophism ae : W -! Gl(L) which is again a pseudo reflection group.
Moreover, they showed these algebraic data can topologically be realized and th*
*at
this topological realization as well as the algebraic data satisfy some uniquen*
*ess
property. To make this explicit we first fix some notation.
A functorial classifying space construction establishes an action of Gl(L) on
BL =: TL ' K(L; 1) and on B2L = BTL ' K(L; 2) which are Eilenberg-MacLane
spaces as well as the p-completion of a torus respectively of the classifying s*
*pace of
a torus. Starting with an action of W on the p-completion BTp^ of the classifyi*
*ng
space of a torus we get the representation back by setting L := H2(BTp^; Z^p).
______________
1991 Mathematics Subject Classification. 55 P 15, 55 R 35.
Key words and phrases. pseudo reflection group, p-compact group, classifying*
* spaces, compact
Lie group, polynomial algebra.
Typeset by AM S-T*
*EX
1
2
1.1 Definition.
(i) A topological space X is called (mod-p) polynomial, if H*(X; Fp) is a po*
*ly-
nomial algebra over the Steenrod algebra.
(ii) A pseudo reflection group W -! Gl(L), L a p-adic lattice, is adapted to
a polynomial space X or called the Weyl group of X, if there exists a map f :
BTL -! X, equivariant up to homotopy, such that f induces an isomorphism
H*(X; Fp) ~=H*(BTL ; Fp)W . Here, W acts trivially on X.
(iii) Let L be a p-adic lattice and W -! Gl(L) a representation. Then L is c*
*alled
polynomial if the ring Z^p[L]W of polynomial invariants is again a polynomial *
*ring.
(iv) Two representations aei : Wi -! Gl(Li), i = 1; 2, are called weakly is*
*o-
morphic, if there exists an isomorphism ff : W1 -! W2 such that L1 and L2 are
isomorphic as W1-modules, where W1 acts on L2 via the composition ae2ff.
Remark 1.2. If H*(X; Fp) is a polynomial algebra, then so is H*(X; Z^p) and,
for an adapted pseudo reflection group W -! Gl(L), we also have an isomorphism
H*(X; Z^p) ~= H*(BTL ^p; Z^p)W (see [No 3; 4.9]. In particular, L is a polyno*
*mial
W -lattice and Z^p[L]W Fp ~=Fp[L=p]W [No 4; 3.3].
Now, the above mentioned results can be stated as follows:
1.3 Theorem. ( [A-W] [D-M-W]) Let p be an odd prime. Let A be a polynomial
algebra over the Steenrod algebra with a a topological realization A ~= H*(X; F*
*p).
Then the following holds:
(i) There exists a lattice L and a pseudo reflection group W -! Gl(L) such t*
*hat
A ~=Fp[L=p]W as algebras over the Steenrod algebra.
(ii) If X is a p-complete space, then there exists an adapted pseudo reflect*
*ion
group WX -! Gl(LX ) such that LX is a polynomial W -lattice. And any two pseu*
*do
reflection groups adapted to X are weakly isomorphic.
(iii) If Y is another p-complete space realizing A with adapted pseudo refl*
*ection
group WY :-! Gl(LY ), then, after reducing mod p, the two pseudo reflection gro*
*ups
WX -! Gl(LX =p) and WY -! Gl(LY =p) are weakly isomorphic.
The first part of this theorem gives a purely algebraic connection between t*
*he
pseudo reflection group W and A, the second and third a geometric connection
between W and X.
Theorem 1.3 is the starting point for our analysis of polynomial spaces. We *
*show
that, for odd primes, the necessary conditions given in part (i) of the above t*
*heorem
are also sufficient for constructing a realization of a polynomial algebra over*
* the
Steenrod algebra.
1.4 Theorem. Let p be an odd prime. Let A be a polynomial algebra over the
Steenrod algebra. Then, A has a topological realzation if and only if there exi*
*sts an
pseudo reflection group W -! Gl(L), such that L is an polynomial W -lattice and
such that A ~=Fp[L=p]W as algebras over the Steenrod algebra. Moreover, we can
choose the realization X in such a way, that W -! Gl(L) is adapted to X.
We also have the following uniqueness result.
1.5 Theorem. Let p be an odd prime. Let X1 and X2 be polynomial p-complete
spaces such that H*(X1; Fp) ~= H*(X2; Fp). For i = 1; 2, let Wi -! Gl(Li) be t*
*he
adapted pseudo reflection groups.
3
(i) If X1 is 2-connected then so is X2. And in this case, the two adapted ps*
*eudo
reflection groups are weakly isomorphic.
(ii) In general, if the two pseudo reflection groups are weakly isomorphic, *
*then
X1 and X2 are homotopy equivalent.
We can draw the following two corollaries.
1.6 Corollary. Let p be an odd prime. Let X and Y be two 2-connected poly-
nomial p-complete spaces. Then X and Y are homotopy equivalent if and only if
H*(X; Fp) ~=H*(Y ; Fp) as algebras over the Steenrod algebra.
1.7 Corollary. Let p be an odd prime. Let A be a polynomial algebra over the
Steenrod algebra. If A has a topological realization, then there exists only a*
* finite
number of homotopy types of p-complete spaces realizing A.
Proof. By Theorem 1.3, for any two realizations H*(X1; Fp) ~=A ~=H*(X2; Fp) the
two adapted pseudo reflection groups Wi -! Gl(Li), i = 1; 2 are weakly isomorph*
*ic
over Fp. In particular, W := W1 ~= W2 and L := L1 ~= L2. As a consequence of
the Jordan-Zassenhaus theorem, for any fixed finite group G and any fixed latti*
*ce
L there exists only a finite number of p-adic integeral representations W -! Gl*
*(L)
(e.g. see [Cu-Re; 24.1, 24.2]). Hence, the statement is a consequence of Theorem
1.5.
The classifying space BG of a connected compact Lie group G has polynomial
mod-p cohomology, if H*(G; Z) is p-torsionfree, which is true for almost all pr*
*imes.
The adapted pseudo reflection group is given by the action of the Weyl group WG
on the classifying space BTG of a maximal torus TG -! G. This is the reason why
adapted pseudo reflection groups are also called Weyl groups. For odd primes, f*
*ur-
ther examples are given by Clark-Ewing [Cl-Ew], by ad hoc constructions of Quil*
*len
[Qu], Zabrodsky [Za], Aguade [Ag] and by Oliver (see [No 3]). Actually, using t*
*he
classification of polynomial lattices over pseudo reflection groups as describe*
*d in
[No 4], the theory of p-compact groups and the above mentioned examples, one
could construct enough polynomial spaces to get a realization for every polynom*
*ial
lattice. That is one could get a proof of Theorem 1.4 along these lines. But we*
* want
to follow a different strategy. A refinement of Oliver's construction will allo*
*w us to
construct all polynomial spaces as a homotopy colimit over a particular diagram.
All pieces are given by products of classifying spaces of unitary and special u*
*nitary
groups and the diagram is given by a full subcategrory of the orbit category of*
* that
pseudo reflection group, which, at the end, is the Weyl group of the constructed
space.
For odd primes, polynomial lattices of pseudo reflection groups are classifi*
*ed
in [No 4]. These calculations are based on the classification of the p-adic ra-
tional pseudo reflection groups by Clark and Ewing [Cl-Ew]. Up to weak iso-
morphism they gave a complete list of all irreducible p-adic rational pseudo re-
flection groups W -! Gl(U), U a Q^p-vector space. For odd primes, almost
all representations of this list contain a polynomial W -lattice, which is uniq*
*ue
up to weak isomorphis. The only exeptions are given by the pairs (W; p) =
(WF4; 3); (WE6 ; 3); (WE7 ; 3); (WE8 ; 3) and (WE8 ; 5), where W denotes the We*
*yl
group of the exeptional Lie group, indicated by the subindex, with the associat*
*ed
representation at the given prime p (for details see [No 4].
4
For connected compact Lie groups, Theorem 1.5, Corollary 1.6 and Corollary
1.7 are already proved in [No 1]. In particular, these results are shown for t*
*he
classifying spaces of unitary groups and special unitary groups. These results *
*and
the particular construction of the polynomial spaces of Theorem 1.4 allow a pro*
*of
of Theorem 1.5.
In [No 3] the analogous theorems are proved for particular pseudo reflection
groups, namely those which are subgroups of the wreath product Z=m o n acting
on (Z^p)n where m divides p - 1. The proofs of Theorem 1.4 and Theorem 1.5
are based on very similar arguments and ideas. We will refer to them quite ofte*
*n,
but explain the construction of polynomial spaces in detail in this paper. Hen*
*ce,
familiarity with that work might be useful for the reader. The proof of Theorem
1.5 uses the theory of p-compact groups, but everything necessary can be found *
*in
[No 3 ; Section 4].
The paper is organized as follows: In the next three section we provide some
technical results necessary for the construction of the spaces with polynomial *
*coho-
mology (Theorem 1.4). In Section 2, we prove vanishing results for higher deriv*
*ed
functors of inverse limits in certain specialized situiations. In Section 3, w*
*e give
an algebraic description of the center of a connected compact Lie group, and in
Section 4 we describe an algberaic decomposition of the polynomial algebras und*
*er
consideration. All this together allows a proof of Theorem 1.4 which is contain*
*ed in
Section 5. In Section 6 we collect some facts about p-adic integral pseudo refl*
*ection
groups, necessary for the proof of the uniqueness properties of polynomial spac*
*es
(Theorem 1.5) which is contained in the final section.
2. Comparison of higher derived limits.
A family H of subgroups of a finite group G is called closed, if H is closed*
* under
conjugation. For each such family, we denote by OH (G) the full subcategory of *
*the
orbit category O(G), whose objects are given by the orbits G=H where H 2 H.
Let P denote the closed family of all p subgroups of G. For a given (covariant)
functor F : Oop(G) -! Ab from the opposite of the orbit category into the categ*
*ory
of abelian groups we want to compare the derived functors of the inverse limit *
*of
F restricted to full subcategories given by closed families of subgroups with *
*the
restriction to the full subcategory given by the family P. To be more exact abo*
*ut
higher derived limits, let C be a small category and let F un(C; Ab) be the cat*
*egory
of (covariant) functors from C to Ab. Then there exist higher limits
limCi: F un(C; Ab) -! Ab
defined as right derived functors of the inverse limit functor limiC: F un(C; A*
*b) -! Ab
(cf. [B-K; XI, 6] or [Ol; Lemma 2]).
The following definition describes one of the assumptions we will have to pu*
*t on
such families of subgroups for this purpose.
2.1 Definition. Let Let H and K be two closed families of subgroups of G. We
say that H is a minimalization of K if each element K 2 K is contained in a uni*
*que
minimal element HK 2 H.
5
Let H be a closed family of subgroups of G and denote by H [ P the union of
H and P. Then we have the following inclusions of full subcategories
OopH(G) -! OopH[P(G)- OopP(G) :
For each object G=K 2 H [ P we denote by G=K -! OopP(G) the under category of
objects in OopP(G): the objects are given by morphisms OE : G=K -! G=P in Oop(G)
such that G=P is an object in OopP(G) and a morphism (G=P1; OE1) -! (G=P2; OE2)
is a morphism : G=P1 -! G=P2 in Oop(G) such that OE1 = OE2.
Taking higher derived limits restriction establishes the maps
limi F- ff lim i F -fi! limi F :
OopH(G) OopH[P(G) OopP(G)
2.2 Proposition. If H is a minimalization of P and if for each object G=H of
OopH(G) not contained in OopP(G)
i j aeF (G=H) if * = 0
lim i F |(G=H-! OopP(G)) =
(G=H-! OopP(G)) 0 if * > 0.
then ff and fi are isomorphisms.
Proof. This is a slightly more general statement as in [No 3 ; Proposition 2.3]*
*, where
a particular functor is considered. But, using the same argument, our statement
also follows from [No 3 ; 2.1, 2.2].
2.3 Remark. We would like to point out the following observation: For each
H G there exists a functor OP (H) -'! (OP (G) -! G=H) given by H=P 7!
(G=P -! G=H). Here, OP (G) -! G=H denotes the over category; i.e. the category
of objects in OP (G) over G=H, which is defined analogously as the under catego*
*ry.
This functor is an equivalence of categories. The inverse maps G=P -! G=H to the
counterimage of 1 . H in G=P . Passing to the opposite of the categrories, for *
*any
subgroup H G, we have an equivalence
OopP(H) -! (G=H -! OopP(G))
of categories.
In the next step we spezialize the functor under consideration. For each G-
module M we denote by H*M : Oop(G) -! Ab the covariant functor given by
H*M(G=H) := H*(H; M); i.e. given by the group cohomology with coefficients
in M.
2.4 Corollary. Let H be a closed family of subgroups of G. If H is a minimaliz*
*a-
tion of P. Then,
ae H*(G; M) if i = 0
lim i H*M ~= lim i H*M ~=
OopH(G) OopP(G) 0 if i > 0.
6
Proof. The second isomorphism follows from [J-M; Section 5]. To prove the first
equivalence we notice that
ae H*(H; M) if i = 0
lim i H*M ~= olimipH*M ~=
G=H-! OopP(G) OP (H) 0 if i > 0.
(Remark 2.3 and again [J-M; Section 5]). Because M is also an H-module, we can
think of H*M as a functor defined on OopP(H).
Next we want to find minimalzations of P. Let V be a vector space over Fp and
let G -! Gl(V ) be a Fp-representation of the finite group G. For each element
P 2 P we consider the isotropy subgroup HP := IsoG (V P). Then, the family
HV := {HP G|P 2 P}
is closed under conjugation.
2.5 Lemma. The family HV is a minimalization of P.
Proof. First of all, for P 2 P, P HP and V P = V HP . Thus, HP IsoG (V HP ) =
IsoG (V P) = HP , in particular HP = IsoG (V HP ). And this is true for all ele*
*ments
of HV . Now, let H 2 HV be any element such that P H. Then, we have
HP = IsoG (V P) IsoG (V H) = H, which shows that HP is the minimal unique
element of HV containing P .
2.6 Corollary. Let G -! Gl(V ) be a Fp-representation of a finite group G and *
*let
M be a G-module. Then,
ae H*(G; M) if i = 0
lim i H*M ~=
OopHV(G) 0 if i > 0.
2.7 Remark. If a family of subgroups of G contains G itself, then the opposite*
* of
the associated full subcategory of the orbit category has an initial object and*
* the
above results become a triviality. To avoid this, one should work with a fixed *
*point
free representation G -! Gl(V ).
Moreover, the same arguments as in the above proofs work, when we consider
an action of G on a set S with the extra condition, that for any p-subgroup P *
*G,
the fixed-point set is non trivial; e.g. it is sufficient to consider G-actions*
* on finite
modules over Z^p.
3. An algebraic description of the center of a connected compact Lie
group.
Let G be a connected compact Lie group. For odd primes, the p-toral part
Zp(G) of the center Z(G) of G can be calculated from the Weyl group data of G.
Let TG G be a maximal torus, WG the Weyl group and L := LG := H2(BTG ; Z^p)
the associated WG -lattice. Passing to fixed points, the short exact sequence L*
* -!
LQ := L Q -! L1 esablishes an exact sequence
0 -! LWG -! (LQ )WG -! (L1 )WG -! H1(WG ; L) -! H1(WG ; LQ ) = 0 :
7
For odd primes, the completed space (B(L1 )WG )^pis nothing but BZp(G)^p[D-W
2; Remark 7.7]. The equivalence is induced from the composition
BLWG1 -! BL1 -! BL1 ^p' BTG ^p-! BG^p
which factors over BZp(G)^p' BZ(G)^p-! BG^p. Since Zp(G) is abelian, Zp(G) ~=
S x P is the product of a torus S and a finite p-group P . The quotient (LWG )1*
* =
(LQ )WG =LWG detects the torus S, that is B(LWG )1 ^p '-! BS^p is a homotopy
equivalence. And H1(WG ; L) detects the finite group P ~= (L1 )WG =(LWG )1 , th*
*at
is P -! H1(WG ; L) is an isomorphism. This proves the following proposition:
3.1 Proposition. Let p be an odd prime. Let G be connected compact Lie group.
Then there exist equivalences
ss2(BZp(G)^p) ~=LWG and ss1(BZp(G)^p) ~=H1(WG ; L) :
Actually, these isomorphisms are natural with respect to monomorphisms be-
tween connected compact Lie groups of the same rank. And this is even true in
more general situation, which is motivated by the theory of p-compact groups and
which we explain next.
Let f : BH^p -! BG^p be a map between the completed classifying spaces of
two connected compact Lie groups G and H with maximal tori TG G and TH
H. Then, there exists a map fT : BTH ^p-! BTG ^p, unique up to homotopy and
composition with elements of WG , such that the diagram
BTH ^p --fT--! BTG ^p
?? ?
y ?y
BH^p ----! BG^p
commutes up to homotopy (e.g. see [D-W 1]). We say that f is a T -equivalence,
if fT is a homotopy equivalence. In particular G and H have to have the same
rank and the homotopy fiber of f has finite mod p cohomology. In fact, this is
a generalization of the notion of monomorphisms between connected compact Lie
groups of the same rank. Anyway, a T -equivalence f : BH^p-! BG^pestablishes a
monomorphism fffT : WH -! WG defined by the equation fffT(w)fT ' fT w, which
makes sense because of the uniqueness properties of fT . The maps
L(fT ) := H2(fT ; Z^p) : LH -! LG and L1 (fT ) : LH;1 -! LG;1
are WH -equivariant with respect to fffT. In particular we get maps
(LG )WG ----! (LH )WH
BZp(G)^p ----! BZp(H)
H*(WG ; LG ) ----! H*(WH ; LH ) :
8
For the second map, our construction only works for odd primes, but using the
theory of p-compact groups, it can also be constructed for p = 2. The first two
maps are independent of the chosen lift fT of f. The third map is induced by
the map fT between the coefficients and the homomorphims fffT. This map is
also independent of the chosen lift since for any element w 2 W the pair of maps
lw : LG -! LG and ffw : W -! W induces the identity H*(W ; LG ) -=!H*(W ; LG )
[Br; 8.3]. Hence, after having fixed a maximal tori of G and H, the isomorphisms
of Proposition 3.1 are natural with respect to T -equivalences. To give a funct*
*orial
formulation of these facts, we consider the category LGp(n) whose objects are g*
*iven
by the set
{BG^p: G a connected compact Lie group of rank n}
where each G comes with a fixed maximal torus. And the morphisms are given
by homotopy classes of T -equivalences. The above consideration establish the
following proposition:
3.2 Proposition. Let p be an odd prime. Then there exists (covariant) functors
BZp : LGp(n)op -! HoT op and H*L: LGp(n)op -! Ab
whose values on the objects are given by BZp(BG^p) = BZp(G) and H*L(BG^p) =
H*(WG ; LG ). Moreover, there exist natural equivalences
ss2(BZp) ~=H0L and ss1(BZp) ~=H1L:
4. The algebraic decomposition.
Let L be a lattice and W -! Gl(L) be a pseudo reflection group, such that L *
*is
a polynomial W -lattice. By [No 4; 3.3], the vector space L=p is also polynomi*
*al,
i.e. the ring of Fp[L=p]W is a polynomial algebra. In this section we constr*
*uct
a decomposition of Z^p[L]W as well as of Fp[L=p]W which, as shown in the next
section, allows the construction of a space with the mod-p cohomology given by
the ring of invariants.
Polynomial lattices are closely connected to connected compact Lie groups, as
the next statement shows.
4.1 Proposition. Let p be an odd prime. Let W -! Gl(L) be a pseudo reflection
group such that L is a polynomial W -lattice. Then, there exists a connected co*
*mpact
Lie group G and a subspace L=p such that the following holds:
(i) WG ~= IsoW ().
(ii) As WG -lattice, L is weakly isomorphic to LG .
(iii) The index [W : WG ] is coprime to p.
(iv) H*(BG; Fp) ~=Fp[L=p]WG and H*(BG; Z^p) ~=Z^p[L]WG .
Proof. All this is stated in [No 4; 5.1].
Actually, as the proof of [No 4; 5.1] shows, the connected compact Lie group*
* G
is a quotient of a product of unitary groups, special unitary groups and tori.
Let H := HL=p = {WP W : WP = Iso(L=pP )} be the closed family of
subgroups as defined in Section 2 and let H0 H be the family of those groups
which are contained in WG where WG is given by Proposition 4.1. The next lemma
is the key for the construction of a topological realization.
9
4.2 Lemma. With the above notation the following holds:
(i) The family H is a minimalization of P.
(ii) For any p-subgroup P W , the group WP is subconjugated to WG and P WG
iff WP WG . In particular, OH0 (W ) -! OH (W ) is an equivalence of categories.
(iii) Let HP := CG (L=pP ) for any p-subgroup P WG . Then, WP ~=WHP and
R[L R]WP ~=H*(BHP ; R) for R = Fp and R = Z^p.
Proof. The first point is already discussed in Lemma 2.5. The second follows fr*
*om
Proposition 4.1 (iii) and the observation that, for P WG , we have (L=p)P
and therefore IsoW (L=pP ) = IsoWG ((L=p)P ).
The isotropy group WP acts on the maximal torus of HP and therefore WP
WH . On the other hand, WH IsoW (L=pP ) = WP by construction. Hence,
WP = WH , which is th first part of (iii). For R = Fp, the second identity foll*
*ows
from [No 1; 10.1, 10.2] since H*(BG; Fp) ~= Fp[L=p]WG . And for R = Z^p, the
identity follows from Remark 1.2.
4.3 Corollary. Let p be odd and L a W -lattice such that L is polynomial. Let M
be a W -module. Then,
ae H*(W ; M) if i = 0
limi H*M ~= limi H*M ~=
OH0 (W) OP (W) 0 if i > 0.
4.4 Remark. Setting M := Fp[L=p] and taking the functor H0M this corollary
gives an algebraic decomposition of the ring of invariants Fp[L=p]W . The piec*
*es
are given by the mod-p cohomology of classifying spaces of connected compact Lie
groups. The same holds for M := Z^p[L].
5. The realization of the algebraic decomposition in the category of
topological spaces.
In this section we want to realize the algebraic diagram of the last section*
* in the
category T op of topological spaces. We do this in two steps. First we realiz*
*e the
algebraic diagram of Section 4 in the homotopy category HoT op (Theorem 5.1)
and then lift this realization to the category T op (Theorem 5.2). We use the s*
*ame
notation as in Section 4.
5.1 Theorem. Let p be an odd prime and L a polynomial W -lattice. Then, there
exists a functor
: OH0 (W ) -! HoT op
such that the following holds:
(i) (W=WP ) = BCG ((L=p)WP ).
(ii) There exist natural equivalences
~= 0 * ^ ~= 0
H*( ; Fp) -! HFp[L=p] and H ( ; Zp ) -! HZ^p[L]:
Proof. Part (i) indicates how we have define the functor on the objects, namely
(W=WP ) := BCG ((L=p)WP )^p=: BHP ^p. Each morphism ff : W=WP -! W=WP0
10
is given by left translation with an element w 2 W such that cw (WP ) := w-1 WP*
* w
WP0 and splits therefore into a composition of the bijection lw : W=WP -! W=cw *
*(WP )
and the projection q : W=cw (WP ) -! W=WP0. We have to say what are the values
of for projection and for isomorphisms.
If q : W=WP -! W=WP0 is a projection, we define (q) : BHP ^p-! BHP0^pto be
the map induced by the inclusions WP WP0 and L=pWP0 L=pWP .
If lw : W=WP -! W=WP0 is an isomorphism, the two compact connected Lie
groups HP and HP0 have p-adically isomorphic Weyl group data; i.e. the p-
adic representations of the two Weyl groups given by the action on the maximal
torus respectively on the associated lattice LHP ~= L ~= LHP0 are weakly isom*
*or-
phic. Since both have isomorphic polynomial mod-p cohomology, this implies that
BHP ^p' BHP0^p [No 1; Theorem 1.2]. Hence, for the analysis of maps between
these two classifying spaces, we can assume that both spaces are equal. The el*
*e-
ment w 2 W induces a map w : L -! L which is admissible in the sense of [A-M];
i.e. for each x0 2 WP0 there exists an element x 2 Wp (namely x = w-1 xw) such
that x0w = wx. Now, by [J-M-O 2; Corollary 2.5], there exists an equivalence
BHP ^p-! BHP0^psuch that the restriction (BTHP )^p-! (BTHP0 )^pof this map to
the maximal tori is given by w.
Finally we have to show that this gives a functor. Let W=WP ff-!W=WP0 -fi!
W=WP00 be a composition of maps of OH (W ). The definition of the values of ff,
fi and fiff may depend on the chosen splittings. But independent of these choic*
*es,
since H*(BH^p; Z^p) Q ~=H*(BTH ; Z^p)WH Q for every connected compact Lie
group H, it is clear from the construction that (fi) (ff) and (fiff) induce t*
*he
same map
H*(BHP00^p; Z^p) Q -! H*(BHP ^p; Z^p) Q
in rational cohomology. Since HP ^pand HP00 are connected (Lemma 4.2 (iii)), we
can apply [No 3 ; Proposition 3.3], which shows that both maps are homotopic.
In particular, this argument also shows that (ff) does not depend on the chosen
splitting of ff. This proves the first part. The second part is obvious from *
*the
construction.
As next step we have to construct a lift of into the category T op of topo*
*logical
spaces. This is done by the next statement. Let Ho : T op -! HoT op denote the
obvious functor.
5.2 Theorem. There exists a functor
OE : OH0 (W ) -! T op
such that the following holds:
(i) Ho OE = .
(ii) For X := hocolim OE, we have H*(X; R) ~=R[L R]W for R = Z^por R = Fp.
OH0 (W)
Proof. We will use the obstruction theory of Dwyer and Kan [D-K] made for such
lifting problems. To apply this theory we have to check that the diagram given *
*by
: OH0 (W ) -! HoT op
11
is centric. That is that for every morphism ff = qlw : W=WP -! W=WP0 the
induced map
map(BHP ^p; BHP ^p)id -! map(BHP ^p; BHP0^p) (ff)
is an equivalence. This is automatic if ff is an isomorphism, so it suffices to*
* check
it for projections given by inclusions WP W=WP0. And follows in this case from
[No 3 ; Proposition 3.3 (a)] (applied with G = HP0). Furthermore, for each WP we
have map(BHP ^p; BHP ^p)id ' BZ(HP )^p[J-M-O 2; Theorem 1.1]. Since Z(HP ) is
a toral group, the only nonvanishing homotopy groups of these mapping spaces are
given by
ae LWP for i = 2
ssi(map(BHP ^p; BHP ^p)id ~=ssi(BZ(HP )^p) ~= :
H1(WP ; L) for i = 1
The second isomorphism folows from Proposition 3.1. Moreover, since all mor-
phisms in OopH0(W ) induce T -equivalences, this isomorphism comes from a natur*
*al
transformation between the two functors (Proposition 3.2). Hence, by Corollary
4.3
lim i ssj(map( (-); (-))id = 0
OopH0(W)
for all i; j > 0. The obstruction groups for lifting are just given by some o*
*f these
higher derived limits [D-K; Theorem 1.1]. Since all these groups vanish, there *
*exists
a lift
OE : OH0 (W ) -! T op
of . This proves the first part of the statement. The second part follows from*
* the
Bousfiel-Kan spectral for calculating the cohomology of hocolim OE, Theorem 5.1
OH0 (W)
and Corollar 4.3.
Finally, we are able to proof Theorem 1.4:
Proof of Theorem 1.4. Let A ~=H*(X; Fp) be a polynomial algebra over the Steen-
rod algebra with topological realization given by the space X. Then, the comple*
*ted
space X^pis p-complete, realizes A too and has therefore an adapted p-adic pseu*
*do
reflection group W -! Gl(L), L a lattice (Theorem 1.3). In particular, L is a
polynomial W -lattice and A ~=Fp[L=p]W .
Now let W -! Gl(L) be a pseudo reflection group, such that L is a polynomial
W -lattice and such that A ~= Fp[L=p]W . Then Theorem 5.2 provides a space X
realizing A. And, by construction, it is obvious that W -! Gl(L) is adapted to
X.
6. p-lattices of pseudo reflection groups.
Any space with polynomial mod-p cohomology has an adapted pseudo reflection
group W -! Gl(L) for a suitable lattice L. And the associated mod-p representa-
tion W -! Gl(L=p) only depends on the mod-p cohomology of the space considered
as an algebraic object, namely as an algebra over the Steenrod algebra (Theorem
1.3). In this section we discuss how much information about the p-adic represen-
tation is already contained in the the mod-p representation.
12
6.1 Definition. Let W -! Gl(L) be a p-adic pseudo reflection group, L a p-adic
lattice. Then we define SL L to be the sublattice generated by elements of the
form l - w(l), l 2 L and W 2 W . The W -lattice L is called simply connected, *
*if
the covariants LW := L=SL vanish.
6.2 Lemma. Let p be an odd prime. Let L be a lattice and let ae1; ae2 : W -! G*
*l(L)
be_faithful representations of a finite group such that the reduced representat*
*ions
ae1; __ae2: W -! Gl(L=p) are isomorphic.
(i) The pair (W; ae1) is a pseudo reflection group if and only if (W; ae2) is.
(ii) If (W; ae1) and (W; ae2) are both pseudo reflection groups, then L is simp*
*ly con-
nected with respect to ae1 if and only if it is with respect to ae2.
(iii) If (W; ae1) is a pseudo reflection group such that L is simply connected *
*and
polynomial or if the order of W and p are coprime, then ae1 and ae2 are isomorp*
*hic.
Proof. Since_p is odd, an element A 2 Gl(L) is a pseudo reflection iff the mod-p
reduction A 2 Gl(L=p) is a pseudo reflection. Hence, for odd primes, the pair
(W; aei) gives a pseudo reflection group iff (W; __aei) does. This proves part *
*(i).
The second part follows from the fact that a lattice L is simply connected i*
*ff
L]=pW = 0 [No 2; 2.2, 4.1], where L] := Hom(L; Z^p) denotes the dual lattice.
The kernel of the homomorphism Gl(L=pk) -! Gl(L=pk-1 ) is given by the
abelian group Hom(L=p; L=p) of endomorphism of L=p. Therefore, the num-
ber of possible lifts of a homomorphism W -! Gl(L=pk-1 ) to Gl(L=pk) is given
by the order of the obstruction group H1(W ; Hom(L=p; L=p)) where W acts on
Hom(L=p; L=p) in the obvious way. If this obstruction group vanishes, which is
obvious for p coprime to the order of W , the statement follows by induction.
If p is odd and if L is simply connected and polynomial, there exists a conn*
*ected
compact Lie group G such that WG W with index coprime to p, such that
WG -! W -! Gl(L) describes the Weyl group data of G and such that L is a
polynomial WG -lattice. Moreover, G is given as a product of unitary and special
unitary groups. All this follows from [No 4; 5.1]. By [No 1; Lemma 8.2] we ha*
*ve
H1(WG ; Hom(L=p; L=p)) = 0 = H1(W ; Hom(L=p; L=p)). The second equality
follows because the index [W : WG ] is coprime to p. This proves the last part*
* of
the statement.
7. Homotopy uniqueness.
In this section we prove the homotopy uniqueness property of polynomial spac*
*es.
For a polynomial W -lattice we denote by XL the polynomial space with adapted
pseudo reflection group W -! Gl(L) constructed in Section 5. The completion of
XL is p-complete and is obviously another realization of H*(XL ; Fp) with adapt*
*ed
pseudo reflection group W -! Gl(L).
7.1 Theorem. Let Y be a polynomial p-complete space with adapted pseudo reflec-
tion group gibven by W -! Gl(L). Then there exists a mod-p equivalence XL -! Y ,
which, in particular, establishes a homotopy equivalence XL ^p'-!Y .
Proof. Since Y has polynomial mod-p cohomology, an Eilenberg-Moore spectral
sequence argument shows that the loop space Y has finite mod-p cohomology.
Hence, we can think of Y as the classifying space of a p-compact group.
13
The construction of the mod-p equivalence is based on the theory of p-compact
groups. For the basic definitions we refer the reader to [D-W 1]. Actually, eve*
*ry-
thing necessary may be found in [No 3 ; Section 4], since the arguments of this
proof run analogously as in Section 5 of [No 3].
As explained in the Section 4, there exist a connected compact Lie group G
and a subspace L=p such that WG ~= IsoW () W -! Gl(L) gives the
Weyl group data of G. By assumption the space Y comes with a W -equivariant
map fT : BTL -! Y . Composing this map with B -! BTL gives a map f :
B -! Y . Since H*(Y ; Fp) ~= Fp(L=p)W an application of [No 1; 10.1, 10.2]
shows that H*(map(B; Y )f ; Fp) ~=Fp[L=p]WG ~=H*(BG; Fp) and that the Weyl
group data of the p-complete space map(B; Y )f are given by the composition
WG W -! Gl(L). The argument runs analogously as in [No 3 ; Proposition 4.8].
But this implies that BG^p' map(B; Y )f [No 1; Theorem 1.3].
Since for each object W=WP 2 OH (W ), the space OE(W=WP ) ' BCG (L=pWP ) =:
BHP allows a map into BG^p. Composition of maps establishes maps fP : BHP ^p-!
BG^p-fG! Y . Now, for for each morphism ff : W=WP -! W=WP0, we show that
fP0OE(ff) ' fP . Argueing analogously as in the proof of [No 3 ; Theorem 5.1] t*
*his
becomes a consequence of [No 3 ; Proposition 4.6]. We can define a map from the
1-skeleton of the homotopy colimit into Y .
Finally we have to extend this map. For doing this we have to consider the
obstruction groups
limi ssj(map(OE; Y )fP )
OopH(W)
[Wo]. Again, analogously as in the proof of [No 3 ; 5.1], one can show that
map(BHP ^p; Y )fP ' map(BHP ^p; BHP ^p)id ' BZ(HP )^p;
where Z(HP ) denotes the center. The calculation of the above obstruction groups
boiles down to the equality
lim i ssj(BZ(HP ^p)) = 0
OopH(W)
as shown in the proof of Theorem 5.2. Hence, there exists a map f : XL -! Y
which, by construction, is a mod-p equivalence and establishes therefore a homo-
topy equivalence XL ^p'-!Y .
The following theorem contains Theorem 1.5.
7.2 Theorem. Let p be an odd prime. Let X and Y be two p-complete polynomial
spaces such that H*(X; Fp) ~= H*(Y ; Fp) as algebras over the Steenrod algebra.
Then X and Y are homotopy equivalent if one of the following three conditions *
*is
satisfied:
(i) X is 2-connected.
(ii)The order of WX is coprime to p.
(ii)X and Y have the same Weyl group data.
14
Proof. Let WX -! Gl(LX ) and WY -! Gl(LY ) be the adapted pseudo reflection
groups. From Theorem 1.3 follows that LX ~= LY =: L as lattices, that WX ~=
WY =: W and that the two representation __aeX; __aeY: W -! Gl(L=p) are conjuga*
*te.
Here, we identified the lattices and the Weyl groups via these isomorphisms.
If X is simply connected, we have 0 = H2(X; Fp) ~=(L]=pW ) and hence, that L
is simply connected W -lattice [No 2]. Thus, Lemma 6.2 shows that the first two
conditions imply the third one.
Starting with the third condition the statement follows from Thorem 7.1.
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e-mail : notbohm at cfgauss.uni-math.gwdg.de