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

§1. Introduction

If G 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, F_{p}) is a finitely generated polynomial algebra (i.e., such that X has a polynomial cohomology ring); at that time, the only examples known were spaces of the form X = BG.

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 exotic 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 that if X is any space with a polynomial cohomology ring then H^{∗}(X, F_{p}) 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 and in 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 A_{p} and K_{poly} the full subcategory of K consisting of objects which as rings are finitely generated polynomial algebras. If X is a p-complete space with H^{∗}(X, F_{p}) ∈ K_{poly} we extend the ideas of [1] by associating to X a finite p-adic linear group W_{X} generated by pseudoreflections (see 1.1); given any such finite group W such that p does not divide the order of W, we then show (1.2) that there is up to homotopy exactly one p-complete space X with H^{∗}(X, F_{p}) ∈ K_{poly} and W_{X} = W. In a 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.

The authors were supported in part by the National Science Foundation.

As a corollary we observe that if G is a connected compact Lie group and p a prime which does not divide the order of the Weyl group of G then the cohomology ring H^{∗}(BG, F_{p}), considered as an object of K, determines the homotopy type of the p-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 the paper.

We will now describe our results in more detail. If R is an object of K_{poly}, let ρ_{R} (the rank of R) be the number of polynomial generators in R and ν_{R} the product ∏_{i}(|x_{i}|/2) indexed by a set {x_{i}} of polynomial generators for R; recall that the prime p is odd, so that the degree |x_{i}| of each polynomial generator x_{i} is even. The integer ν_{R} does not depend upon a choice of polynomial generators for R. If X is a space with H^{∗}(X, F_{p}) ∈ K_{poly}, we will write ρ_{X} and ν_{X} for ρ_{R} and ν_{R} with R = H^{∗}(X, F_{p}). If H^{∗}(X, F_{p}) ∈ K_{poly} and X = BG for a connected compact Lie group G, then ρ_{X} is the Lie-theoretic rank of G and ν_{X} the order of the Weyl group of G.

Let Z_{p} denote the ring of p-adic integers. The Eilenberg-MacLane space K((Z_{p})^{r}, 2) is the p-completion of the classifying space of the rtorus T^{r}, and we will denote it B̂T^{r} or B̂T if r is understood; it is clear that the general linear group GL(r, Z_{p}) acts on B̂T^{r} in a natural way.

Given a commutative domain D, an element g ∈ GL(r, D) of finite order is said to be a pseudoreflection if the r × r matrix (g − I_{r}) has rank at most one (here I_{r} ∈ GL(r, D) is the identity matrix). If X is a space, say that a subgroup W ⊂ GL(r, Z_{p}) is adapted to X if there is a map τ : B̂T^{r} → X, equivariant up to homotopy with respect to the natural action of W on B̂T and the trivial action of W on X, such that τ induces an isomorphism

H^{∗}(X, F_{p}) ≅ H^{∗}(B̂T, F_{p})^{W} .

1.1 Theorem. Let p be an odd prime and let X be a p-complete space with H^{∗}(X, F_{p}) in K_{poly}. Then there exists up to conjugacy a unique finite subgroup W_{X} ⊂ GL(ρ_{X}, Z_{p}) adapted to X. The group W_{X} is generated by pseudoreflections and |W_{X}| = ν_{X}.

Remark: Theorem 1.1 does not require the assumption p ν_{X} and for this reason leads to non-realizability results for many objects of K_{poly}. The image in GL(ρ_{X}, F_{p}) of the group W_{X} ⊂ GL(ρ_{X}, Z_{p}) is the “Galois group” of H^{∗}(X, F_{p}) constructed in [1] (see §2). If H^{∗}(X, F_{p}) ∈ K_{poly} and X is the p-completion of BG for a connected compact Lie group G, then W_{X} is the image in GL(ρ_{X}, Z_{p}) of the Lie-theoretic Weyl group W_{G} ⊂ GL(ρ_{X}, Z). In general the group W_{X} of 1.1 must be a product of the irreducible p-adic pseudoreflection groups enumerated in [8]. This perhaps gives the list in [8] a richer significance.

1.2 Theorem. Let p be an odd prime and let W be a finite subgroup of GL(r, Z_{p}) generated by pseudoreflections. Suppose that p does not divide the order of W. Then there exists up to homotopy exactly one p-complete space X such that H^{∗}(X, F_{p}) ∈ K_{poly} and W is adapted to X.

It is clear from 1.1 that if H^{∗}(X, F_{p}) ∈ K_{poly} then p divides |W_{X}| iff p divides ν_{X}. This leads to the following corollary.

1.3 Corollary. Let p be an odd prime. Then for any r ≥ 1 there is a bijective correspondence between the following two sets:

(1) Homotopy equivalence classes of p-complete spaces X such that H^{∗}(X, F_{p}) ∈ K_{poly}, ρ_{X} = r and p ν_{X}.

(2) Conjugacy classes of finite subgroups W ⊂ GL(r, Z_{p}) such that W is generated by pseudoreflections and p |W|.

If r ≤ p − 2 then GL(r, Z_{p}) does not contain any non-trivial elements of order p, so Corollary 1.3 takes the following simpler form in this case.

1.4 Corollary. Let p be an odd prime. Then for any r such that 1 ≤ r ≤ p − 2 there is a bijective correspondence between the following two sets:

(1) Homotopy equivalence classes of p-complete spaces X such that H^{∗}(X, F_{p}) ∈ K_{poly} and ρ_{X} = r.

(2) Conjugacy classes of finite subgroups W ⊂ GL(r, Z_{p}) such that

W is generated by pseudoreflections. Moreover, if X is a space as in (1) then p ν_{X}.

The following algebraic theorem is both a summary and an extension of some of the results of [8] and [1]; it lists several ways of looking at the groups W which appear in 1.2–1.4. Let Q_{p} denote the quotient field of Z_{p}.

1.5 Theorem. Let p be an odd prime. Then for any positive integers r and n with p n the following sets are in bijective correspondence:

(1) Isomorphism classes of objects R ∈ K_{poly} such that ρ_{R} = r and ν_{R} = n.

(2) Conjugacy classes of subgroups W ⊂ GL(r, F_{p}) such that W is generated by pseudoreflections and |W| = n.

(3) Conjugacy classes of finite subgroups W ⊂ GL(r, Z_{p}) such that W is generated by pseudoreflections and |W| = n.

(4) Conjugacy classes of finite subgroups W ⊂ GL(r, C) such that W is generated by pseudoreflections, |W| = n, and the character field K_{W} (see §5) can be embedded in Q_{p}.

Combining 1.3 with 1.5 gives the following result.

1.6 Corollary. Let p be an odd prime. Then for any positive integers r and n with p n there is a bijective correspondence between the following two sets:

(1) Homotopy equivalence classes of p-complete spaces X such that H^{∗}(X, F_{p}) ∈ K_{poly}, ρ_{X} = r and ν_{X} = n.

(2) Isomorphism classes of objects R ∈ K_{poly} such that ρ_{R} = r and ν_{R} = n.

Of course, the bijection (1)→(2) in 1.6 is given by the cohomology functor, so in effect 1.6 states that a space X of the indicated type is determined up to homotopy by H^{∗}(X, F_{p}) ∈ K_{poly}. This immediately leads to the following homotopical uniqueness theorem for classifying spaces of Lie groups.

1.7 Theorem. Let G be a connected compact Lie group with Weyl group W and assume that p |W|. Suppose X is a space with H^{∗}(X, F_{p}) isomorphic to H^{∗}(BG, F_{p}) as an object of K. Then the p-completion of X is homotopy equivalent to the p-completion of BG.

Proof: Let r be the rank of G and let T ⊂ G be a maximal torus. The conjugation action of W on T exhibits W as a finite subgroup of GL(r, Z) generated by (pseudo)reflections. Since p |W| there is an isomorphism H^{∗}(BG, F_{p}) ≅ H^{∗}(BT, F_{p})^{W}, where the action of W on H^{∗}(BT, F_{p}) can be interpreted as the algebra action induced under the composite W → GL(r, Z) → GL(r, F_{p}) by the natural action of GL(r, F_{p}) on H^{2}(BT, F_{p}) = Hom((F_{p})^{r}, F_{p}). The second part of [1, 1.2] (cf. (2)↔(1) in 1.5) now shows that H^{∗}(BG, F_{p}) ∈ K_{poly} and p ν_{BG}, so the theorem follows immediately from 1.6.

We will give one concrete example of the calculations possible with 1.5 and 1.6. Let F_{p}[x_{24}, x_{48}] denote a graded polynomial algebra over F_{p} on generators of the indicated dimensions.

1.8 Theorem. If p > 3 then the number of isomorphism classes of objects of K with underlying ring F_{p}[x_{24}, x_{48}] is 1 if p ≡ 1 mod 48, 3 if p ≡ 25 mod 48, 2 if p ≡ 13, 37 mod 48 and zero otherwise. This number is also the number of homotopy types of p-complete spaces X such that H^{∗}(X, F_{p}) ≅ F_{p}[x_{24}, x_{48}].

This follows from a check of the group tables in [8]. The following more qualitative result is proved by combining 1.5, 1.6, the spectral sequence argument of [3, 13.1], and the observation that a finite group has only a finite number of conjugacy classes of subgroups.

1.9 Theorem. Let p be an odd prime and let Y be a one-connected pcomplete space such that H^{∗}(Y, F_{p}) is an exterior algebra on generators of degree 2d_{i} − 1, i = 1, …, n. Suppose that p d_{i}, i = 1, …, n. Then up to homotopy there are at most finitely many connected spaces X such that ΩX is equivalent to Y.

Further work: The results in this paper deal mostly with odd primes. For several reasons the results do not extend directly to cover the case p = 2; one of these reasons is that there are more elements of Hopf invariant one mod 2 than there are mod p for odd p (cf. proof of 4.1). Removing the restriction p |W | in Theorem 1.2 will probably require a new technique. Nevertheless, one example in which p = 2 and |W | = 2 (i.e. X = BSU(2)) has been treated successfully in [12]; there is some hope that the techniques of that paper, as extended by [16] and [17], may lead to further progress in this area.

Organization of the paper: Our main technique is to take a pcomplete space X with H^{∗}(X, F_{p}) ∈ K_{poly} and find a homotopical “maximal torus” for X; this is essentially a map t : BT → X with certain homological properties (see 2.6). We describe the theory of homotopical maximal tori and of their associated “Weyl groups” in §2, where we also detail how this theory leads to proofs of 1.1 and 1.2.

If X is the p-completion of BG for a compact Lie group G, then a homotopical maximal torus for X can be constructed from the map BT → BG induced by the inclusion T ⊂ G of an ordinary maximal torus (cf. proof of 1.7). If p is an odd prime, though, it is easy to see that T is in fact the centralizer in G of the subgroup V ⊂ T generated by elements of order p, and so [14] the map BT → BG can also be obtained (up to p-completion) by restricting the domain of the basepoint evaluation map Map(BV, BG) → BG to an appropriate component. It turns out that a corresponding function space construction gives a homotopical maximal torus for any p-complete X with H^{∗}X ∈ K_{poly}; this is proved in §4 with the help of some properties of Lannes’ functor “T^{V} ” which are established in §3. Section 5 contains the proof of 1.5.

1.10 Notation and terminology: The letter p will stand for a fixed rational prime, which except in §3 is assumed to be odd. For a space X, H^{∗}X will denote the object H^{∗}(X, F_{p}) of K and X̂ the p-completion of X in the sense of [6]. For a topological group or group-like topological monoid G, EG will denote some functorial contractible G-CW complex on which G acts freely, BG = EG/G the classifying space of G, and B̂G the p-completion of BG; if X is some space on which G acts, then B(G, X) will denote EG ×_{G} X and B̂(G, X) the p-completion of B(G, X). If f : X → Y is a map of spaces, we will let Map(X, Y)_{f} denote the path component of the space of maps from X to Y containing f.

An object of K is of finite type if in each dimension it is of finite rank as an F_{p} vector space. An object R ∈ K which is a domain is said to have transcendence degree r if its (graded) field of fractions Frac(R) has transcendence degree r over F_{p}. We will let K_{ftd} (resp. K_{ftd}^{r}) denote the full subcategory of K consisting of domains of finite transcendence degree (resp. transcendence degree r). Note that K_{ftd} ⊃ K_{poly}; many of our methods give information about objects of K_{ftd} as well as about objects of K_{poly}. Let K_{poly}^{r} denote K_{poly} ∩ K_{ftd}^{r}.

We are grateful to J. Lannes for telling us of his results and showing us how to simplify some early cumbersome proofs. Much of the work in this paper was discussed by the third author in a talk at the 1986 Barcelona Conference on Algebraic Topology.

§2. Maximal tori

In this section we will describe the properties of homotopical maximal tori and then show how these properties lead to 1.1 and 1.2. The proofs of the theorems in this section will appear in §4.

It is convenient to begin by recalling some material from [1] and [28] which deals with properties of an algebraic analogue of a maximal torus for rings R ∈ K_{ftd}^{r}.

2.1 Theorem. [1] For any object R of K_{ftd}^{r} there exists a K-map t : R → H^{∗}B̂T^{r} which is an embedding (i.e. monomorphism). If t_{1} and t_{2} are two such embeddings, then there is an automorphism w of H^{∗}B̂T^{r} such that w · t_{1} = t_{2}.

According to 2.1, considering general objects of K_{ftd}^{r} is equivalent to considering suitable A_{p} subalgebras of H^{∗}B̂T^{r}.

The group GL(r, F_{p}) can be identified with the group of automorphisms of H^{∗}B̂T^{r} (in the category K). Given an A_{p} subalgebra R of H^{∗}B̂T^{r}, let W_{R} denote the subgroup of GL(r, F_{p}) consisting of automorphisms of H^{∗}B̂T which leave R elementwise fixed. The geometric multiplication on B̂T gives a Hopf algebra structure on H^{∗}B̂T which is preserved by the action of GL(r, F_{p}), and we will let R_{sep} denote the smallest Hopf subalgebra of H^{∗}B̂T which is closed under the action of A_{p} and contains R. The action of W_{R} on H^{∗}B̂T clearly restricts to an action of W_{R} on R_{sep}.

Remark: The ring R_{sep} is in some sense the separable closure of R in H^{∗}B̂T and in generated by iterated p’th powers of two-dimensional classes (cf. proof of 4.1).

2.2 Theorem. [28] Let R ∈ K_{ftd}^{r} be an A_{p}-subalgebra of H^{∗}B̂T^{r}. Assume that as an algebra R is noetherian and integrally closed. Then the natural map

R → (R_{sep})^{W_{R}}

is an isomorphism.

Theorem 2.2 of course applies if R is a polynomial algebra, but even more is true in this case.

2.3 Theorem. [1] [28] Let R ∈ K_{poly}^{r} be an A_{p} subalgebra of H^{∗}B̂T^{r}. Assume that p ν_{R}. Then R_{sep} = H^{∗}B̂T^{r}, the group W_{R} ⊂ GL(r, F_{p}) is generated by pseudoreflections, and |W_{R}| = ν_{R}.

Finally, there is an algebraic existence theorem.

2.4 Theorem. [1] [28] Let W be a subgroup of GL(r, F_{p}) and R the ring (H^{∗}B̂T^{r})^{W}. Then R ∈ K_{ftd}^{r}, R is noetherian and integrally closed, R_{sep} = H^{∗}B̂T^{r}, and the natural map W → W_{R} is an isomorphism. Moreover, if W is generated by pseudoreflections and p |W| then R ∈ K_{poly} and ν_{R} = |W|.

What we will now do is to give geometric strengthenings of the above theorems under the assumption that the ring R in question is the cohomology ring of a space. The basic philosophy behind our approach is due to Rector [22].

Let B̂T^{r} stand for a CW-complex with some chosen homotopy equivalence to B̂T^{r}. We introduce this notation because below we would like to take a map f : B̂T^{r} → X and consider for instance self-maps of B̂T^{r} over X; to do this in a homotopy invariant way it is necessary to replace f by an equivalent Serre fibration t : B̂T^{r} → X.

2.5 Theorem. For any p-complete space X with H^{∗}X ∈ K_{ftd}^{r} there exists a space B̂T^{r} and a Serre fibration t : B̂T^{r} → X such that the induced cohomology homomorphism t^{∗} is an embedding. If t_{i} : B̂T_{i}^{r} → X, (i = 1, 2) are two such maps, then there is a homotopy equivalence w : B̂T_{1}^{r} → B̂T_{2}^{r} such that t_{1} = w · t_{2}.

2.6 Definition: A space X with a maximal r-torus is a pair (X, t) in which X is a p-complete space with H^{∗}X ∈ K_{ftd}^{r} and t : B̂T^{r} → X is a Serre fibration such that t^{∗} is an embedding.

The above definition of “maximal r-torus” is not the most useful general one but it is convenient for the purposes of this paper. According to 2.5, considering a general p-complete X with H^{∗}X ∈ K_{ftd}^{r} is more or less equivalent to considering a space X with a maximal r-torus.

2.7 Theorem. Let X be a space with a maximal r-torus t : B̂T^{r} → X and let R be the algebra t^{∗}(H^{∗}X) ⊂ H^{∗}B̂T^{r}. Then R_{sep} = H^{∗}B̂T^{r} and hence R = (H^{∗}B̂T^{r})^{W_{R}}.

In the above theorem, the geometric assumption that R is the cohomology of a space has allowed us to sidestep requirements (cf. 2.3) of the type p ν_{R}. This theorem is false for p = 2 (for instance, let X be the 2-completion of BSU(2)). 2.8 Definition: Let X be a space with a maximal r-torus t : B̂T^{r} → X. The Weyl space W_{X} = W_{X}(t) of X is the space of self-equivalences w : B̂T → B̂T such that t · w = t. The Weyl group W_{X} = W_{X}(t) of X is the component group π_{0}W_{X}. Remark: Composition of maps gives the Weyl space W_{X} the structure of a group-like topological monoid, and actions of W_{X} can be used in forming suitable Borel constructions (1.10).

Given X as in 2.8, let R stand for the subalgebra t^{∗}(H^{∗}X) of H^{∗}B̂T^{r}. The group of homotopy classes of self homotopy equivalences of B̂T^{r} is GL(r, Z_{p}), so that there is a commutative diagram

W_{X} ⟶ W_{R}
↓  ↓
GL(r, Z_{p}) ⟶ GL(r, F_{p})

in which the horizontal arrows are constructed by taking induced cohomology homomorphisms. In this diagram, it is not a priori clear that W_{X} is finite or that the left hand vertical arrow is a monomorphism, although by definition the right-hand vertical arrow is a monomorphism.

2.9 Theorem. Let X be a space with a maximal r-torus t : B̂T^{r} → X and let R be the algebra t^{∗}(H^{∗}X) ⊂ H^{∗}B̂T^{r}. Then the Weyl space W_{X} is homotopically discrete (i.e., each component of W_{X} is contractible). and the natural map W_{X} → W_{R} is an isomorphism.

If B̂T^{r} → X is a space with a maximal r-torus, there is an induced map B(W_{X}, B̂T^{r}) → X and up to homotopy a map τ : B̂(W_{X}, B̂T^{r}) → X (see 1.10 for the notation here). Note that the homomorphism W_{X} → GL(r, Z_{p}) gives by naturality an action of W_{X} on the Eilenberg-MacLane space K((Z_{p})^{r}, 2) = B̂T^{r}.

2.10 Theorem. Suppose that X is a space with a maximal r-torus and that as an algebra H^{∗}X is noetherian and integrally closed. Assume that p |W_{X}|. Then the above map

τ : B̂(W_{X}, B̂T^{r}) → X

is a homotopy equivalence. Moreover, the space B̂(W_{X}, B̂T^{r}) is homotopy equivalent to B̂(W_{X}, B̂T^{r}).

As in 2.3, there are additional algebraic restrictions in the case of a space with a polynomial cohomology ring. Note (2.9) that under the hypotheses of the following theorem the group W_{X} can be identified with a subgroup of GL(r, Z_{p}), well defined up to conjugacy.

2.11 Theorem. Suppose that X is a space with a maximal r-torus and that H^{∗}X is noetherian and integrally closed. Then the natural maps

H^{∗}X → H^{∗}(B̂T^{r})^{W_{X}}
H^{∗}(X, Z_{p}) → H^{∗}(B̂T^{r}, Z_{p})^{W_{X}}
Q_{p} ⊗_{Z_{p}} H^{∗}(X, Z_{p}) → (Q_{p} ⊗_{Z_{p}} H^{∗}(B̂T^{r}, Z_{p}))^{W_{X}}

are isomorphisms. If H^{∗}X ∈ K_{poly}, then W_{X} ⊂ GL(r, Z_{p}) is generated by pseudoreflections.

Finally, there is a geometric existence theorem.

2.12 Theorem. Let W be a finite subgroup of GL(r, Z_{p}) and let X be the space B̂(W, B̂T^{r}) formed with respect to the natural action of W on B̂T^{r} = K((Z_{p})^{r}, 2). Assume that p |W|. Then H^{∗}X = (H^{∗}B̂T^{r})^{W} ∈ K_{ftd}^{r}, H^{∗}X is noetherian and integrally closed, and, given a maximal r-torus t : B̂T^{r} → X for X (2.5, 2.6), the group W_{X}(t) is conjugate as a subgroup of GL(r, Z_{p}) to W. Moreover, if W is generated by pseudoreflections then H^{∗}X ∈ K_{poly} and ν_{X} = |W|.

We will finish this section by giving proofs for the homotopy theoretic theorems of the introduction. Proof of 1.1: Let r = ρ_{X} (so that H^{∗}X ∈ K_{poly}^{r}) and let t : B̂T^{r} → X be a maximal r-torus for X (2.5). By 2.9, 2.7 and 2.2, the group W_{X}(t) defined in 2.8 is adapted to X. By 2.11 the group W_{X}(t) is generated by pseudoreflections; the equality ν_{X} = |W_{X}(t)| follows from 2.4. Suppose that W ⊂ GL(r, Z_{p}) is some other subgroup which is adapted to X. Let S be the set of homotopy classes of maps f : B̂T^{r} → X such that f^{∗} is an embedding. It follows from 2.5 that GL(r, Z_{p}) acts transitively on S and that the isotropy subgroup of any element of S is conjugate to the image of the natural map π_{0}W_{X}(t) → GL(r, Z_{p}). As a consequence, W is conjugate to a subgroup of W_{X}(t) and so (2.9) the reduction mod p map W → GL(r, F_{p}) is a monomorphism. By 2.4 and 2.1 the image of W in GL(r, F_{p}) is conjugate to the image of W_{X}(t), which by the above implies that |W_{X}(t)| = |W| and finishes the proof. Proof of 1.2: This is a direct consequence of 2.4 and 2.2.

§3. The functor T^{V}

In this section V will denote an F_{p}-vector space of dimension r. The purpose of this section is to describe some properties of Lannes’ functor T^{V} [20].

Recall [20, §2] that T^{V} is left adjoint to the functor K → K which which sends R ∈ K to H^{∗}(BV) ⊗ R. The functor T^{V} is exact [20, 2.4.1] and preserves tensor products [20, 2.4.3].

If f : R → H^{∗}BV is a map of K, then corresponding to f under adjointness is a map T^{V} (R) → F_{p} or in other words a ring homomorphism f̄: T^{V} (R)^{0} → F_{p}. We will let T_{f}^{V} (R) denote the tensor product T^{V} (R) ⊗_{T^{V} (R)^{0}} F_{p}, where the action of T^{V} (R)^{0} on F_{p} is given by f̄. If f : R ⟶ S and g : S ⟶ H^{∗}BV are maps of K with g · f = h then f induces a natural map T_{h}^{V} R → T_{g}^{V} S. We will denote this map T_{g}^{V} (f); using the notation T_{h}^{V} (f) would leave some ambiguity as to the range of the map, since there might be several maps g : S ⟶ H^{∗}BV with g · f = h. The inclusion {0} ⊂ V induces for any R ∈ K a natural map ϵ : R = T^{{0}}R → T^{V} R which for f : R → H^{∗}BV passes to a map ϵ_{f} : R → T_{f}^{V} R.

For any space X, the evaluation map BV ×Map(BV, X) → X induces a cohomology homomorphism

H^{∗}X → H^{∗}(BV) ⊗ H^{∗} Map(BV, X)

which has as adjoint a natural map

λ : T^{V} (H^{∗}X) → H^{∗} Map(BV, X) .

For each map f : BV → X, λ induces a map λ_{f} : T_{f}^{V} (H^{∗}X) → H^{∗} Map(BV, X)_{f}, which fits into a commutative diagram

H^{∗}X ⟶^{=}  H^{∗}X
ϵ_{f} ↓  ↓
T_{f}^{V} (H^{∗}X) ⟶^{λ_{f}} H^{∗} Map(BV, X)_{f}

in which the right vertical map is induced by evaluation at the basepoint of BV. (Here and subsequently we write T_{f}^{V} (H^{∗}X), etc., instead of T_{f^{∗}}^{V}(H^{∗}X).)

Lannes has proved the following two theorems.

3.1 Theorem. [20, 3.1.4] If X is a simply-connected p-complete space such that H^{∗}X is of finite type, then the natural map

[BV, X] → Hom_{K}(H^{∗}X, H^{∗}BV)

is a bijection.

3.2 Theorem. [20, 3.2.2] Let X be a simply-connected p-complete space such that H^{∗}X is of finite type and let f : BV → X be a map. Assume that T_{f}^{V} (H^{∗}X) is of finite type and vanishes in degree 1. Then the natural map

λ_{f} : T_{f}^{V} (H^{∗}X) → H^{∗} Map(BV, X)_{f}

is an isomorphism

We will need to establish some algebraic properties of T^{V}. Recall that the topological multiplication on BV gives H^{∗}BV the structure of a Hopf algebra with comultiplication Δ : H^{∗}BV → H^{∗}BV ⊗ H^{∗}BV. For any g : H^{∗}BV → H^{∗}BV, the map T^{V} (H^{∗}BV) → H^{∗}BV which is adjoint to (g ⊗ 1) · Δ induces a map μ_{g} : T_{g}^{V} (H^{∗}BV) → H^{∗}BV. For any map f : BV → BV, μ_{f} fits into a commutative diagram

T_{f}^{V} (H^{∗}BV)  ⟶^{μ_{f}} H^{∗}BV
λ_{f} ↓  ↓^{=}
H^{∗} Map(BV, BV)_{f} ⟶_{τ} H^{∗}BV

in which τ is induced by the map BV → Map(BV, BV)_{f} which sends a point x ∈ BV to the map obtained by composing f with (right) multiplication by x.

3.3 Lemma. For any map g : H^{∗}BV → H^{∗}BV the maps ϵ_{g} : H^{∗}BV → T_{g}^{V} (H^{∗}BV) and μ_{g} : T_{g}^{V} (H^{∗}BV) → H^{∗}BV are isomorphisms. If f is an endomorphism of H^{∗}BV then μ_{g} · T_{g}^{V} (f) = f · μ_{gf}.

Proof: The first two statements can be proved by using the elementary fact that λ : T^{V} (H^{∗}BV) → H^{∗} Map(BV, BV) is an isomorphism [20, 3.4.4] and then calculating with function space cohomology; for instance, the statement about ϵ_{g} follows from the calculation that for any h : BV → BV the map Map(BV, BV)_{h} → BV given by evaluation at the basepoint is an equivalence. The final assertion is proved by showing that both maps involved correspond under adjointness to ((gf) ⊗ f) · Δ.

The following lemma will be used in §4. It is proved by using the fact (cf. [14] or [19]) that λ : T^{V} (H^{∗}B̂T^{s}) → H^{∗} Map(BV, B̂T^{s}) is an isomorphism.

3.4 Lemma. For any s ≥ 0 and map f : H^{∗}B̂T^{s} → H^{∗}BV, the map ϵ_{f} : H^{∗}B̂T^{s} → T_{f}^{V} (H^{∗}B̂T^{s}) is an isomorphism.

3.5 Proposition. Let f : R → H^{∗}BV be a monomorphism and let ι denote the identity map of H^{∗}BV. Then the map T_{ι}^{V} (f) : T_{f}^{V} R → T_{ι}^{V} (H^{∗}BV) ≅ H^{∗}BV is an monomorphism.

Proof: Denote the tensor product T_{f}^{V} (R) ⊗_{T^{V} (R)} T^{V} (H^{∗}BV) by S. The algebra T_{f}^{V} R is flat as a module over T^{V} R (this is a consequence of the associativity of tensor product and the fact that every module over the p-boolean ring (T^{V} R)^{0} is flat[20, 1.8]). It follows from the exactness property of T^{V} [20, 2.4.1] that the natural map T_{f}^{V} R → S is a monomorphism. It is easy to calculate (cf. 3.3) that S is the product ∏_{g} T_{g}^{V} (H^{∗}BV) indexed by g : H^{∗}BV → H^{∗}BV such that g · f = f; under this identification the monomorphism T_{f}^{V} R → S is ∏_{g} T_{g}^{V} (f). To complete the proof, then, it is enough to show that if g : H^{∗}BV → H^{∗}BV is a map with g · f = f, then the kernel of T_{g}^{V} (f) contains the kernel of T_{ι}^{V} (f).

Let u denote the composite map μ_{ι} · T_{ι}^{V} (f) and v the corresponding composite μ_{g} · T_{g}^{V} (f). By 3.3, we will be done if we can show that the kernel of v contains the kernel of u. The map g : H^{∗}BV → H^{∗}BV is (Bγ)^{∗} for a unique homomorphism γ : V → V. Write V as a direct sum V_{1} ⊕V_{2}, where V_{1} is the subgroup killed by some power of γ and V_{2} is the intersection of the images of all powers of γ. Note that the restriction of γ to V_{2} gives an automorphism of V_{2}. Let ϵ : V → V_{2} be the projection map, and choose N large enough that the restriction of γ^{N} to V_{1} is zero. Let e : H^{∗}BV → H^{∗}BV be the map (Bϵ)^{∗}. Since e · g^{N} = g^{N}, it follows from 3.3 that μ_{g} · T_{g}^{V} (g^{N}) = e · μ_{g} · T_{g}^{V} (g^{N}). Consequently, there are equalities

v = μ_{g} · T_{g}^{V} (f)
= μ_{g} · T_{g}^{V} (g^{N}) · T_{g^{N+1}}^{V}(f)
= e · μ_{g} · T_{g}^{V} (g^{N}) · T_{g^{N+1}}^{V}(f)
= e · μ_{g} · T_{g}^{V} (f)

where the first equality is the definition of v, the second and fourth follow from g · f = f, and the third is the above remark. Let γ^{−1} denote the self-map of V which is zero on V_{1} and the inverse of γ|_{V_{2}} on V_{2}; let g^{−1} denote (Bγ^{−1})^{∗}. The identity γ ·γ^{−1} ·ϵ = ϵ gives e = e·g^{−1} ·g. It follows from 3.3 that e · μ_{g} = e · g^{−1} · μ_{ι} · T_{ι}^{V} (g). Combining this with the above gives that v agrees with the map g^{−1}·μ_{ι}·T_{ι}^{V} (g)·T_{g}^{V} (f) = g^{−1}·μ_{ι}·T_{ι}^{V} (f), where the equality follows from g · f = f. In other words, v = g^{−1} · u, which proves the desired result.

3.6 Proposition. Suppose that R ∈ K is a subalgebra of H^{∗}BV. Let f : R → H^{∗}BV be the inclusion map and let ι be the identity map of H^{∗}BV. Then (ϵ_{ι})^{−1} · T_{ι}^{V} (f) gives an isomorphism from T_{f}^{V} R to the smallest Hopf subalgebra of H^{∗}BV which is closed under the action of A_{p} and contains R.

This proposition does not require the assumption that R is noetherian. Proof of 3.6: Adjoint to the identity map of T^{V} R is a map R → H^{∗}(BV) ⊗ T^{V} R. Compose this with the diagonal map of H^{∗}BV to get a map

R → H^{∗}(BV) ⊗ H^{∗}(BV) ⊗ T^{V} R

and reverse the process of taking adjoints; this gives a map

Δ : T^{V} R → H^{∗}(BV) ⊗ T^{V} R

which makes T^{V} R into a comodule over H^{∗}BV. (This comodule structure is an algebraic counterpart to the action of BV on Map(BV, X) induced by the left translation action of BV on itself). The comodule structure passes to a quotient comodule stucture Δ_{f} : T_{f}^{V} R → H^{∗}(BV) ⊗ T_{f}^{V} R which fits into a commutative diagram

R ⟶^{ϵ_{f}} T_{f}^{V} R ⟶^{Δ_{f}} H^{∗}(BV) ⊗ T_{f}^{V} R
f↓  ^{g}↓  ↓^{1⊗g}
H^{∗}BV ⟶^{=} H^{∗}BV ⟶^{Δ_{ι}} H^{∗}(BV) ⊗ H^{∗}(BV)

where we have used ϵ_{ι} (3.3) to identify T_{ι}^{V} (H^{∗}BV) with H^{∗}BV and g = (ϵ_{ι})^{−1}·T_{ι}^{V} (f). Since f is a monomorphism the map g is a monomorphism (3.5). It is easy to see that Δ_{ι} is the Hopf algebra comultiplication map on H^{∗}BV. It now follows from the fact that the comultiplication on H^{∗}BV is cocommutative that Δ_{f}(T_{f}^{V} R) is contained in T_{f}^{V} R ⊗ T_{f}^{V} R and thus that T_{f}^{V} R is a Hopf subalgebra of H^{∗}BV. On the other hand, if S is a Hopf subalgebra of H^{∗}BV closed under the action of A_{p} and h : S → H^{∗}BV is the inclusion, then ϵ_{h} : S → T_{h}^{V} S is an isomorphism [13, 4.8]. The proposition follows.

§4. Maximal tori, continued

In this section we will give proofs of the theorems from §2. We will continue to use the notation of §2 and §3.

4.1 Lemma. Let R ∈ K_{ftd}^{r} be an A_{p} subalgebra of H^{∗}B̂T^{r}. Then there exists a space X with H^{∗}X ≅ R_{sep} iff R_{sep} = H^{∗}B̂T^{r}.

Proof: Define a filtration of H^{2}B̂T^{r}

F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ H^{2}B̂T

by declaring that x ∈ H^{2}B̂T belongs to F_{s} iff the element x^{p^{s}} of the field (1.10) Frac(H^{∗}B̂T) is separable [28] over Frac(R). According to [28] R_{sep} ⊃ R is the subalgebra of H^{∗}B̂T generated by ∑_{s≥0}(F_{s})^{p^{s}}. Sup-

pose that there exists such a space X of the indicated type, so that in particular H^{2}X ≅ F_{0}. Let s be the F_{p} rank of F_{0}. Since H^{3}X = 0 the group H^{2}(X, Z_{p}) has no torsion, so there exists a map f : X → B̂T^{s} such that H^{2}(f) is an isomorphism. Let Y be the homotopy fibre of f. It is clear that H^{∗}Y is isomorphic to the (polynomial) subalgebra of H^{∗}B̂T^{r} generated by ∑_{s≥1}(F_{s})^{p^{s}}, and thus that H^{∗}Y is non-zero only in degrees divisible by 2p. An argument [15] involving the non-existence of higher elements of Hopf invariant one mod p implies that this fibre Y is contractible.

Proof of 2.5: Let V be the group_{p}T^{r} generated by the elements of order p in the torus T^{r}. The map BV → B̂T^{r} induces a monomorphism on cohomology which we will use to identify H^{∗}B̂T^{r} with the subalgebra of H^{∗}BV generated by the images under the Bockstein of one-dimensional classes. By 2.1 and 3.1 there is a map f : BV → X such that f^{∗} embeds H^{∗}X into H^{∗}B̂T^{r} ⊂ H^{∗}BV. Let R ⊂ H^{∗}B̂T^{r} denote f^{∗}(H^{∗}X). It follows from 3.6 and the fact that H^{∗}B̂T^{r} is a Hopf subalgebra of H^{∗}BV that T_{f}^{V} (H^{∗}X) is isomorphic to R_{sep}. On the other hand, by 3.2 there is an isomorphism λ_{f} : T_{f}^{V} (H^{∗}X) → H^{∗} Map(BV, X)_{f} and so T_{f}^{V} (H^{∗}X) ≅ R_{sep} is realizable as the cohomology of a space. Lemma 4.1 now shows that R_{sep} = H^{∗}B̂T^{r} and thus that H^{∗} Map(BV, X)_{f} ≅ H^{∗}B̂T^{r}. There is a map c : Map(BV, X)_{f} → B̂T^{r} which induces this cohomology isomorphism, and, since the space Map(BV, X)_{f} is H_{∗}(−, F_{p})- local [5, 12.9], the map c is an equivalence. Let ι denote the identity map of BV. There is a commutative diagram

Map(BV, BV)_{ι} ⟶^{f·(−)} Map(BV, X)_{f}
e_{ι}↓  ↓^{e_{f}}
BV  ⟶^{f}  X

in which the vertical maps are given by basepoint evaluation and the left-hand vertical map is an equivalence. It follows that the map e_{f} induces an embedding (e_{f})^{∗} : H^{∗}X → H^{∗} Map(BV, X)_{f} and that the desired map t : B̂T^{r} → X can be obtained by replacing e_{f} by a weakly equivalent Serre fibration.

Now suppose that g : Y → X is a Serre fibration with Y ≅ B̂T^{r} and g^{∗} an embedding. By the fact [20] that H^{∗}BV is an injective object of K, the map f^{∗} : H^{∗}X → H^{∗}BV extends over g^{∗} to a map χ : H^{∗}Y → H^{∗}BV (i.e. χ · g^{∗} = f^{∗}). If h : BV → Y is a map with h^{∗} = χ, then the composite g · h is homotopic to f (3.1). Since g is a fibration it is possible by the covering homotopy property to adjust h so that g · h = f. There is then a commutative diagram

Map(BV, Y)_{h} ⟶^{g·(−)} Map(BV, X)_{f}
e_{h}↓  ↓^{e_{f}}
Y  ⟶^{g}  X

in which the vertical arrows are given by basepoint evaluation. The map e_{h} is an equivalence by a simple direct argument or by a combination of 3.2 and 3.4. To complete the proof it is enough by standard homotopy theory to show that the upper horizontal arrow in this diagram is an equivalence, or, since the spaces involved are H_{∗}(−, F_{p})- local [5, 12.9], that the corresponding cohomology map T_{h}^{V} (g) (see 3.2) is an isomorphism. The map T_{ι}^{V} (f) : T_{f}^{V} (H^{∗}X) → T_{ι}^{V} (H^{∗}BV) is a monomorphism by 3.5; since T_{ι}^{V} (f) = T_{ι}^{V} (h) · T_{h}^{V} (g), this implies that T_{h}^{V} (g) is a monomorphism. Since T_{h}^{V} (H^{∗}Y) ≅ H^{∗}B̂T^{r} by 3.4 and T_{f}^{V} (H^{∗}X) ≅ H^{∗}B̂T^{r} by the argument above, dimension counting in fact shows that T_{h}^{V} (g) must be an isomorphism.

Proof of 2.7: Let V = (F_{p})^{r} and choose a map g : BV → B̂T^{r} such that g^{∗} is a monomorphism. Let f = t · g. It is possible to argue as above in the proof of 2.5 that R_{sep} is realizable as the cohomology of the space Map(BV, X)_{f}. The desired conclusion follows from 4.1.

4.2 Lemma. Let X be a space with a maximal r-torus t : B̂T^{r} → X and V the vector space (F_{p})^{r}. Suppose that f : BV → B̂T^{r} is a map such that f^{∗} is an embedding, and let g = t · f. Then the natural map

Map(BV, B̂T^{r})_{f} ⟶^{t·(−)} Map(BV, X)_{g}

is an equivalence.

The proof of this is contained in the proof above of 2.5. We now head towards a proof of 2.9. First we have an algebraic lemma, whose proof comes down to a straightforward calculation.

4.3 Lemma. Suppose that R is an object of K_{ftd}^{r} and that t : R → H^{∗}B̂T^{r} is an embedding. Let V be the vector space (F_{p})^{r}, and identify H^{∗}B̂T^{r} with the subalgebra of H^{∗}BV generated by the images under the Bockstein of one-dimensional classes. Let G be the group of K automorphisms g of H^{∗}BV such that g · t = t. Then each element of G carries H^{∗}B̂T^{r} to itself, and the subsequent induced homomorphism G → W_{R} is an isomorphism.

Proof of 2.9: Let V = _{p}T^{r} as above. The main idea of this proof is to find a relationship between maps BV → X and maps B̂T^{r} → X by using the fibration sequence BV → B̂T^{r} ⟶ B̂T^{r}. We begin with some general considerations.

Suppose that q : E → B is a (Serre) fibration of connected CWcomplexes with fibre F, and that Y is a space. By elementary homotopy theory there is a space Map(E/B, Y) and a fibration q̇ : Map(E/B, Y) → B such that the space of sections Γ(q̇) is naturally weakly equivalent to Map(E, Y). The fibre of q̇ is Map(F, Y).

Suppose in addition that f : E → X and g : Y → X are maps, with g a fibration. Let Map_{X}(E, Y) denote the space of all maps h : E → Y such that f = g · h. The map Map(E, Y) ⟶ Map(E, X) given by composition with g is a fibration with fibre over f ∈ Map(E, X) equal to Map_{X}(E, Y). It is clear that there is a space Map_{X}(E/B, Y) and a fibration q̇_{X} : Map_{X}(E/B, Y) → B such that Γ(q̇_{X}) ≅ Map_{X}(E, Y); the map q̇_{X} can be obtained by constructing a (homotopy) fibre square

Map_{X}(E/B, Y) ⟶ Map(E/B, Y)
q̇_{X} ↓  ↓
B  ⟶ Map(E/B, X)

in which the lower horizontal arrow corresponds to f. It follows from this fibre square that the (homotopy) fibre of q̇_{X} is Map_{X}(F, Y).

Suppose that X is a space with a maximal torus t : B̂T^{r} → X. Let Y be the space B̂T^{r} and g : Y → X the map t. Let q : E → B be a fibration equivalent to the map K((Z_{p})^{r}, 2) → K((Z_{p})^{r}, 2) induced by the homomorphism (Z_{p})^{r} → (Z_{p})^{r} given by multiplication by p, and let f : E → X be the composite of t with an equivalence E ≅ B̂T^{r}. Up to homotopy we will identify the fibre of q with BV, where V is the vector space (F_{p})^{r} = _{p}T^{r}. By the above considerations, Map_{X}(Y, Y) ≅ Map_{X}(E, Y) is equivalent to the space of sections of a fibration q̇_{X} over B with fibre Map_{X}(BV, Y).

We will use a superscript “+” (e.g. “Map^{+}(X, Y)”) to denote the subspace of a mapping space consisting of maps with induce a monomorphism on cohomology. It is clear that there are equivalences W_{X} ≅ Map_{X}^{+}(Y, Y) ≅ Map_{X}^{+}(E, Y); in effect, a map Y → Y is a homotopy equivalence iff it induces a monomorphism on cohomology. A map h : E → Y induces a monomorphism on cohomology iff the composite of h with the map BV → E induces a monomorphism H^{∗}Y → H^{∗}BV. It follows that Map_{X}^{+}(E, Y) is equivalent to the space of sections of a fibration q̇_{X}^{+} : Map_{X}^{+}(E/B, Y) → B with fibre Map_{X}^{+}(BV, Y).

The space Map_{X}^{+}(BV, Y) is the fibre of the map Map^{+}(BV, Y) → Map^{+}(BV, X) given by composition with g over the point represented by the composite h : BV ⟶ E ⟶ X. Thus by 4.2 the space Map_{X}^{+}(BV, Y) is homotopically discrete and the component set π_{0} Map_{X}^{+}(BV, Y) is isomorphic in a natural way to the set of homotopy classes of maps a : BV → Y such that a^{∗} is a monomorphism and g · a ∼ h. This last set is isomorphic (3.1) to the set of monomorphisms a′ : H^{∗}Y → H^{∗}BV such that a′·g^{∗} = h^{∗}. It now follows easily that the group W_{R}, considered (4.3) as a group of automorphisms of H^{∗}BV or equivalently as a group of homotopy classes of self-equivalences of BV, acts simply transitively on π_{0} Map_{X}^{+}(BV, Y).

The fibration q̇_{X}^{+} is necessarily fibre-homotopically trivial, since it is a fibration with a homotopically discrete fibre over a simply connected base. The basepoint evaluation map Γ(q̇_{X}^{+}) → Map_{X}^{+}(BV, Y) is thus an equivalence. But Γ(q̇_{X}^{+}) ∼ Map_{X}^{+}(E, Y), so, by naturality, the restriction map Map_{X}^{+}(E, Y) → Map_{X}^{+}(BV, Y) is an equivalence. This shows that Map_{X}^{+}(E, Y) ≅ W_{X} is homotopically discrete; the desired statement about W_{X}(≅ π_{0}W_{X}) follows in a straightforward way from the above identification of π_{0} Map_{X}^{+}(BV, Y).

Proof of 2.10: Let R ⊂ H^{∗}B̂T^{r} be the image of H^{∗}X. By 2.2 the natural restriction map gives an isomorphism H^{∗}X ≅ (H^{∗}B̂T^{r})^{W_{R}}, while by a twisted coefficient Serre spectral sequence argument the corresponding restriction map gives an isomorphism H^{∗}B(W_{X}, B̂T^{r}) ≅ H^{∗}(B̂T^{r})^{W_{X}}. Since W_{R} is naturally isomorphic to W_{X}, it follows immediately that the map B(W_{X}, B̂T^{r}) → X induces an isomorphism on cohomology and thus that the completed map τ is an equivalence.

The spaces B(W_{X}, B̂T^{r}) and B(W_{X}, B̂T^{r}) are each two-stage Postnikov systems with fundamental group W_{X} and second homotopy group (Z_{p})^{r}; by construction both spaces have the same action of the fundamental group on the second homotopy group. The difference between the two spaces can therefore be measured by a k-invariant in H^{3}(BW_{X}, (Z_{p})^{r}); since this group vanishes (p |W_{X}|) the two spaces (and hence their completions) must be equivalent.

The following lemmas will be used in proving 2.11.

4.4 Lemma. Let W be a finite group and f : A → B a map of modules over the group ring Z_{p}[W]. Suppose that A and B are finitely generated free modules over Z_{p}, that W acts trivially on A, and that f induces an isomorphism A/pA ≅ (B/pB)^{W}. Then f induces isomorphisms A ≅ B^{W} and Q_{p} ⊗_{Z_{p}} A ≅ (Q_{p} ⊗_{Z_{p}} B)^{W}.

Proof: Let C denote B^{W}. Pick x ∈ B and suppose that px ∈ C. For each w ∈ W the element px − wpx = p(x − wx) vanishes; since B is p-torsion free, this implies that x = wx and thus that x ∈ C. Consequently, the maps C/pC → B/pB and C/pC → (B/pB)^{W} are monomorphisms and hence (since the composite A/pA → C/pC → (B/pB)^{W} is an isomorphism) the map A/pA → C/pC is an isomorphism. The fact that A → C is an isomorphism now follows from the fact that A and C are both finitely generated free modules over Z_{p}. For the second isomorphism, note that since Q_{p} is flat as a module over Z_{p} (i.e. Q_{p} ⊗_{Z_{p}} - is an exact functor) and the fixed point functor (-)^{W} can be expressed naturally as a kernel, the natural map Q_{p} ⊗_{Z_{p}} (B^{W}) → (Q_{p} ⊗_{Z_{p}} B)^{W} is an isomorphism.

4.5 Lemma. If H^{∗}X ∈ K_{poly} then the ring Q_{p} ⊗_{Z_{p}} H^{∗}(X, Z_{p}) is a polynomial algebra over Q_{p} on a finite number of generators.

Proof: We can assume that X is p-complete and in particular simplyconnected. The Eilenberg-Moore spectral sequence (or Serre spectral sequence) shows that H^{∗}ΩX is finite and hence that H^{∗}(ΩX, Z_{p}) is finitely generated over Z_{p}. This implies that Q_{p} ⊗_{Z_{p}} H^{∗}(ΩX, Z_{p}) is a finite connected commutative Hopf algebra over Q_{p} and thus isomorphic as an algebra to an exterior algebra over Q_{p} (see [21, Appendix and 5.15]). (Another way to obtain this last conclusion is to use obstruction theory and the fact that the k-invariants of ΩX are torsion to produce a map

ΩX → ∏_{i} K(π_{i}ΩX/torsion, i)

which induces an isomorphism on Q_{p} ⊗_{Z_{p}} H^{∗}(–, Z_{p}). The fact that H^{∗}(ΩX, Z_{p}) is finitely generated implies that except for a finite number of odd integers i the spaces in the above product are contractible.) By the Samelson-Leray theorem [21, 7.20], the dual of the Hopf algebra Q_{p} ⊗_{Z_{p}} H^{∗}(ΩX, Z_{p}) is also an exterior algebra over Q_{p}. The spectral sequence argument of Borel [3, 13.1] (or an easy argument using the bar construction spectral sequence) now shows that Q_{p} ⊗_{Z_{p}} H^{∗}(X, Z_{p}) is the desired polynomial algebra.

Proof of 2.11: If Y is a space such that H^{∗}Y is of finite type and vanishes in odd dimensions, then for each i the group H^{i}(Y, Z_{p}) is a finitely generated free module over Z_{p} and there is a natural isomorphism H^{i}Y ≅ F_{p}⊗_{Z_{p}} H^{i}(Y, Z_{p}). This remark applies to the spaces X and B̂T^{r}, where t : B̂T^{r} → X is the maximal torus for X. The three isomorphisms listed in the theorem follow from 2.2 (cf. proof of 2.10) and 4.4. By 4.5, then, the ring of invariants (Q_{p} ⊗_{Z_{p}} H^{∗}(B̂T^{r}, Z_{p}))^{W_{X}} is a polynomial algebra. Under the monomorphism (2.9) W_{X} → GL(r, Z_{p}) → GL(r, Q_{p}) the action of W_{X} on Q_{p} ⊗_{Z_{p}} H^{∗}(B̂T^{r}, Z_{p}) can be identified with the natural induced action of W_{X} on the symmetric algebra of the dual of (Q_{p})^{r}, so the desired result is a consequence of 5.1.

Proof of 2.12: It is clear (cf. proof of 2.10) that the natural map H^{∗}X → (H^{∗}B̂T^{r})^{W} is an isomorphism, so it follows immediately from 2.4 that H^{∗}X ∈ K_{ftd}^{r}, that H^{∗}X is noetherian and integrally closed, and that H^{∗}X ∈ K_{poly} and ν_{X} = |W| if W is generated by pseudoreflections. Let f : B̂T^{r} → X be the maximal r-torus for X obtained by replacing the evident map B̂T^{r} → X with a Serre fibration, and let R ⊂ H^{∗}B̂T^{r} denote f^{∗}(H^{∗}X). By construction W ⊂ W_{X}(f), by 2.4 the natural map W → W_{R} is an isomorphism, and by 2.9 the natural map W_{X}(f) → W_{R} is an isomorphism. It follows that W = W_{X}(f). The desired conjugacy statement is a consequence of 2.5.

§5. Pseudoreflection groups

In this section we will prove Theorem 1.5.

If D is a commutative domain, a subgroup of GL(n, D) is said to be a pseudoreflection (p. r.) subgroup if it is generated by pseudoreflections (see §1). The main property of p. r. subgroups that makes them interesting for our purposes is the following one.

5.1 Theorem. [10] [7] [4, V, Sect 6, Exer. 8(a)] Suppose that F is a field and that W is a finite subgroup of GL(n, F). Let W act on the polynomial algebra R = F [t_{1}, …, t_{n}] in the natural way. If the ring of invariants R^{W} is a polynomial algebra, then W is a p. r. subgroup. If the characteristic of F is zero or prime to |W |, then the converse holds as well.

5.2 Lemma. If W ⊂ GL(n, F_{p}) is a subgroup of order prime to p, then W lifts to a subgroup W̄ ⊂ GL(n, Z_{p}) and the conjugacy class of W̄ depends only on that of W. If W̄ ⊂ GL(n, Z_{p}) is a subgroup of finite order prime to p and W̄ reduces mod p to a p. r. subgroup, then W̄ is a p. r. subgroup.

5.3 Remark: It is possible to have a finite p. r. subgroup W of GL(n, F_{p}) with the following paradoxical combination of properties:

(1) F_{p}[t_{1}, …, t_{n}]^{W} is a polynomial ring,

(2) W lifts to a subgroup W̄ of GL(n, Z_{p}), but

(3) W does not lift to a p. r. subgroup of GL(n, Z_{p}). Of course, by 5.2 the prime p must divide the order of W, and by 5.1 the ring of invariants (Z_{p}[t_{1}, …, t_{n}])W̄ must fail to be a polynomial ring. For example, let p be 3, let W ⊂ GL(2, F_{p}) be the cyclic group of strictly upper-triangular matrices, and let W̄ ⊂ GL(2, Z_{p}) the group generated

by the matrix ( \begin{matrix}1 & 1 \\ −3 & −2 \end{matrix} ). It is easy to check that (F_{p}[t_{1}, t_{2}])^{W} is the

polynomial algebra on t_{1} and t_{2}^{3} − t_{1}^{2}t_{2}. No generator of W can lift to a pseudoreflection, since Q_{p} does not contain a cube root of unity. Proof of 5.2: The fact that a group W of the indicated type lifts to a subgroup of GL(n, Z_{p}) in a way which is unique up to conjugacy is proven in [1, pp. 141–142]. To finish the proof we show in fact that if A ∈ GL(n, Z_{p}) is an element of finite order prime to p which reduces mod p to a pseudoreflection then A itself is a pseudoreflection. Pick such an A, let s be the order of A, and let N denote s^{−1}(∑ _{i=0}^{s−1}A^{i}). Choose linearly independent elements v_{1}, …, v_{n−1} in (F_{p})^{n} which are fixed by A, and let v̄_{i} ∈ (Z_{p})^{n} be a lift of v_{i}. Then for each i ≤ n − 1 the element Nv̄_{i} is fixed by A and reduces mod p to v_{i}. Since the v_{i} are linearly independent, it follows easily that the set {Nv̄_{1}, …, Nv̄_{n−1}} generates a rank (n − 1) submodule of (Z_{p})^{n}; by construction the elements of this submodule are in the kernel of (A − I_{n}) and so the existence of this submodule shows that A is a pseudoreflection.

If F is a field of characteristic zero and χ : W → GL(n, F) is a representation of the finite group W, then the character field K_{χ} is by definition the extension field of Q generated by the values of the character of χ (i.e. by the algebraic numbers trace(χ(w)), w ∈ W). If W is a finite subgroup of GL(n, F), we will let K_{W} denote the character field of the inclusion representation W ⊂ GL(n, F). Note that K_{W} ⊂ F. For any w ∈ W the algebraic number trace(w) is the sum of the eigenvalues of w and therefore a sum of roots of unity; this implies that K_{W} lies in a cyclotomic, hence abelian, extension of Q and thus that K_{W} itself is a Galois extension of Q.

5.4 Proposition. [8] [2] If F is a field of characteristic 0 and W ⊂ GL(n, F) is a p. r. subgroup, then W is conjugate in GL(n, F) to a subgroup of GL(n, K_{W}).

Remark: This proposition amounts to the statement that the Schur index of a reflection representation is one [2].

5.5 Proposition. If W ⊂ GL(n, C) is a p. r. subgroup and σ is a field automorphism of C, then the group σ(W) is conjugate as a subgroup of GL(n, C) to W.

Remark: If i : W → GL(n, C) is the inclusion of a p. r. subgroup it is not necessarily true in the situation of 5.5 that the representation σ · i of W is conjugate to the representation i; for instance, the dihedral group D_{10} of order 10 has two distinct p. r. representations over Q(\sqrt{5}) which are interchanged by the non-identity automorphism of Q(\sqrt{5}). According to 5.5, these two representations must also be interchanged by an outer automorphism of D_{10}. Proof of 5.5: We may assume that the action of W on C^{n} gives an irreducible representation of W. The action of σ(W) on C^{n} is then also irreducible. It is clear that W and σ(W) are irreducible p. r. groups of the same rank (i.e., both W and σ(W) lie in GL(n, C)) and the same order. Since the character field K_{W} is Galois and hence carried to itself by σ, K_{σ(W)} = K_{W}. Moreover, W is primitive [9] as a p. r. group iff σ(W) is primitive. The fact that W and σ(W) have all of these properties in common now implies, by inspection of the tables in [8], that W and σ(W) are conjugate.

5.6 Proposition. Let W be a finite group of order prime to p. Then the inclusion GL(n, Z_{p}) → GL(n, Q_{p}) induces a bijection from conjugacy classes of homomorphisms W → GL(n, Z_{p}) to conjugacy classes of homomorphisms W → GL(n, Q_{p}).

Proof: The argument in [11, proof of 73.5] shows that every homomorphism W → GL(n, Q_{p}) is derived up to conjugacy from a homomorphism W → GL(n, Z_{p}). The proposition now follows from [11, 76.17] 5.7 Proposition. Let W be a finite group, F a field of characteristic 0, M and N finitely generated F [W] modules, and L a field containing F. Then L ⊗_{F} M is isomorphic to L ⊗_{F} N as a module over L[W] iff M is isomorphic to N as a module over F [W].

Proof: A finite-dimensional representation of a finite group over a field of characteristic 0 is determined up to isomorphism by the character which it affords [11, 30.14]. This character is not changed by extension of scalars. Proof of 1.5: The bijection (1)↔(2) is obtained from 2.4. The bijection (2)↔(3) is obtained from 5.2.

It remains to give the bijection (3)↔(4). Suppose that W is a p. r. subgroup of GL(n, Z_{p}). The character field K_{W} ⊂ Q_{p} is a finite extension of Q, and by 5.4 the group W is conjugate in GL(n, Q_{p}) to a subgroup W′ of GL(n, K_{W}). By 5.7 the subgroup W′ is well-defined up to conjugacy in GL(n, K_{W}). Pick an embedding j : K_{W} → C and observe that j(W′) ⊂ GL(n, C) is a p. r. subgroup. Since K_{W} is a Galois extension of Q any two embeddings K_{W} → C differ by an automorphism of C; in this way 5.5 shows that the conjugacy class of the subgroup j(W′) ⊂ GL(n, C) does not depend on the choice of j. This gives the desired function from the set of conjugacy classes of p. r. subgroups of GL(n, Q_{p}) to the set of conjugacy classes of p. r. subgroups of GL(n, C) with character field embeddable in Q_{p}.

To construct an inverse to this function, reverse the procedure. Let W ⊂ GL(n, C) be a p. r. group with character field K_{W}, and assume that K_{W} can be embedded in Q_{p}. By 5.4 the group W is conjugate to a subgroup W′ of GL(n, K_{W}); by 5.7 there is only one such W′ up to conjugacy. Choose an embedding k : K_{W} → Q_{p}, and observe that k(W′) is a p. r. subgroup of GL(n, Q_{p}). Since K_{W} is a Galois extension of Q any two such embeddings differ by a field automorphism of K_{W} and thus by 5.5 the conjugacy class of the subgroup k(W′) ⊂ GL(n, Q_{p}) does not depend on the choice of k. Finally, by 5.6 there is up to conjugacy a unique subgroup W′′ ⊂ GL(n, Z_{p}) such that W′′ is conjugate in GL(n, Q_{p}) to k(W′).

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University of Notre Dame, Notre Dame, Indiana 46556 Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Purdue University, West Lafayette, Indiana 47907

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