SPACES WITH POLYNOMIAL MOD–p COHOMOLOGY

D. Notbohm

Abstract. In the early seventies, Steenrod posed the question which polynomial algebras over the Steenrod algebra appear as the cohomology ring of a topological space. For odd primes, work of Adams and Wilkerson and Dwyer, Miller and Wilkerson 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–adic integers. We show that this necessary condition is also sufficient for finding a realization of such a polynomial algebra. We also show that, up to completion, every space with polynomial mod–p cohomology is equivalent to the product of the classifying space of a connected compact Lie group and some spaces determined by irreducible p–adic rational pseudo reflection groups.

1. Introduction.

In 1970, Steenrod [St] posed the question, which polynomial algebras over the field 𝔽_{p} of p elements appear as the mod–p cohomology of a topological space. Work of Adams and Wilkerson [2] and Dwyer, Miller and Wilkerson [10] 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; 𝔽_{p}) is a polynomial algebra, there exits a 𝔽_{p}–vector space V and a monomorphism \overline{ρ} : W ⟶ Gl(V) representing W as pseudo reflection group such that H^{∗}(X; 𝔽_{p}) ≅ 𝔽_{p}[V]^{W}. Actually, this is an isomorphism of algebras over the Steenrod algebra. Here, 𝔽_{p}[V] denotes the ring of polynomial functions on V with values in 𝔽_{p}. Following the topological conventions, the elements of V get the degree 2. And a (mod–p) pseudo reflection group is a faithful representation \overline{ρ} : W ⟶ Gl(V) of a finite group such that the image is generated by pseudo reflections, that are elements of finite order fixing a hyperplane of codimension 1. This definition works for vector spaces over any field and for lattices over the p–adic integers. In fact, Dwyer, Miller and Wilkerson also showed, still for odd primes, that the representation \overline{ρ} has a p–adic integral lift, i.e. there exists a p–adic lattice L, such that L/p := L ⊗ 𝔽_{p} ≅ V and such that \overline{ρ} lifts to a homomorphism ρ : W ⟶ Gl(L) which is again a pseudo reflection group. Moreover, they showed that these algebraic data can topologically be realized and that this topological realization as well as the algebraic data satisfy some uniqueness property. To make this explicit we first fix some notation.

A functorial classifying space construction establishes an action of Gl(L) on BL =: T_{L} ≃ K(L, 1) and on B^{2}L = BT_{L} ≃ K(L, 2) which are Eilenberg–MacLane

Typeset by AMS-TEX

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.

spaces as well as the p–completion of a torus respectively of the classifying space of a torus. Starting with an action of W on the p–completion BT_{p}^{∧} of the classifying space of a torus we get the representation back by setting L := H_{2}(BT_{p}^{∧}; ℤ_{p}^{∧}).

1.1 Definition.

(i) A topological space X is called (mod–p) polynomial, if H^{∗}(X, 𝔽_{p}) is a polynomial 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 : BT_{L} ⟶ X, equivariant up to homotopy, such that f induces an isomorphism H^{∗}(X, 𝔽_{p}) ≅ H^{∗}(BT_{L}; 𝔽_{p})^{W}. Here, W acts trivially on X.

(iii) Let L be a p–adic lattice and W ⟶ Gl(L) a representation. Then L is called polynomial if the ring ℤ_{p}^{∧}[L]^{W} of polynomial invariants is again a polynomial ring.

(iv) Two representations ρ_{i} : W_{i} ⟶ Gl(L_{i}), i = 1, 2, are called weakly isomorphic, if there exists an isomorphism α : W_{1} ⟶ W_{2} such that L_{1} and L_{2} are isomorphic as W_{1}–modules, where W_{1} acts on L_{2} via the composition ρ_{2}α.

(v) A p–adic rational pseudo reflection group W ⟶ Gl(U) is called of Lie type, if there exists a connected compact Lie group G with maximal torus T_{G} ⊂ G such that W ⟶ Gl(U) and W_{G} ⟶ Gl(L_{T_{G}} ⊗ ℚ) are weakly isomorphic.

1.2 Remark 1.2. If H^{∗}(X; 𝔽_{p}) 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^{∗}(BT_{L}_{p}^{∧}; ℤ_{p}^{∧})^{W} (see [18; 4.9]. In particular, L is a polynomial W –lattice and ℤ_{p}^{∧}[L]^{W} ⊗ 𝔽_{p} ≅ 𝔽_{p}[L/p]^{W} [19; 2.3].

Now, the above mentioned results can be stated as follows:

1.3 Theorem. ( [2] [10]) Let p be an odd prime. Let A be a polynomial algebra over the Steenrod algebra with 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 that A ≅ 𝔽_{p}[L/p]^{W} as algebras over the Steenrod algebra.

(ii) If X is a p–complete space, then there exists an adapted pseudo reflection group W_{X} ⟶ Gl(L_{X}) such that L_{X} is a polynomial W –lattice. And any two pseudo reflection groups adapted to X are weakly isomorphic.

(iii) If Y is another p–complete space realizing A with adapted pseudo reflection group W_{Y} :⟶ Gl(L_{Y}), then, after reducing mod p, the two pseudo reflection groups W_{X} ⟶ Gl(L_{X}/p) and W_{Y} ⟶ Gl(L_{Y} /p) are weakly isomorphic.

The first part of this theorem gives a purely algebraic connection between the 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 theorem 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 realization if and only if there exists an pseudo reflection group W ⟶ Gl(L), such that L is an polynomial W –lattice and such that A ≅ 𝔽_{p}[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.

Theorem 1.4 connects realizable polynomial algebras over 𝔽_{p} with polynomial W –lattices. But it also turns out that every polynomial W –lattice gives rise to a realizable polynomial algebra over the Steenrod algebra.

1.5 Theorem. Let W ⟶ Gl(L) be a pseudo reflection group such that L is a polynomial W –lattice. Then the following holds:

(i) ℤ_{p}^{∧}[L]^{W} ⊗ 𝔽_{p} ≅ 𝔽_{p}[L]^{W}, and both are polynomial algebras over the Steenrod algebra. (ii) There exists a p–complete polynomial space X such that W ⟶ Gl(L) is adapted to X.

Proof. The isomorphism of the first is stated in [19; Lemma 2.3 (ii)]. And, since the action of the Steenrod algebra commutes with the W –action on BT_{L}_{p}^{∧}, the ring 𝔽_{p}[L]^{W} ≅ H^{∗}(BT_{L}; 𝔽_{p})^{W} is also an algebra over the Steenrod algebra.

The second part is a consequence of Theorem 1.4. □

We also have the following uniqueness result.

1.6 Theorem. Let p be an odd prime. Let X_{1} and X_{2} be polynomial p–complete spaces such that H^{∗}(X_{1}, 𝔽_{p}) ≅ H^{∗}(X_{2}; 𝔽_{p}). For i = 1, 2, let W_{i} ⟶ Gl(L_{i}) be the adapted pseudo reflection groups.

(i) If X_{1} is 2–connected then so is X_{2}. And in this case, the two adapted pseudo reflection groups are weakly isomorphic.

(ii) In general, if the two pseudo reflection groups are weakly isomorphic, then X_{1} and X_{2} are homotopy equivalent.

We can draw the following two corollaries.

1.7 Corollary. Let p be an odd prime. Let X and Y be two 2–connected polynomial p–complete spaces. Then, the following conditions are equivalent:

(i) The spaces X and Y are homotopy equivalent. (ii) H^{∗}(X, 𝔽_{p}) ≅ H^{∗}(Y; 𝔽_{p}) as algebras over the Steenrod algebra. (iii) The two adapted pseudo reflection groups W_{X} ⟶ Gl(L_{X}) and W_{Y} ⟶ Gl(L_{Y}) are rationally weakly isomorphic.

Proof. The equivalence of the first two conditions is a corollary of Theorem 1.6. And, by Theorem 1.3, (i) implies (iii). Since X and Y are two connected, the adapted pseudo reflection groups are both fixed–point free and, by by Lemma 6.3 (ii), p–adic integrally weakly isomorphic. Hence, (iii) implies (i) by Theorem 1.6. □

1.8 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^{∗}(X_{1}; 𝔽_{p}) ≅ A ≅ H^{∗}(X_{2}, 𝔽_{p}) the two adapted pseudo reflection groups W_{i} ⟶ Gl(L_{i}), i = 1, 2 are weakly isomorphic over 𝔽_{p}. In particular, W := W_{1} ≅ W_{2} and L := L_{1} ≅ L_{2}. As a consequence of the Jordan–Zassenhaus theorem, for any fixed finite group G and any fixed lattice L there exists only a finite number of p–adic integral representations W ⟶ Gl(L) (e.g. see [8; 24.1, 24.2]). Hence, the statement is a consequence of Theorem 1.6. □

A classical result of Borel says that, for odd primes, the classifying space BG of a connected compact Lie group G has polynomial mod–p cohomology, if H^{∗}(G, ℤ) is p–torsion free (see also Proposition 1.11 below), which is true for almost all primes. The adapted pseudo reflection group is given by the action of the Weyl group W_{G} on the classifying space BT_{G} of a maximal torus T_{G} ⟶ G. In fact, it is given by the action on L_{G} := H_{2}(BT_{G}; ℤ_{p}^{∧}). This is the reason why adapted pseudo reflection groups are also called Weyl groups. For odd primes, further examples are given by Clark–Ewing [7], by ad hoc constructions of Quillen [21], Zabrodsky [23], Aguadé [3] and by Oliver (see [18]). In particular, these further examples contain all polynomial spaces whose adapted pseudo reflection group is rationally of non Lie type.

We are able to give a complete list of all spaces with polynomial mod–p cohomology. They are given by products of classifying spaces of connected compact Lie groups and some spaces associated to pseudo reflection groups of non Lie type. We need the following definition and remark.

1.9 Definition and Remark. Let p be an odd prime. Let W ⟶ Gl(U) be an irreducible p–adic rational pseudo reflection group of non Lie type. Up to weak isomorphism, there exists a unique W –lattice L ⊂ U which is simply connected and polynomial [19; Proposition 1.6]. In particular, by Theorem 1.5, there exists a p–complete polynomial space X_{W} with adapted pseudo reflection group given by W ⟶ Gl(L). And by Corollary 1.7, the homotopy type of X_{W} only depends on the rational representation W ⟶ Gl(U).

1.10 Theorem. Let p be an odd prime. Let X be a p–complete polynomial space. Then, there exists a connected compact Lie group G and a finite number of irreducible p–adic rational pseudo reflection groups W_{i} ⟶ Gl(U_{i}) of non Lie type such that X and BG_{p}^{∧} × ∏_{i} X_{W_{i}} are homotopy equivalent.

Finally we want to characterize those connected compact Lie groups for which the classifying space has polynomial mod–p cohomology.

1.11 Proposition. Let p be an odd prime. Let G be a connected compact Lie group. The following conditions are equivalent.

(i) H^{∗}(G; ℤ) is p–torsion free. (ii) H^{∗}(BG; 𝔽_{p}) is a polynomial algebra. (iii) L_{G} is a polynomial W_{G}–lattice.

The equivalence of the first two conditions is a classical result of Borel [4].

Actually, using the classification of polynomial lattices over pseudo reflection groups as described in [19], the theory of p–compact groups and the above mentioned further examples, one could construct enough polynomial spaces to get a realization for every polynomial 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 allow 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 unitary groups and the diagram is given by a full subcategory 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 classified in [19]. These calculations are based on the classification of the p–adic rational pseudo reflection groups by Clark and Ewing [7]. Up to weak isomorphism they gave a complete list of all irreducible p–adic rational pseudo reflection 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 unique up to weak isomorphism. The only exceptions are given by the pairs (W, p) = (W_{F_{4}}, 3), (W_{E_{6}}, 3), (W_{E_{7}}, 3), (W_{E_{8}}, 3) and (W_{E_{8}}, 5), where W denotes the Weyl group of the exceptional Lie group, indicated by the subindex, with the associated representation at the given prime p (for details see [19].

For connected compact Lie groups, Theorem 1.6, Corollary 1.7 and Corollary 1.8 are already proved in [16]. In particular, these results are shown for the 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 proof of Theorem 1.6.

In [18] the analogous theorems are proved for particular pseudo reflection groups, namely those which are subgroups of the wreath product ℤ/m ≀ Σ_{n} acting on (ℤ_{p}^{∧})^{n} where m divides p − 1. The proofs of Theorem 1.4 and Theorem 1.6 are based on very similar arguments and ideas. We will refer to them quite often, but explain the construction of polynomial spaces in detail in this paper. Hence, familiarity with that work might be useful for the reader. The proof of Theorem 1.6 uses the theory of p–compact groups, but everything necessary can be found in [18 ; Section 4].

The paper is organized as follows: In the next three sections we provide some technical results necessary for the construction of the spaces with polynomial cohomology (Theorem 1.4). In Section 2, we prove vanishing results for higher derived functors of inverse limits in certain specialized situations. In Section 3, we give an algebraic description of the center of a connected compact Lie group, and in Section 4 we describe an algebraic decomposition of the polynomial algebras under consideration. All this together allows a proof of Theorem 1.4 which is contained in Section 5. In Section 6 we collect some facts about p–adic integral pseudo reflection groups, necessary for the proof of the uniqueness properties of polynomial spaces (Theorem 1.6) which is contained in Section 7. The final section is devoted to proofs of Theorem 1.10 and Proposition 1.11.

We would like to thank Allan Hatcher, whose questions about the main results of this paper, gave me the chance to add Theorem 1.10 and Proposition 1.11.

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 O_{H}(G) the full subcategory of the orbit category O(G), whose objects are given by the orbits G/H where H ∈ H. Let P denote the closed family of all p–subgroups of G. For a given (covariant) functor F : O^{op}(G) ⟶ Ab from the opposite of the orbit category into the category 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 about higher derived limits, let C be a small category and let F un(C, Ab) be the category of (covariant) functors from C to Ab. Then there exist higher limits

lim^{i}_{C} : F un(C, Ab) ⟶ Ab

defined as right derived functors of the inverse limit functor lim^{i}_{C} : F un(C, Ab) ⟶ Ab

(cf. [5; XI, 6] or [20; Lemma 2]).

The following definition describes one of the assumptions we will have to put 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 ∈ K is contained in a unique minimal element H_{K} ∈ H.

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

O_{H}^{op}(G) ⟶ O_{H∪P}^{op}(G) ⟵ O_{P}^{op}(G) .

For each object G/K ∈ H ∪ P we denote by G/K ⟶ O_{P}^{op}(G) the under category of objects in O_{P}^{op}(G): the objects are given by morphisms ϕ : G/K ⟶ G/P in O^{op}(G) such that G/P is an object in O_{P}^{op}(G) and a morphism (G/P_{1}, ϕ_{1}) ⟶ (G/P_{2}, ϕ_{2}) is a morphism ψ : G/P_{1} ⟶ G/P_{2} in O^{op}(G) such that ψϕ_{1} = ϕ_{2}.

Taking higher derived limits restriction establishes the maps

lim^{i}_{O_{H}^{op}(G)} F ⟵^{α} lim^{i}_{O_{H∪P}^{op} (G)} F ⟶^{β} lim^{i}_{O_{P}^{op}(G)} F .

2.2 Proposition. If H is a minimalization of P and if for each object G/H of O_{H}^{op}(G) not contained in O_{P}^{op}(G), we have

lim^{i}_{(G/H⟶O_{P}^{op}(G))} (F |(G/H⟶O_{P}^{op}(G))) = \{ \begin{matrix}F (G/H) & if ∗ = 0 \\ 0 & if ∗ > 0, \end{matrix}

then α and β are isomorphisms.

Proof. This is a slightly more general statement as in [18 ; Proposition 2.3], where a particular functor is considered. But, using the same argument, our statement also follows from [18 ; 2.1, 2.2]. □

2.3 Remark. We would like to make the following observation: For each H ⊂ G there exists a functor O_{P}(H) ⟶^{≃} (O_{P}(G) ⟶ G/H) given by H/P ↦ (G/P ⟶ G/H). Here, O_{P}(G) ⟶ G/H denotes the over category; i.e. the category of objects in O_{P}(G) over G/H, which is defined analogously as the under category. 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 categories, for any subgroup H ⊂ G, we have an equivalence

O_{P}^{op}(H) ⟶ (G/H ⟶ O_{P}^{op}(G))

of categories.

In the next step we specialize the functor under consideration. For each G– module M we denote by H_{M}^{∗} : O^{op}(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. Let M be a G–modul. If H is a minimalization of P. Then,

lim^{i}_{O_{H}^{op}(G)} H_{M}^{∗} ≅ lim^{i}_{O_{P}^{op}(G)} H_{M}^{∗} ≅ \{ \begin{matrix}H^{∗}(G; M) & if i = 0 \\ 0 & if i > 0. \end{matrix}

Proof. The second isomorphism follows from [13; Section 5]. To prove the first equivalence we notice that

lim^{i}_{G/H⟶O_{P}^{op}(G)} H_{M}^{∗} ≅ lim^{i}_{O_{P}^{op}(H)} H_{M}^{∗} ≅ \{ \begin{matrix}H^{∗}(H; M) & if i = 0 \\ 0 & if i > 0. \end{matrix}

(Remark 2.3 and again [13; Section 5]). Because M is also an H–module, we can think of H_{M}^{∗} as a functor defined on O_{P}^{op}(H). □

Next we want to find minimalzations of P. Let V be a vector space over 𝔽_{p} and let G ⟶ Gl(V) be a 𝔽_{p}–representation of the finite group G. For each element P ∈ P we consider the isotropy subgroup H_{P} := Iso_{G}(V^{P}). Then, the family

H_{V} := {H_{P} ⊂ G|P ∈ P}

is closed under conjugation.

2.5 Lemma. The family H_{V} is a minimalization of P.

Proof. First of all, for P ∈ P, P ⊂ H_{P} and V^{P} = V^{H_{P}}. Thus, H_{P} ⊂ Iso_{G}(V^{H_{P}}) = Iso_{G}(V^{P}) = H_{P}, in particular H_{P} = Iso_{G}(V^{H_{P}}). And this is true for all elements of H_{V}. Now, let H ∈ H_{V} be any element such that P ⊂ H. Then, we have H_{P} = Iso_{G}(V^{P}) ⊂ Iso_{G}(V^{H}) = H, which shows that H_{P} is the minimal unique element of H_{V} containing P. □

2.6 Corollary. Let G ⟶ Gl(V) be a 𝔽_{p}–representation of a finite group G and let M be a G–module. Then,

lim^{i}_{O_{H_{V}}^{op} (G)} H_{M}^{∗} ≅ \{ \begin{matrix}H^{∗}(G; M) & if i = 0 \\ 0 & if i > 0. \end{matrix}

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 Z_{p}(G) of the center Z(G) of G can be calculated from the Weyl group data of G. Let T_{G} ⊂ G be a maximal torus, W_{G} the Weyl group and L := L_{G} := H_{2}(BT_{G}; ℤ_{p}^{∧}) the associated W_{G}–lattice. Passing to fixed points, the short exact sequence L ⟶ L_{ℚ} := L ⊗ ℚ ⟶ L_{∞} establishes an exact sequence

0 ⟶ L^{W_{G}} ⟶ (L_{ℚ})^{W_{G}} ⟶ (L_{∞})^{W_{G}} ⟶ H^{1}(W_{G}; L) ⟶ H^{1}(W_{G}; L_{ℚ}) = 0 .

For odd primes, the completed space (B(L_{∞})^{W_{G}})_{p}^{∧} is nothing but BZ_{p}(G)_{p}^{∧} [12; Remark 7.7]. The equivalence is induced from the composition

BL_{∞}^{W_{G}} ⟶ BL_{∞} ⟶ BL_{∞}_{p}^{∧} ≃ BT_{G}_{p}^{∧} ⟶ BG_{p}^{∧}

which factors over BZ_{p}(G)_{p}^{∧} ≃ BZ(G)_{p}^{∧} ⟶ BG_{p}^{∧}. Since Z_{p}(G) is abelian, Z_{p}(G) ≅ S × P is the product of a torus S and a finite p–group P. The quotient (L^{W_{G}})_{∞} = (L_{ℚ})^{W_{G}}/L^{W_{G}} detects the torus S, that is B(L^{W_{G}})_{∞}_{p}^{∧} ⟶^{≃} BS_{p}^{∧} is a homotopy equivalence. And H^{1}(W_{G}; L) detects the finite group P ≅ (L_{∞})^{W_{G}}/(L^{W_{G}})_{∞}, that is P ⟶ H^{1}(W_{G}; 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

π_{2}(BZ_{p}(G)_{p}^{∧}) ≅ L^{W_{G}} and π_{1}(BZ_{p}(G)_{p}^{∧}) ≅ H^{1}(W_{G}; L) .

Actually, these isomorphisms are natural with respect to monomorphisms between 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 T_{G} ⊂ G and T_{H} ⊂ H. Then, there exists a map f_{T} : BT_{H} _{p}^{∧} ⟶ BT_{G}_{p}^{∧}, unique up to homotopy and composition with elements of W_{G}, such that the diagram

BT_{H} _{p}^{∧} ⟶^{f_{T}} BT_{G}_{p}^{∧}
↓  ↓
BH_{p}^{∧} ⟶ BG_{p}^{∧}

commutes up to homotopy (e.g. see [11]). We say that f is a T –equivalence, if f_{T} 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_{p}^{∧} establishes a monomorphism α_{f_{T}} : W_{H} ⟶ W_{G} defined by the equation α_{f_{T}} (w)f_{T} ≃ f_{T} w, which makes sense because of the uniqueness properties of f_{T}. The maps

L(f_{T}) := H_{2}(f_{T}; ℤ_{p}^{∧}) : L_{H} ⟶ L_{G} and L_{∞}(f_{T}) : L_{H,∞} ⟶ L_{G,∞}

are W_{H}–equivariant with respect to α_{f_{T}}. In particular we get maps

\begin{matrix}(L_{G})^{W_{G}} & ⟶ & (L_{H})^{W_{H}} \\ BZ_{p}(G)_{p}^{∧} & ⟶ & BZ_{p}(H) \\ H^{∗}(W_{G}; L_{G}) & ⟶ & H^{∗}(W_{H}; L_{H}) . \end{matrix}

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 f_{T} of f. The third map is induced by the map f_{T} between the coefficients and the homomorphism α_{f_{T}}. This map is also independent of the chosen lift since for any element w ∈ W the pair of maps l_{w} : L_{G} ⟶ L_{G} and α_{w} : W ⟶ W induces the identity H^{∗}(W; L_{G}) ⟶^{=} H^{∗}(W; L_{G}) [6; 8.3]. Hence, after having fixed a maximal torus of G and H, the isomorphisms of Proposition 3.1 are natural with respect to T –equivalences. To give a functorial formulation of these facts, we consider the category LG_{p}(n) whose objects are given 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

BZ_{p} : LG_{p}(n)^{op} ⟶ HoT op and H_{L}^{∗} : LG_{p}(n)^{op} ⟶ Ab

whose values on the objects are given by BZ_{p}(BG_{p}^{∧}) = BZ_{p}(G) and H_{L}^{∗}(BG_{p}^{∧}) = H^{∗}(W_{G}; L_{G}). Moreover, there exist natural equivalences

π_{2}(BZ_{p}) ≅ H_{L}^{0} and π_{1}(BZ_{p}) ≅ H_{L}^{1} .

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 [19; 2.3], the vector space L/p is also polynomial, i.e. the ring of 𝔽_{p}[L/p]^{W} is a polynomial algebra. In this section we construct a decomposition of ℤ_{p}^{∧}[L]^{W} as well as of 𝔽_{p}[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 compact Lie group G and a subspace Δ ⊂ L/p such that the following holds:

(i) W_{G} ≅ Iso_{W} (Δ). (ii) As W_{G}–lattice, L is weakly isomorphic to L_{G}. (iii) The index [W : W_{G}] is coprime to p. (iv) H^{∗}(BG; 𝔽_{p}) ≅ 𝔽_{p}[L/p]^{W_{G}} and H^{∗}(BG; ℤ_{p}^{∧}) ≅ ℤ_{p}^{∧}[L]^{W_{G}}.

Proof. All this is stated in [19; 4.1]. □

Actually, as the proof of [19; 4.1] shows, the connected compact Lie group G is a quotient of a product of unitary groups, special unitary groups and tori.

Let H := H_{L/p} = {W_{P} ⊂ W : W_{P} = Iso(L/p^{P})} be the closed family of subgroups as defined in Section 2 and let H_{0} ⊂ H be the family of those groups which are contained in W_{G} where W_{G} is given by Proposition 4.1. The next lemma is the key for the construction of a topological realization.

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 W_{P} is subconjugated to W_{G} and P ⊂ W_{G} iff W_{P} ⊂ W_{G}. In particular, O_{H_{0}}(W) ⟶ O_{H}(W) is an equivalence of categories. (iii) Let H_{P} := C_{G}(L/p^{P}) for any p–subgroup P ⊂ W_{G}. Then, W_{P} ≅ W_{H_{P}} and R[L ⊗ R]^{W_{P}} ≅ H^{∗}(BH_{P}; R) for R = 𝔽_{p} and R = ℤ_{p}^{∧}.

Proof. The first point is already discussed in Lemma 2.5. The second follows from Proposition 4.1 (iii) and the observation that, for P ⊂ W_{G}, we have Δ ⊂ (L/p)^{P} and therefore Iso_{W} (L/p^{P}) = Iso_{W_{G}}((L/p)P).

The isotropy group W_{P} acts on the maximal torus of H_{P} and therefore W_{P} ⊂ W_{H}. On the other hand, W_{H} ⊂ Iso_{W} (L/p^{P}) = W_{P} by construction. Hence, W_{P} = W_{H}, which is th first part of (iii). For R = 𝔽_{p}, the second identity follows from [16; 10.1, 10.2] since H^{∗}(BG; 𝔽_{p}) ≅ 𝔽_{p}[L/p]^{W_{G}}. 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,

lim^{i}_{O_{H_{0}} (W)} H_{M}^{∗} ≅ lim^{i}_{O_{P} (W)} H_{M}^{∗} ≅ \{ \begin{matrix}H^{∗}(W; M) & if i = 0 \\ 0 & if i > 0. \end{matrix}

4.4 Remark. Setting M := 𝔽_{p}[L/p] and taking the functor H_{M}^{0} this corollary gives an algebraic decomposition of the ring of invariants 𝔽_{p}[L/p]^{W}. The pieces 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 realize 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 same 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

ψ : O_{H_{0}}(W) ⟶ HoT op

such that the following holds:

(i) ψ(W/W_{P}) = BC_{G}((L/p)^{W_{P}}). (ii) There exist natural equivalences

H^{∗}(ψ; 𝔽_{p}) ⟶^{≅} H_{𝔽_{p}[L/p]}^{0} and H^{∗}(ψ; ℤ_{p}^{∧}) ⟶^{≅} H_{ℤ_{p}^{∧}[L]}^{0} .

Proof. Part (i) indicates how we have to define the functor on the objects, namely ψ(W/W_{P}) := BC_{G}((L/p)^{W_{P}})_{p}^{∧} =: BH_{P} _{p}^{∧}. Each morphism α : W/W_{P} ⟶ W/W_{P′} is given by left translation with an element w ∈ W such that c_{w}(W_{P}) := w^{−1}W_{P} w ⊂ W_{P′} and splits therefore into a composition of the bijection l_{w} : W/W_{P} ⟶ W/c_{w}(W_{P}) and the projection q : W/c_{w}(W_{P}) ⟶ W/W_{P′}. We have to say what are the values of ψ for projection and for isomorphisms.

If q : W/W_{P} ⟶ W/W_{P′} is a projection, we define ψ(q) : BH_{P} _{p}^{∧} ⟶ BH_{P′} _{p}^{∧} to be the map induced by the inclusions W_{P} ⊂ W_{P′} and L/p^{W_{P′}} ⊂ L/p^{W_{P}}.

If l_{w} : W/W_{P} ⟶ W/W_{P′} is an isomorphism, the two compact connected Lie groups H_{P} and H_{P′} 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 L_{H_{P}} ≅ L ≅ L_{H_{P′}} are weakly isomorphic. Since both have isomorphic polynomial mod-p cohomology, this implies that BH_{P} _{p}^{∧} ≃ BH_{P′} _{p}^{∧} [16; Theorem 1.2]. Hence, for the analysis of maps between these two classifying spaces, we can assume that both spaces are equal. The element w ∈ W induces a map w : L ⟶ L which is admissible in the sense of [1]; i.e. for each x′ ∈ W_{P′} there exists an element x ∈ W_{p} (namely x = w^{−1}x′w) such that x′w = wx. Now, by [15; Corollary 2.5], there exists an equivalence BH_{P} _{p}^{∧} ⟶ BH_{P′} _{p}^{∧} such that the restriction (BT_{H_{P}})_{p}^{∧} ⟶ (BT_{H_{P′}})_{p}^{∧} of this map to the maximal tori is given by w.

Finally we have to show that this gives a functor. Let W/W_{P} ⟶^{α} W/W_{P′} ⟶^{β} W/W_{P′′} be a composition of maps of O_{H}(W). The definition of the values of α, β and βα may depend on the chosen splittings. But independent of these choices, since H^{∗}(BH_{p}^{∧}; ℤ_{p}^{∧}) ⊗ ℚ ≅ H^{∗}(BT_{H}; ℤ_{p}^{∧})^{W_{H}} ⊗ ℚ for every connected compact Lie group H, it is clear from the construction that ψ(β)ψ(α) and ψ(βα) induce the same map

H^{∗}(BH_{P′′} _{p}^{∧}; ℤ_{p}^{∧}) ⊗ ℚ ⟶ H^{∗}(BH_{P} _{p}^{∧}; ℤ_{p}^{∧}) ⊗ ℚ

in rational cohomology. Since H_{P} _{p}^{∧} and H_{P′′} are connected (Lemma 4.2 (iii)), we can apply [18 ; Proposition 3.3], which shows that both maps are homotopic. In particular, this argument also shows that ψ(α) does not depend on the chosen splitting of α. This proves the first part. The second part is obvious from the construction. □

As next step we have to construct a lift of ψ into the category T op of topological 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

ϕ : O_{H_{0}}(W) ⟶ T op

such that the following holds:

(i) Ho ϕ = ψ.

(ii) For X := hocolim_{O_{H_{0}} (W)} ϕ, we have H^{∗}(X; R) ≅ R[L ⊗ R]^{W} for R = ℤ_{p}^{∧} or R = 𝔽_{p}.

Proof. We will use the obstruction theory of Dwyer and Kan [9] made for such lifting problems. To apply this theory we have to check that the diagram given by

ψ : O_{H_{0}}(W) ⟶ HoT op

is centric. That is that for every morphism α = ql_{w} : W/W_{P} ⟶ W/W_{P′} the induced map

map(BH_{P} _{p}^{∧}, BH_{P} _{p}^{∧})_{id} ⟶ map(BH_{P} _{p}^{∧}, BH_{P′} _{p}^{∧})_{ψ(α)}

is an equivalence. This is automatic if α is an isomorphism, so it suffices to check it for projections given by inclusions W_{P} ⊂ W/W_{P′}. And follows in this case from [18 ; Proposition 3.3 (a)] (applied with G = H_{P′}). Furthermore, for each W_{P} we have map(BH_{P} _{p}^{∧}, BH_{P} _{p}^{∧})_{id} ≃ BZ(H_{P})_{p}^{∧} [15; Theorem 1.1]. Since Z(H_{P}) is a toral group, the only non vanishing homotopy groups of these mapping spaces are given by

π_{i}(map(BH_{P} _{p}^{∧}, BH_{P} _{p}^{∧})_{id} ≅ π_{i}(BZ(H_{P})_{p}^{∧}) ≅ \{ \begin{matrix}L^{W_{P}} & for i = 2 \\ H^{1}(W_{P}; L) & for i = 1 \end{matrix} .

The second isomorphism follows from Proposition 3.1. Moreover, since all morphisms in O_{H_{0}}^{op}(W) induce T –equivalences, this isomorphism comes from a natural transformation between the two functors (Proposition 3.2). Hence, by Corollary 4.3

lim^{i}_{O_{H_{0}}^{op} (W)} π_{j}(map(ψ(−), ψ(−))_{id} = 0

for all i, j > 0. The obstruction groups for lifting ψ are just given by some of these higher derived limits [9; Theorem 1.1]. Since all these groups vanish, there exists a lift

ϕ : O_{H_{0}}(W) ⟶ T op

of ψ. This proves the first part of the statement. The second part follows from the Bousfield–Kan spectral for calculating the cohomology of hocolim_{O_{H_{0}} (W)} ϕ, Theorem 5.1

and Corollary 4.3. □

Finally, we are able to proof Theorem 1.4:

Proof of Theorem 1.4. Let A ≅ H^{∗}(X; 𝔽_{p}) be a polynomial algebra over the Steenrod algebra with topological realization given by the space X. Then, the completed space X_{p}^{∧} is p–complete, realizes A too and has therefore an adapted p–adic pseudo reflection group W ⟶ Gl(L), L a lattice (Theorem 1.3). In particular, L is a polynomial W –lattice and A ≅ 𝔽_{p}[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 ≅ 𝔽_{p}[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. 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 representation 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 representation is already contained in the the mod–p representation and how much in the associated rational representation.

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 ∈ L and W ∈ W. The W –lattice L is called simply connected, if the covariants L_{W} := L/SL vanish.

6.2 Lemma. Let p be an odd prime. Let L be a lattice and let ρ_{1}, ρ_{2} : W ⟶ Gl(L) be faithful representations of a finite group such that the reduced representations \overline{ρ}_{1}, \overline{ρ}_{2} : W ⟶ Gl(L/p) are isomorphic.

(i) The pair (W, ρ_{1}) is a pseudo reflection group if and only if (W, ρ_{2}) is. (ii) If (W, ρ_{1}) and (W, ρ_{2}) are both pseudo reflection groups, then L is simply connected with respect to ρ_{1} if and only if it is with respect to ρ_{2}. (iii) If (W, ρ_{1}) is a pseudo reflection group such that L is simply connected and polynomial or if the order of W and p are coprime, then ρ_{1} and ρ_{2} are isomorphic.

Proof. Since p is odd, an element A ∈ Gl(L) is a pseudo reflection iff the mod–p reduction \overline{A} ∈ Gl(L/p) is a pseudo reflection. Hence, for odd primes, the pair (W, ρ_{i}) gives a pseudo reflection group iff (W, \overline{ρ}_{i}) does. This proves part (i).

The second part follows from the fact that a lattice L is simply connected iff (L^{♯}/p)^{W} = 0 [17; 2.2, 4.1], where L^{♯} := Hom(L, ℤ_{p}^{∧}) denotes the dual lattice.

The kernel of the homomorphism Gl(L/p^{k}) ⟶ Gl(L/p^{k−1}) is given by the abelian group Hom(L/p, L/p) of endomorphism of L/p. Therefore, the number of possible lifts of a homomorphism W ⟶ Gl(L/p^{k−1}) to Gl(L/p^{k}) is given by the order of the obstruction group H^{1}(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 connected compact Lie group G such that W_{G} ⊂ W with index coprime to p, such that W_{G} ⟶ W ⟶ Gl(L) describes the Weyl group data of G and such that L is a polynomial W_{G}–lattice. Moreover, G is given as a product of unitary and special unitary groups. All this follows from [19; 4.1]. By [16; Lemma 8.2] we have H^{1}(W_{G}; Hom(L/p, L/p)) = 0 = H^{1}(W; Hom(L/p, L/p)). The second equality follows because the index [W : W_{G}] is coprime to p. This proves the last part of the statement. □

We finish this section with the following lemma.

6.3 Lemma. Let p be an odd prime. Let W ⟶ Gl(U) be a fixed–point free p–adic rational pseudo reflection group. If U contains a polynomial W –lattice, then this is unique up weak isomorphism.

Proof. Let L ⊂ U be the polynomial W –lattice. By [19; Theorem 1.3], the associated simply connected lattice SL is polynomial and the covariants L_{W} are torsion free. Since L_{W} ⊗ ℚ ⊂ U^{W}, the covariants L_{W} vanish and SL ≅ L as W –lattice. By [17; Theorem 1.2], the W –lattice L is uniquely determined up to weak isomorphism. □

7. Homotopy uniqueness.

In this section we prove the homotopy uniqueness property of polynomial spaces. For a polynomial W –lattice we denote by X_{L} the polynomial space with adapted pseudo reflection group W ⟶ Gl(L) constructed in Section 5. The completion of X_{L} is p–complete and is obviously another realization of H^{∗}(X_{L}; 𝔽_{p}) with adapted pseudo reflection group W ⟶ Gl(L).

7.1 Theorem. Let Y be a polynomial p–complete space with adapted pseudo reflection group given by W ⟶ Gl(L). Then there exists a mod–p equivalence X_{L} ⟶ Y, which, in particular, establishes a homotopy equivalence X_{L}_{p}^{∧} ⟶^{≃} Y.

Proof. Since Y has polynomial mod–p cohomology, an Eilenberg–Moore spectral sequence argument shows that the loop space ΩY has finite mod–p cohomology. Hence, we can think of Y as the classifying space of a p–compact group.

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 [11]. Actually, everything necessary may be found in [18 ; Section 4], since the arguments of this proof run analogously as in Section 5 of [18].

As explained in the Section 4, there exist a connected compact Lie group G and a subspace Δ ⊂ L/p such that W_{G} ≅ Iso_{W} (Δ) ⊂ W ⟶ Gl(L) gives the Weyl group data of G. By assumption the space Y comes with a W –equivariant map f_{T} : BT_{L} ⟶ Y. Composing this map with BΔ ⟶ BT_{L} gives a map f_{Δ} : BΔ ⟶ Y. Since H^{∗}(Y; 𝔽_{p}) ≅ 𝔽_{p}(L/p)^{W} an application of [16; 10.1, 10.2] shows that H^{∗}(map(BΔ, Y)_{f_{Δ}}; 𝔽_{p}) ≅ 𝔽_{p}[L/p]^{W_{G}} ≅ H^{∗}(BG; 𝔽_{p}) and that the Weyl group data of the p–complete space map(BΔ, Y)_{f_{Δ}} are given by the composition W_{G} ⊂ W ⟶ Gl(L). The argument runs analogously as in [18 ; Proposition 4.8]. But this implies that BG_{p}^{∧} ≃ map(BΔ, Y)_{f_{Δ}} [16; Theorem 1.3].

Since for each object W/W_{P} ∈ O_{H}(W), the space ϕ(W/W_{P}) ≃ BC_{G}(L/p^{W_{P}}) =: BH_{P} allows a map into BG_{p}^{∧}. Composition of maps establishes maps f_{P} : BH_{P} _{p}^{∧} ⟶

BG_{p}^{∧} ⟶^{f_{G}} Y. Now, for for each morphism α : W/W_{P} ⟶ W/W_{P′}, we show that f_{P′}ϕ(α) ≃ f_{P}. Arguing analogously as in the proof of [18 ; Theorem 5.1] this becomes a consequence of [18 ; 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

lim^{i}_{O_{H}^{op}(W)} π_{j}(map(ϕ, Y)_{f_{P}})

[22]. Again, analogously as in the proof of [18 ; 5.1], one can show that

map(BH_{P} _{p}^{∧}, Y)_{f_{P}} ≃ map(BH_{P} _{p}^{∧}, BH_{P} _{p}^{∧})_{id} ≃ BZ(H_{P})_{p}^{∧} ,

where Z(H_{P}) denotes the center. The calculation of the above obstruction groups boils down to the equality

lim^{i}_{O_{H}^{op}(W)} π_{j}(BZ(H_{P} _{p}^{∧})) = 0

as shown in the proof of Theorem 5.2. Hence, there exists a map f : X_{L} ⟶ Y which, by construction, is a mod–p equivalence and establishes therefore a homotopy equivalence X_{L}_{p}^{∧} ⟶^{≃} Y. □

The following theorem contains Theorem 1.6.

7.2 Theorem. Let p be an odd prime. Let X and Y be two p–complete polynomial spaces such that H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(Y; 𝔽_{p}) 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 W_{X} is coprime to p.

(ii) X and Y have the same Weyl group data.

Proof. Let W_{X} ⟶ Gl(L_{X}) and W_{Y} ⟶ Gl(L_{Y}) be the adapted pseudo reflection groups. From Theorem 1.3 follows that L_{X} ≅ L_{Y} =: L as lattices, that W_{X} ≅ W_{Y} =: W and that the two representation \overline{ρ}_{X}, \overline{ρ}_{Y} : W ⟶ Gl(L/p) are conjugate. Here, we identified the lattices and the Weyl groups via these isomorphisms.

If X is simply connected, we have 0 = H^{2}(X; 𝔽_{p}) ≅ (L^{♯}/p^{W}) and hence, that L is a simply connected W –lattice [17]. Thus, Lemma 6.2 shows that the first two conditions imply the third one.

Starting with the third condition the statement follows from Theorem 7.1. □

8. A list of all polynomial spaces.

In this section we prove Theorem 1.10 and Proposition 1.11.

Proof of Theorem 1.10.

Let X be a p–complete space with adapted p-adic pseudo reflection group W ⟶ Gl(L). Then, L is a polynomial W –lattice. And by [19; Theorem 1.5], there exits a compact connected Lie group G and a finite number of irreducible p–adic rational pseudo reflection groups W_{i} ⟶ Gl(U_{i}) of non Lie type, such that W ⟶ Gl(L) and W_{G} × ∏_{i} W_{i} ⟶ Gl(L_{G} ⊕ ⨁_{i} L_{i}) are weakly isomorphic, where L_{i} ⊂ U_{i} is the uniquely determined W_{i}–lattice as described in Remark 1.9. By Theorem 1.5 follows that X ≃ BG_{p}^{∧} × ∏_{i} X_{W_{i}}. □

Proof of Proposition 1.11.

The equivalence of the first two conditions is a classical result of Borel [4]. So we only have to prove the equivalence of the second and the third condition.

For every connected compact Lie group G there exists a diagram of fibrations

BT_{H} ⟶ BT_{G} ⟶ BT
↓  ↓  ∥
BH ⟶ BG ⟶ BT ,

where T is a torus and where H is semi simple connected compact Lie group with maximal torus T_{H} (e.g. see [16; Section 4]). Moreover we can identify the Weyl groups W_{H} = W_{G} =: W. Since L_{G} ≅ H_{2}(BT_{G}ℤ_{p}^{∧}) ≅ π_{2}(BT_{G}) ⊗ ℤ_{p}^{∧}, passing to homology or homotopy groups establishes a commutative diagram

0  0
0 ⟶ S(L_{H}) ⟶^{≅} S(L_{G}) =: S ⟶ 0
↓  ↓  ↓
0 ⟶ L_{H} ⟶  L_{G}  ⟶ L_{T} ⟶ 0
↓  ↓  ∥
0 ⟶ (L_{H})_{W} ⟶ (L_{G})_{W} ⟶ L_{T} ⟶ 0

of exact rows and columns. The bottom row is exact since W acts trivially on L_{T} and since, for odd primes, H_{1}(W, ℤ_{p}^{∧}) = 0 [17; Lemma 1.9]. The inclusion S(L_{H}) ⟶ S(L_{G}) has a cokernel K with trivial W –action. The epimorphism S(L_{G}) ⟶ K factors over the covariants S(L_{G})_{W} which vanish, since the lattice is simply connected. Thus, the above inclusion is an isomorphism and the top row is exact. Because H is semi simple, we have (L_{H} ⊗ ℚ)^{W} = 0 and (L_{H})_{W} is a finite group.

If L_{G} is a polynomial W –lattice, then so is S = S(L_{G}) and (L_{G})_{W} is torsion free [19; Theorem 1.3]. Hence, (L_{H})_{W} = 0 and S ≅ L_{H} as W –lattices. In particular, π_{1}(H) ⊗ ℤ_{p}^{∧} = 0 [19; Lemma 4.5] and L_{H} is polynomial. Let H_{s} be the simply connected cover of H. Then, the kernel of the homomorphism H_{s} ⟶ H is coprime to p, and a similar diagram as above shows that L_{H_{S}} ≅ L_{H} as W –lattices. That is, we can assume that H is simply connected. Let H ≅ ∏_{i} H_{i} be the splitting into a product of simple simply connected factors. And each W_{H_{i}}–lattice L_{H_{i}} is polynomial. By [19; Theorem 1.4] non of the pairs (F_{4}, 3), (E_{6}, 3), (E_{7}, 3), (E_{8}, 3) or (E_{8}, 5) appears among the (H_{i}, p). For simple simply connected Lie groups, the above cases are the only one’s, where p–torsion appears in the cohomology. Therefore, H^{∗}(H; ℤ) is p–torsion free and H^{∗}(BH; 𝔽_{p}) a polynomial algebra. This proves the implication from (iii) to (ii).

Finally, let us assume that BG is a polynomial space. Then, again by results of Borel [4], H^{∗}(BG; 𝔽_{p}) ≅ H^{∗}(BT_{G}; 𝔽_{p})^{W_{G}} and L_{G} is a polynomial W_{G}–lattice (Theorem 1.5), which finishes the proof. □

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Mathematisches Institut, Bunssenstr. 3-5, 37073 Göttingen, Germany. e-mail : notbohm at cfgauss.uni-math.gwdg.de