TOPOLOGICAL REALIZATION OF A FAMILY

OF PSEUDO REFLECTION GROUPS

by Dietrich Notbohm

Abstract. We are interested in a topological realization of a family of pseudoreflection groups G ⊂ GL(n, 𝔽_{p}); i.e. we are looking for topological spaces whose mod–p cohomology is isomorphic to the ring of invariants 𝔽_{p}[x_{1}, ..., x_{n}]^{G}. Spaces of this type give partial answers to a problem of Steenrod, namely which polynomial algebras over 𝔽_{p} might appear as the mod-p cohomology of a space. The family under consideration is given by pseudoreflection groups which are subgroups of the wreath product ℤ/q ≀ Σ_{n} where q divides p − 1 and where p is odd. Let G be such a subgroup acting on the polynomial algebra A := 𝔽_{p}[x_{1}, ..., x_{n}]. We show, that there exists a space X such that H^{∗}(X; 𝔽_{p}) ≅ A^{G} which is again a polynomial algebra. Examples of polynomial algebras of this form are given by the mod–p cohomology of the classifying spaces of special orthogonal groups. or of symplectic groups.

The construction uses products of classifying spaces of unitary groups as building blocks which are glued together via information encoded in a full subcategory of the orbit category of the group G. Using this construction we also show that the homotopy type of the p–adic completion of these spaces is completely determined by the mod–p cohomology considered as algebra over the Steenrod algebra. Moreover we calculate the set of homotopy classes of self maps of the completed spaces.

1. Introduction.

In 1970, Steenrod [33] posed the question of which polynomial algebras over the field 𝔽_{p} of p elements can occur as the mod-p cohomology of a topological space. Later work of Adams and Wilkerson [2] and Dwyer, Miller, and Wilkerson [10] show that, at least at odd primes, such a polynomial algebra is always the ring of invariants of a pseudoreflection group acting on a polynomial algebra with generators in degree 2. More precisely, for every space X for which H^{∗}(X; 𝔽_{p}) is a polynomial algebra on n generators, there exists a pseudoreflection group W ⊆ GL(n, 𝔽_{p}) such that H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(BT^{n}; 𝔽_{p})^{W}, where T^{n} denotes the n–dimensional torus. Actually, this isomorphism is an isomorphism of algebras over the Steenrod algebra. Here, a pseudoreflection group means a finite subgroup W ⊆ GL(n; 𝔽_{p}) which is generated by pseudoreflections, i.e. by elements of finite order fixing a hyperplane of codimension one. Moreover, Dwyer, Miller, and Wilkerson also showed (still for odd p) that any such W lifts to a subgroup of GL(n, ℤ̂_{p}), that this lift depends only on the geometric realization of the 𝔽_{p}–algebra, and that W as a subgroup of GL(n, ℚ̂_{p}) is again a pseudoreflection group [10].

1991 Mathematics Subject Classification. 55 P 15, 55 R 35. Key words and phrases. pseudoreflection group, p–compact group, classifying spaces, compact

Lie group, polynomial algebra.

Typeset by AMS-TEX

Conversely, we can ask for a realization of a given pseudoreflection group W ⟶ GL(n, ℤ̂_{p}); e.g. for a p–complete space X with polynomial mod–p cohomology such that H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(BT^{n}; 𝔽_{p})^{W} (as algebras over the Steenrod algebra) and such that the lift to GL(n, ℤ̂_{p}), determined by X, is given by the representation W ⟶ GL(n, 𝔽_{p}) we started with. Actually, for odd primes the work of Dwyer, Miller and Wilkerson shows that such a space X even satisfies some stronger conditions. These conditions we will use for a definition.

Definition 1:1. A realization of a pseudoreflection group W ⟶ GL(n, ℤ̂_{p}) consists of a p–complete space X with polynomial mod–p cohomology and a map f : BT_{p}^{∧} ⟶ X (T := T^{n}) such that f is equivariant up to homotopy with respect to the induced W –action on BT_{p}^{∧} and the trivial action on X such that f induces an isomorphism H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(BT; 𝔽_{p})^{W}. Here, GL(n, ℤ̂_{p}) acts on BT_{p}^{∧} ≃ K((ℤ̂_{p})^{n}, 2) in the obvious way. A realization of a p–adic rational pseudoreflection group is given by a realization of a p–adic integral lift.

The classifying space BG of a compact connected Lie group G has polynomial mod–p cohomology if the integral cohomology of G is p–torsion free, which is true for almost all primes. In this case, the completion of BG together with the map BT_{G}_{p}^{∧} ⟶ BG_{p}^{∧} (T_{G} ⊂ G a maximal torus) realizes the Weyl group W_{G} when regarded as a subgroup of GL(n, ℤ̂_{p}) (n = rk(G)). For a slightly more general definition of the realization of a pseudoreflection group (see Remark 4.10).

If X is a realization of the pseudoreflection group W ⟶ GL(n, ℤ̂_{p}), one can easily show that H^{∗}(X; ℤ̂_{p}) ≅ H^{∗}(BT_{p}^{∧}; ℤ̂_{p})^{W} and that H^{∗}(X; ℤ̂_{p})⊗ℚ ≅ (H^{∗}(BT_{p}^{∧}; ℤ̂_{p})⊗ℚ)^{W} (see Proposition 4.9).

A complete classification of all irreducible pseudoreflection groups over ℚ̂_{p} has been made by Clark and Ewing [8], based on the classification by Shepard and Todd of irreducible pseudoreflection groups over ℂ [32]. Most of the groups on this list have already been shown to be realizable by spaces with polynomial cohomology (see [8], [36], [3], [12]); i.e. there exists a p–adic integral lift which has a realization in the above sense. In fact, there is only one case on the Clark-Ewing list for which the problem is not completely solved. Partial results for this case are achieved by Quillen [31, §10] and by Xu [35], who constructed the spaces in the cases where p^{2} does not divide the order of the group. Our results here fill this last gap, in that we realize all of the groups in this class. In addition, we show that the realization is unique up to homotopy (in a sense made more precise below), and describe the monoid of self maps of the spaces.

We now set up the notation needed to describe our results explicitly. For any q > 1, let μ_{q} ⊆ ℂ be the group of q-th roots of unity. We fix an identification μ_{q} ≅ ℤ/q.

Definition 1:2. For any q > 1, any r|q, and any n > 1, we set

A(q, r; n) = {(z_{1}, …, z_{n}) ∈ (μ_{q})^{n} | z_{1}⋯z_{n} ∈ μ_{r}}

and define

G(q, r; n) ⊆ U(n)

to be the subgroup generated by A(q, r; n) (regarded as a group of diagonal matrices) and the group Σ_{n} of permutation matrices.

Thus G(q, r; n) is a semidirect product of A(q, r; n) and Σ_{n}, and |G(q, r; n)| = q^{n−1}r·n!. Also, G(q, q; n) ≅ ℤ/q ≀ Σ_{n} (the wreath product).

TOPOLOGICAL REALIZATION OF A FAMILY OF PSEUDO REFLECTION GROUPS 3

Remark 1:3. Clearly, G(q, r; n) can be considered as a subgroup of GL(n, R) for any commutative ring R which contains the group of q-th roots of unity. Note that G(q, r; n) is always a pseudoreflection group in this situation: it is generated by the pseudoreflections

σ ( \begin{matrix}u & 0 & ⋯ & 0 \\ 0 & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & 1 \end{matrix} ) σ^{−1}  and  σ ( \begin{matrix}0 & v & 0 & ⋯ & 0 \\ v^{−1} & 0 & 0 & ⋯ & 0 \\ 0 & 0 & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 \end{matrix} ) σ^{−1},

for all u ∈ μ_{r}, all v ∈ μ_{q}, and all σ ∈ Σ_{n} (regarded as a permutation matrix). In particular, this applies when R = 𝔽_{p}, ℤ̂_{p}, or ℚ̂_{p}, for any prime p ≡ 1 (mod q). In this situation (and when p is understood), we denote by V (q, r; n) this associated representation on (ℤ̂_{p})^{n}.

If G(q, r; n) is any of the groups G(2, 1; n), G(2, 2; n), G(3, 1; 2), G(4, 1; 2), or G(6, 1; 2), then it is in fact conjugate to a subgroup of GL(n, ℤ). Actually, they are the only pseudoreflection groups of the form G(q, r; n) which can be realized over ℚ̂_{2} [8]. Moreover, in these cases, G(q, r; n) is the Weyl group of the compact connected Lie group SO(2n), SO(2n + 1), SU(3), Sp(2), or G_{2}, respectively, with representation given by its action on the integral lattice of the maximal torus. Hence, these groups all have topological realizations. For p = 2 and G(2, 1; n), G(2, 2; n) and G(6, 1; 2) this is not true in the sense of Definition 1.1 but in the extended sense of Remark 4.10.

We therefore can focus our attention in this paper on the groups G = G(q, r; n) realized at odd primes p such that q|(p−1). These are exactly the pseudoreflection groups described in case no. 2a in the list of Clark and Ewing [8].

For convenience, throughout the rest of this section, we fix r|q > 1, n ≥ 2, and let p be an odd prime such that q|(p−1). We first consider the ring of invariants for G(q, r; n) acting on the appropriate polynomial algebras.

Proposition 1:4. Set R = 𝔽_{p}, ℤ̂_{p}, or ℚ̂_{p}, and fix an identification of μ_{q} with the group of q-th roots of unity in R. Then

R[x_{1}, …, x_{n}]^{G(q,r;n)} ≅ R[y_{1}, …, y_{n−1}, e],

where for each i y_{i} = σ_{i}(x_{1}^{q}, …, x_{n}^{q}) (the i-th elementary symmetric polynomial), and where e = (x_{1}⋯x_{n})^{r}. In particular, the ring of invariants is a polynomial algebra.

Proof. Set G = G(q, r; n) and A = A(q, r; n), for short. Any element α ∈ R[x_{1}, …, x_{n}]^{A} can be written uniquely in the form α = a_{0}+a_{1}e+…+a_{k−1}e^{k−1}, where k = q/r and a_{i} ∈ R[(x_{1})^{q}, …, (x_{n})^{q}] for each i. And α is G-invariant if and only if each a_{i} lies in

R[(x_{1})^{q}, …, (x_{n})^{q}]^{Σ_{n}} ≅ R[y_{1}, …, y_{n−1}, y_{n}]

(cf. [22]), where each y_{i} is the i-th elementary symmetric polynomial on the elements (x_{1})^{q}, …, (x_{n})^{q} (and y_{n} = e^{q/r}). □

The polynomial ring 𝔽_{p}[x_{1}, ..., x_{n}] can be identified with the mod–p cohomology H^{∗}(BT, 𝔽_{p}) of the classifying space BT of a n-dimensional torus T. Thus the last proposition shows that H^{∗}(BT; 𝔽_{p})^{G} is a polynomial algebra. We can state our main results: the existence and homotopy uniqueness of spaces which realize the pseudoreflection groups G(q, r; n).

Theorem 1:5. Fix q > 1, r|q, and n > 1, and let p be an odd prime such that q|(p−1). Then G = G(q, r; n) ⊂ GL(n, ℤ̂_{p}) can be realized by a p–complete space BX = BX(q, r; n).

The space BX(q, r; n) satisfies a strong homotopy uniqueness property as the next theorem says.

Theorem 1:6. Fix q > 1, r|q, and n > 1, and let p be an odd prime such that q|(p−1). Let BX = BX(q, r; n) be the space constructed in Theorem 1:5. Then for any p–complete space Y such that H^{∗}(BX; 𝔽_{p}) ≅ H^{∗}(Y; 𝔽_{p}) as algebras over the Steenrod algebra, the spaces Y and BX are homotopy equivalent.

It turns out that the space BX(q, r; n) behaves in many ways like the classifying space of a compact Lie group with Weyl group G(q, r; n) acting on a “p-complete n-torus” via the representation V (q, r; n) (Proposition 4:4). That is the reason why we switched to the notation BX (instead of X). We think think of BX as the classifying space of the loop space X := ΩBX, which is a p–compact group (a p–complete generalization of a compact Lie group) in the sense of Dwyer and Wilkerson. This interpretation is not necessary for the construction of the spaces BX, but is very helpful in the proof of the homotopy uniqueness property (Theorem 1:6), and in the description of the monoid of self maps [BX, BX].

The description of [BX, BX] is given in Theorem 7:2 below, in terms of the normalizer of G in GL(n, ℚ̂_{p}). In particular, we prove that the homotopy classes of self maps of BX are detected by its rational p–adic cohomology:

Theorem 1:7. Let p be an odd prime, q > 2 and r|q|p − 1; and set BX = BX(q, r; n). Then the obvious map

[BX, BX] ⟶ Hom(H^{∗}(BX; ℤ̂_{p})⊗ℚ , H^{∗}(BX; ℤ̂_{p})⊗ℚ)

is a monomorphism.

For q = 2, the group G(q, r; n) is the Weyl group of SO(2n) or SO(2n + 1), and Theorem 1:6 shows that the space BX(q, r; n) is homotopy equivalent to BSO(2n)_{p}^{∧} or BSO(2n + 1)_{p}^{∧}. For these spaces, the monoid of self maps has already been calculated in [20]. In that paper, Theorem 1:7 is also shown for q = 2.

Theorem 1:5 also gives new decompositions of the classifying spaces of SO(n), Sp(n), SU(3) and G_{2} at odd primes; or alternatively, reconstructs the classifying spaces of these groups at odd primes. For these compact connected Lie groups, Theorem 1:6 has already been proven with different methods in [10] (for p coprime to the order of G) and in [29] (for all odd primes).

We sketch here the construction of BX(q, r; n). Let C be the full subcategory of the orbit category of G = G(q, r; n) with objects G/H for all subgroups H conjugate to a product of symmetric groups contained in Σ_{n} (considered as a subgroup of G). By sending each orbit G/H of this form to the ring of invariants R[x_{1}, ..., x_{n}]^{H}, we get a functor from C into the category of algebras over the Steenrod algebra.

This is then realized as the cohomology of a functor on C, first to the homotopy category, and then to the category of spaces: a functor which sends G/H to an appropriate product of classifying spaces of unitary groups. Finally, BX is defined to be the homotopy direct limit of this topological functor. The mod–p cohomology of BX is calculated using the Bousfield–Kan spectral sequence, and shown to be isomorphic to the ring of invariants R[x_{1}, …, x_{n}]^{G}. This particular construction is also used in the proof of uniqueness (Theorem 1:6), and when determining the monoid [BX, BX].

The paper is organized as follows. In Section 2, we prove a vanishing result for higher derived functors of inverse limits in certain very specialized situations. This is then used in Section 3, when constructing the spaces BX. The definitions and basic results about the p–compact groups of Dwyer and Wilkerson [11] are recalled in Section 4, where we also study particular maps between p–compact groups, and show that the space BX constructed in Section 3 are p-compact groups. These results, in this generality, are necessary for the proof of the homotopy uniqueness of the space BX (Theorem 1:6) in Section 5, and for the calculation of the monoid [BX, BX] in Section 7. The calculation of the Weyl group of G ⊆ GL(n, ℚ̂_{p}), needed to determine [BX, BX], is done in Section 6.

This paper appears only under my name, But actually, this is joint work with R.Oliver. He found this very simple construction of the spaces realizing the pseudoreflection groups in question. My only contibution consists in the idea how one can use his construction for a proof of the homotopy uniqueness property and for the calculation of the set of homotopy classes of the self maps of these spaces. It proved impossible for us to agree on a way of presenting the results, and Oliver eventually suggested that I publish a version of the work under my own name. It is a pleasure for me to thank him here for all his contributions.

I also would like to thank the referee for some clarification in the statements and the proofs of the propositions 2.1 and 2.2 and the Centre de Recerca Matemàtica in Barcelona for their hospitality when this joint work was started.

2. Inverse limits of functors on subcategories.

Let C be a small category and F un(C, Ab) the category of (covariant) functors from C to Ab. Then there exist higher limits

\underleftarrow{lim}_{C}^{i} : F un(C, Ab) ⟶ Ab

defined as right derived functors of the inverse limit functor

\underleftarrow{lim}_{C} : F un(C, Ab) ⟶ Ab

(cf. [5; XI, 6] or [30; Lemma 2]).

For any pair of categories C ⊇ D and any object x in C, we let (x⟶D) denote the category of objects in D “under” x: the objects are the morphisms φ : x ⟶ y in C such that y is an object in D, and a morphism (y_{1}, φ_{1}) ⟶ (y_{2}, φ_{2}) is an element in Mor_{D}(y_{1}, y_{2}) which makes the obvious triangle commute in C. The category (D ⟶ x) of objects ”over” x is defined analogously.

Proposition 2:1. Let C be a small category, and let D ⊆ C be a full subcategory. Let F : C ⟶ Ab be a (covariant) functor such that for each object y∈ Ob(C)∖ Ob(D),

\underleftarrow{lim}^{∗}_{(y⟶D)}(F |(y⟶D)) = \{ \begin{matrix}F (y) & if ∗ = 0 \\ 0 & if ∗ > 0. \end{matrix}

Then

\underleftarrow{lim}_{C}^{∗}(F) ≅ \underleftarrow{lim}_{D}^{∗}(F).

Proof. : Let I : D ⟶ C be the inclusion functor. The right Kan extension

I^{∗} : F un(D, Ab) ⟶ F un(C, Ab)

is defined by I^{∗}(F)(y) = \underleftarrow{lim}_{y⟶D}(F |y ⟶ D) There exists a composition of functors

spectral sequence

\underleftarrow{lim}_{C}^{i}(R^{j}I^{∗}(F I)) ⟹ \underleftarrow{lim}_{D}^{i+j}(F I)

converging to the higher limits of F restricted to D (cf. [7; XVI, 3] or [16; Appendix II, 3.6]). Here, R^{j}I^{∗}(F I) ≅ \underleftarrow{lim}^{j}_{y⟶D}(F |(y ⟶ D)) denotes the right derived functor of

I^{∗} evaluated at F I. For y ∈ C ∖ D all higher right derived functors vanish and R^{0}I^{∗}(F)(y) = F (y) by assumption. For y ∈ D the identity id : y ⟶ y is an initial object of the over category y ⟶ D and the same formulas hold. Consequently we have \underleftarrow{lim}_{C}^{∗}(F) ≅ \underleftarrow{lim}_{D}^{∗}(F I) ≅ \underleftarrow{lim}_{D}^{∗}(F). □

Proposition 2:1 dealt with limits over a “terminal” subcategory. In contrast, Proposition 2:2 deals with limits over an “initial” subcategory.

Proposition 2:2. Let C be a small category, and let E ⊆ C be a full subcategory with the following property: For any y∈ Ob(C)∖ Ob(E) there exists R(y)∈ Ob(E) and φ_{y}∈ Mor_{C}(R(y), y), such that any morphism φ : x ⟶ y, where x∈ Ob(E), factors uniquely through φ_{y}. (In other words, the category (E⟶y) has a final object.) Then for any (covariant) functor F : C ⟶ Ab,

\underleftarrow{lim}_{C}^{∗}(F) ≅ \underleftarrow{lim}_{E}^{∗}(F).

Proof. If y ∈ E then the over category E ⟶ y has the identity id : y ⟶ y as final object. Therefore and by assumption all over categories are contractible. That is to say that the inclusion E ⟶ C is left cofinal in the ense of Bousfield and Kan [5; XI, 9.1]. The argument of the proof of [5; XI, 9.2] shows that, if you have a left cofinal functor I : E ⟶ C, the higher limits of F I are canonically isomorphic to the higher limits of F, which is the statement. □

Now, for any finite group G, we let O(G) denote the orbit category for G. Thus, Ob(O(G)) is the set of orbits G/H for all subgroups H ⊆ G, and Mor_{O(G)}(G/H, G/K) is the set of all G-maps between the orbits. If H is any set of subgroups of G, then O_{H}(G) will denote the full subcategory of O(G) whose objects are those orbits G/H for H∈H. Also, O_{p}(G) will denote the full subcategory whose objects are the orbits G/P for p-subgroups P.

Proposition 2:3. Fix a group G and a prime p. Let H be a family of subgroups of G, closed under conjugation, and with the property that each p-subgroup of G is contained in a unique minimal element of H (minimal with respect to inclusions). Then for any ℤ_{(p)}[G]-module M,

\underleftarrow{lim}^{∗}_{G/H∈O_{H}(G)} M^{H} ≅ \{ \begin{matrix}M^{G} & if ∗ = 0 \\ 0 & if ∗ > 0. \end{matrix}

Proof. Let P be the family of p-subgroups of G, and set H̄ = H∪P. Let

F_{M} : OH̄(G)^{op} ⟶ Ab  be defined by  F_{M}(G/H) = M^{H}.

We first check that the hypotheses of Propositions 2:1 and 2:2 hold, when applied to the categories C = (OH̄(G))^{op}, D = (O_{P}(G))^{op}, and E = (O_{H}(G))^{op}, and the functor F.

For each H∈H, we can identify the categories

(G/H ⟶ O_{P}(G)^{op}) = ((G/H) ⟵ O_{P}(G))^{op} ≅ O_{P}(H)^{op} :

by identifying a G-map G/P ⟶^{φ} G/H (for P ∈P) with the H-orbit φ^{−1}(eH). Also, for any H∈H,

\underleftarrow{lim}^{∗}_{O_{P} (H)}(F |O_{P}(H)) = \underleftarrow{lim}^{∗}_{H/P ∈O_{P} (H)} M^{P} ≅ \{ \begin{matrix}M^{H} = F (G/H) & if ∗ = 0 \\ 0 & if ∗ > 0. \end{matrix}

The hypotheses of Proposition 2:1 are thus satisfied; and so

\underleftarrow{lim}^{∗}_{OH̄(G)}(F) ≅ \underleftarrow{lim}^{∗}_{O_{P} (G)}(F) = \underleftarrow{lim}^{∗}_{G/P ∈O_{P} (G)}(M^{P}) ≅ \{ \begin{matrix}M^{G} & if ∗ = 0 \\ 0 & if ∗ > 0. \end{matrix}  (1)

Here, the last step follows from [18, Theorem 5.14].

Now, for each p-subgroup P ⊆ G, let H_{P} ∈H be the unique minimal subgroup in H which contains P. Let φ_{P} : G/P ⟶ G/H_{P} be the map φ_{P} (gP) = gH_{P}. Then for any H∈H, any φ : G/P ⟶ G/H factors uniquely through φ_{P}. Also, by hypothesis, no element of H is contained in any subgroup of P∖H. In other words, the hypotheses of Proposition 2:2 are satisfied for the pair of categories (O_{H}(G))^{op} ⊆ (OH̄(G))^{op}. Thus, we have

\underleftarrow{lim}^{∗}_{G/H∈O_{H}(G)} M^{H} = \underleftarrow{lim}^{∗}_{O_{H}(G)}(F) ≅ \underleftarrow{lim}^{∗}_{OH̄(G)}(F);

and the result now follows from (1). □

3. The construction of the spaces BX(q, r; n).

By a partition of a set S will be meant here a collection of nonempty subsets S_{1}, …, S_{k} ⊆ S such that S = ∐_{i=1}^{k} S_{i}. Let P(n) denote the set of partitions of {1, …, n}. For any π = {S_{1}, …, S_{k}}∈P(n), set

Σ_{π} = Σ(S_{1})×…×Σ(S_{k}) ⊆ Σ_{n} :

the product of the corresponding symmetric groups. For any prime p, let P_{p}(n) be the set of those partitions all of whose elements have p-power order.

For any r|q and any n, let H(q, r; n) denote the set of all subgroups of G = G(q, r; n) conjugate to subgroups of the form Σ_{π} for π∈P(n). And for any prime p such that (p, q) = 1, let H_{p}(q, r; n) ⊆ H(q, r; n) denote the set of those subgroups conjugate to Σ_{π} for some π∈P_{p}(n).

Lemma 3:1. If r|q and (p, q) = 1, then any p-subgroup of G = G(q, r; n) is contained in a unique minimal element of H_{p}(q, r; n). And for any ℤ_{(p)}[G]–module M we have

\underleftarrow{lim}^{∗}_{G/H∈O_{H}(G)} M^{H} ≅ \{ \begin{matrix}M^{G} & if ∗ = 0 \\ 0 & if ∗ > 0. \end{matrix}

Proof. Since (q, p) = 1, any p-subgroup is conjugate to a subgroup of Σ_{n}. So it suffices to consider a p-subgroup P ⊆ Σ_{n} ⊆ G(q, r; n). Regard P as a group of permutations of the set {1, 2, …, n}, let π_{P} ∈P_{p}(n) be the set of orbits of P, and set H_{P} = Σ_{π_{P}} ∈H_{p}(q, r; n). Note in particular that P ⊆ H_{P}. It remains to show that any subgroup in H_{p}(q, r; n) which contains P also contains H_{P}.

Fix any H∈H_{p}(q, r; n) such that H ⊇ P. Let pr : G(q, r; n) ⟶ Σ_{n} be the projection. Then, by construction, pr(H) ⊇ pr(H_{P}). Also (by definition of H_{p}(q, r; n)) there is some g∈G(q, r; n) such that gHg^{−1} ⊆ Σ_{n}, and we can clearly take g∈A(q, r; n). Then pr(gHg^{−1}) = pr(H) ⊇ pr(H_{P}) and both are contained in Σ_{n}, and it follows that gHg^{−1} ⊇ H_{P}. Also, gP g^{−1} ⊆ gHg^{−1} ⊆ Σ_{n}; and since P ⊆ Σ_{n} it follows that [g, P] ⊆ A(q, r; n)∩Σ_{n} = 1. Thus [g, P] = 1, and if we write g = (z_{1}, …, z_{n}), then this means that the z_{i} are constant on orbits of P. This in turn implies that [g, H_{P}] = 1, and hence that H_{P} = g^{−1}(H_{P})g ⊆ H. This proves the first part. The second part follows from Proposition 2:3. □

For any partition π = {S_{1}, …, S_{k}}∈P(n), let U(π) be the corresponding product of unitary groups:

U(π) = U(S_{1})×…×U(S_{k}) ⊆ U(n).

If π_{1}, π_{2}∈P(n) are two partitions, we write π_{1} ≤ π_{2} if each element of π_{1} is contained in an element of π_{2}. In particular, if π_{1} ≤ π_{2}, then Σ_{π_{1}} ⊆ Σ_{π_{2}} and U(π_{1}) ⊆ U(π_{2}).

Theorem 3:2. Fix any prime p, any r|q|(p−1), and any n > 1. Set G = G(q, r; n) and H = H_{p}(q, r; n), for short. Then there exists a functor

Ψ : O_{H}(G) ⟶ Top

with the following properties:

(a) For any partition π∈P_{p}(n),

Ψ(G/Σ_{π}) ≃ BU(π)_{p}^{∧}.

(b) The composite H^{∗}(−; ℤ̂_{p})∘Ψ is isomorphic to the fixed point functor

(G/H) ↦ ℤ̂_{p}[x_{1}, …, x_{n}]^{H}.

(c) If we set BX(q, r; n) := (\underrightarrow{hocolim}_{O_{H}(G)}(Ψ))_{p}^{∧}, then

H^{∗}(BX; 𝔽_{p}) ≅ 𝔽_{p}[x_{1}, …, x_{n}]^{G}.

For the proof we need the following result about maps between classifying spaces of unitary groups.

Proposition 3:3. Let G be a product of unitary groups. Let f : BG ⟶ BG be a homotopy equivalence. Let T ⊆ G be a maximal torus and Z ⊆ T ⊆ G the center of G. Then the following holds:

(a) Fix any abelian p-toral subgroup S ⊆ T and set H = C_{G}(S) ⊆ G. Then the maps

BZ(H)_{p}^{∧} ⟶^{β_{H}} map(BH, BG_{p}^{∧})_{incl} ⟶^{f∘−} map(BH, BG_{p}^{∧})_{f|BH},

and

β′_{H} : BH_{p}^{∧} ⟶ map(BS, BG_{p}^{∧})_{f|BS}

(where β_{H} and β′_{H} are adjoint to the maps induced by multiplication) are homotopy equivalences.

(b) Let S and H be as in (a). Then for any map g : BH ⟶ BG, g ≃ f|BH iff H^{∗}(g; ℤ̂_{p})⊗ℚ = H^{∗}(f|BH; ℤ̂_{p})⊗ℚ iff g|BT ≃ f|BT.

This theorem is a special situation of Proposition 4:6, where we prove a more general statement about p–compact groups. This also will show that the theorem holds for every compact connected Lie group G. In this more general case, for part (b), we only have to assume in adition that H is connected. If G is a product of unitary groups, this extra condition always holds because the centralizer of a subgroup of a unitary group is always a product of unitary groups and therefore connected.

Actually to prove Theorem 4.6 only for compact connected Lie groups the theory of p–compact groups is not necessary as Remark 4:7 explains.

Proof of Theorem 3:2. It will be convenient to consider the subset

H_{0} = {Σ_{π} | π∈P_{p}(n)} ⊆ H.

Then H_{0} is the set of elements of H contained in Σ_{n}, and every element of H is conjugate to an element of H_{0}. Hence O_{H_{0}}(G) contains objects from all isomorphism classes of O_{H}(G); and so limits over O_{H_{0}}(G) and over O_{H}(G) are the same.

Any morphism in O_{H_{0}}(G) is a composite

(G/Σ_{π_{1}}) ⟶^{σ·a} (G/Σ_{σ(π_{1})}) ⟶^{proj} (G/Σ_{π_{2}}),  (1)

where

σ∈Σ_{n}, a = (z_{1}, …, z_{n})∈A(q, r; n), [a, Σ_{π_{1}}] = 1, and σ(π_{1}) ≤ π_{2}.

Note that the condition [a, Σ_{π_{1}}] = 1 is equivalent to the condition that the z_{i} are constant on elements in π_{1}.

We first define a functor

Ψ′_{0} : O_{H_{0}}(G) ⟶ hTop

taking values in the homotopy category of Top which satisfies conditions (a) and (b). On objects, we set

Ψ′_{0}(G/Σ_{π}) = BU(π)_{p}^{∧}.

And a morphism of the type (1) above is sent to a composite of the form

BU(π_{1})_{p}^{∧} ⟶^{ψ_{π_{1}}^{α}} BU(π_{1})_{p}^{∧} ⟶^{σ_{∗}} BU(σ(π_{1}))_{p}^{∧} ⟶^{incl} BU(π_{2})_{p}^{∧}.

Here, ψ_{π_{1}}^{α} means the product of the corresponding unstable Adams operations on the unitary group factors (recall that the coordinates of α are constant on each element of the partition π_{1}). And σ_{∗} is induced by the homomorphism which sends each factor of U(π_{1}) to the corresponding factor of U(σ(π_{1})). When showing that these maps compose correctly (i.e., that Ψ′_{0} is a functor), the only difficult parts are to show that ψ^{r}∘ψ^{s} ≃ ψ^{rs} (this follows from the homotopy uniqueness of the unstable Adams operations, shown in [19,Theorem 4.3]); and that the unstable Adams operations commute with inclusions. This last point holds since (ψ_{π}^{α})^{−1}∘incl∘ψ_{π}^{α} is homotopic to the inclusion by Proposition 3:3(b). (Note, that for U(π_{1}) ⊆ U(π_{2}) there exists a subtorus S = Z(U(π_{1})) ⊆ U(π_{1}) such that U(π_{1}) = C_{U(π_{2})}(S).)

Thus, Ψ′_{0} is a well defined functor to the homotopy category, and can be extended to a functor

Ψ′ : O_{H}(G) ⟶ hTop.

Conditions (a) and (b) hold for Ψ′ by construction.

We now claim that Ψ′ is a centric functor in the sense of Dwyer & Kan [9]. That is that, for any morphism φ : G/H_{1} ⟶ G/H_{2} in O_{H}(G), the map

map(Ψ′(G/H_{1}), Ψ′(G/H_{1}))_{Id} ⟶^{Ψ′(φ)∘−} map(Ψ′(G/H_{1}), Ψ′(G/H_{2}))_{Ψ′(φ)}

is a homotopy equivalence. This is automatic when φ is an isomorphism, so it suffices to check it for inclusions of the form Σ_{π_{1}} ⊆ Σ_{π_{2}}. And follows from Proposition 3:3(a) (applied with G = Ψ′(G/H_{2}) and with and G = H = ψ′(G/H_{1})). Furthermore, for each H∈H,

map(Ψ′(G/H), Ψ′(G/H))_{Id} ≃ BZ(H)_{p}^{∧},

again by Lemma 3:3(b). Since Z(ψ′(G/H)) is a torus (ψ′(G/H) is a product of unitary groups), the only nonvanishing homotopy group in these mapping spaces is

π_{2}(map(Ψ′(G/H), Ψ′(G/H))_{Id}) ≅ π_{2}(BZ(ψ′(G/H)))_{p}^{∧} ≅ [(ℤ̂_{p})^{n}]^{H}.

So by Lemma 3:1,

\underleftarrow{lim}^{i}_{O_{H}(G)} π_{j}(map(Ψ′(−), Ψ′(−))_{Id}) ≅ 0

for all i, j > 0. The obstruction groups for lifting Ψ′ to a functor to topological spaces are given by some of these higher derived limits [9, Theorem 1.1]. Since all these groups vanish, we obtain a lifting

Ψ : O_{H}(G) ⟶ Top

which satisfies conditions (a) and (b) above.

Finally, by Lemma 3:1 again,

\underleftarrow{lim}^{i}_{O_{H}(G)}(H^{∗}(Ψ(−); 𝔽_{p})) ≅ \underleftarrow{lim}^{i}_{G/H∈O_{H}(G)}(𝔽_{p}[x_{1}, …, x_{n}]^{H})

≅ \{ \begin{matrix}𝔽_{p}[x_{1}, …, x_{n}]^{G} & if i = 0 \\ 0 & if i > 0. \end{matrix}

So by the spectral sequence for the cohomology of a homotopy colimit (cf. [5, XII.4.5] or [21]),

H^{∗}(\underrightarrow{hocolim}_{O_{H}(G)}(Ψ); 𝔽_{p}) ≅ 𝔽_{p}[x_{1}, …, x_{n}]^{G}.

The homotopy colimit is thus mod–p simply connected, and hence its p-completion BX(q, r; n) has the same mod–p cohomology. □

Proof of Theorem 1:5: Let T ⊂ U(n) be the maximal torus given by the diagonal. Making the identification H^{∗}(BT^{n}_{p}^{∧}; ℤ̂_{p}) ≅ ℤ̂_{p}[x_{1}, …, x_{n}] the composition BT_{p}^{∧} ⟶ BU(n)_{p}^{∧} ⟶ BX := BX(q, r; n) becomes equivariant with respect to the action of G(q, r; n) and realizes the isomorphism H^{∗}(BX; 𝔽_{p}) ≅ H^{∗}(BT; 𝔽_{p})^{G}. All this follows directly from the construction of the space BX. □

4. p–compact groups.

The concept of p–compact groups was introduced by Dwyer and Wilkerson in [13], where they showed that p–compact groups behave very much like compact Lie groups. In particular, a p-compact group always has a maximal torus and a Weyl group, which satisfy properties analogous to those of the maximal torus and Weyl group of a compact Lie group. We recall here the basic definitions and results from [13].

A p–compact group is a triple X = (X, BX, e), where BX is a p–complete pointed space, where X is mod–p finite (i.e., H^{∗}(X; 𝔽_{p}) is finite), and where e : ΩBX ⟶^{≃} X is a homotopy equivalence. The space BX is thought of as the classifying space of the p–compact group X.

The motivating examples of p-compact groups come from compact Lie groups. If G is any compact Lie group for which π_{0}(G) is a finite p–group, then the triple (G_{p}^{∧}, BG_{p}^{∧}, G_{p}^{∧} ≃ ΩBG_{p}^{∧}) is a p–compact group. Particular compact Lie groups establish particular p–compact groups. A p-compact torus is a triple of the form (T_{p}^{∧}, BT_{p}^{∧}, ≃) where T is a torus and a finite p–compact group is of the form (π, Bπ, ≃) where π is a finite p–group. And p–toral groups give p–compact toral groups which in general are p–compact groups whose component of the unit is a p–compact torus. (The component of the unit is given by the universal cover of BX or by the component of X related to the constant loop in BX.) Note that for any p–compact group (X, BX, e), π_{1}(BX) ≅ π_{0}(X) is a finite p-group (since BX is p-complete).

A homomorphism f : X ⟶ Y between to p–compact groups is a pointed map Bf : BX ⟶ BY. A homomorphism f is called an isomorphism if Bf is a homotopy equivalence; and a monomorphism if the homotopy fiber of Bf, denoted by Y/X, is mod–p finite. Two homomorphisms f, g : X ⟶ Y are conjugate if Bf and Bg are freely homotopic.

Let P be a p–toral or p–compact toral group. Dwyer and Wilkerson showed that for every homomorphism f : P ⟶ X of p–compact groups, the mapping space BC_{X}(f) := map(BP, BX)_{Bf} is the classifying space of another p–compact group; i.e. that BC_{X}(f) is p–complete and its loop space C_{X}(f) is mod–p finite. The triple C_{X}(f) = (C_{X}(f), BC_{X}(f), Id) is called the centralizer of f. Evaluation at the base point induces a map BC_{X}(f) ⟶ BX respectively a homomorphism C_{X}(f) ⟶ X which is always a monomorphism [13, 5.2, 6.1]. If P is an abelian compact Lie group, or if map(BP, BP)_{Id} ≃ BP, then there is an obvious map BP ⟶ BC_{X}(f) [13, 8.2]. The notion of the centralizer is motivated by:

Proposition 4:1. (Dwyer–Zabrodsky [15] & Notbohm [28]) For any homomorphism ρ : P ⟶ G from a p-toral group P into a compact connected Lie group G, let C_{G}(ρ) denote the centralizer of Im(ρ) (in the group theoretic sense). Then the map

BC_{G}(ρ)_{p}^{∧} ⟶^{≃} map(BP, BG_{p}^{∧})_{Bρ},

adjoint to B(ρ, incl) : BC_{G}(ρ) × BP ⟶ BG, is a homotopy equivalence.

Note that this form of the statement, which will be used later, is only contained implicitly in the two mentioned references. It is derived explicitly in [19; Theorem 3.2].

Now we recall the notion of maximal tori and Weyl groups. A maximal torus of a p–compact group X is a monomorphism f : T ⟶ X from a p–compact torus into X such that the induced homomorphism T ⟶ C_{X}(T) is an equivalence to the identity component of the centralizer. The rank of X is defined as the dimension of T.

Theorem 4:2. [13, 8.11, 8.13, 9.1] Every p–compact group X has a maximal torus T_{X} ⟶ X, and any two maximal tori of X are conjugate.

When defining the Weyl group W_{X} of a p-compact group X, we will assume here for simplicity that X is connected. For a fixed maximal torus f : T_{X} ⟶ X of X, W_{X} is defined to be the set of all homotopy classes of self maps of w : BT_{X} ⟶ BT_{X} such that Bf ∘w and Bf are homotopic. It turns out that W_{X} has many of the well known properties of Weyl groups of a compact connected Lie group. This definition is equivalent to the one given by Dwyer and Wilkerson in [13, 9.11] (see [27, 1.8]); their definition also applies to nonconected p–compact groups.

Proposition 4:3. [13, 9.5 & 9.7] Let T_{X} ⟶ X be a maximal torus of a connected p–compact group X of rank n. Then the following hold:

(1) W_{X} is a finite group.

(2) The action of W_{X} on BT_{X} induces a faithful representation

W_{X} ⟶ GL(n, ℚ̂_{p})

which represents W_{X} as a pseudoreflection group.

(3) The map T_{X} ⟶ X induces an isomorphism

H^{∗}(BX; ℤ̂_{p}) ⊗ ℚ ⟶ (H^{∗}(BT_{X}; ℤ̂_{p}) ⊗ ℚ)^{W_{X}} .

The next proposition describes the structure of the spaces BX constructed in Section 3, as p–compact groups.

Proposition 4:4. Fix r|q|(p − 1) and n ≥ 2 and set G = G(q, r; n). Let BX be any p–complete space such that H^{∗}(BX; 𝔽_{p}) ≅ 𝔽_{p}[x_{1}, ..., x_{n}]^{G}. Then the following holds:

(1) The triple X = (ΩBX, BX, Id) is a connected p–compact group.

(2) There exists a maximal torus T_{X} ⟶ X and an identification H^{∗}(BT_{X}; 𝔽_{p}) ≅ 𝔽_{p}[x_{1}, ⋯, x_{n}] such that the composition

𝔽_{p}[x_{1}, ⋯, x_{n}]^{G} ≅ H^{∗}(BX; 𝔽_{p}) ⟶ H^{∗}(BT_{X}; 𝔽_{p}) ≅ 𝔽_{p}[x_{1}, ⋯, x_{n}]

is the obvious iclusion.

(3) W_{X} ≅ G.

Proof. To prove the first part we only have to show that X = ΩBX is mod–p finite. This follows easily from an Eilenberg-Moore spectral sequence argument.

Using Lannes’ T –functor, one can show that there exists a realization BT_{X} ⟶ BX of the given algebraic map. The following construction is similar to that in [10] (see also [29, Section 7]). Set V ≅ (ℤ/p)^{n}. By [23, Théorème 0.4] the composite

H^{∗}(BX; 𝔽_{p}) ≅ 𝔽_{p}[x_{1}, …, x_{n}]^{G} ⟶ 𝔽_{p}[x_{1}, …, x_{n}] ⟶ H^{∗}(BV; 𝔽_{p})

can be realized by a map f_{0} : BV ⟶ BX. A calculation of the mod–p cohomology of map(BV, BX)_{f_{0}}, using Lannes’ T –functor, shows that this mapping space has the same mod–p cohomology as the classifying space BT_{X} of a p–compact torus T_{X} and hence, that they can be identified. The evaluation at the basepoint

BT_{X} = map(BV; BX)_{f_{0}} ⟶^{ev} BX

induces the inclusion 𝔽_{p}[x_{1}, …, x_{n}]^{G} ⊆ 𝔽_{p}[x_{1}, …, x_{n}] (in mod–p cohomology). Because H^{∗}(BT_{X}, 𝔽_{p}) is finitely generated over H^{∗}(BX; 𝔽_{p}), the map BT_{X} ⟶ BX is a monomorphism of p–compact groups [13, 9.11]. Moreover, the sequence of monomorphisms T_{X} ↪ C_{X}(T_{X}) ↪ C_{X}(V) ≅ T_{X} [13, 5.2, 6.1] shows that T_{X} ≅ C_{X}(T_{X}) and that T_{X} ⟶ X is a maximal torus. This proves the second part.

The group G acts basepoint preserving on the space BV and fixes the induced map H^{∗}(f_{0}; 𝔽_{p}). Hence, again by [23, Théorème 0.4], this action also fixes the component of map(BV, BX)_{f_{0}} ≃ BT_{X} of the mapping space map(BV, BX) and is therefore a subgroup of W_{X}.

Passing to invariants the inclusion of the maximal torus T_{X} ⟶ X establishes the composition

H^{∗}(BX; 𝔽_{p}) ⟶ H^{∗}(BT_{X}; 𝔽_{p})^{W_{X}} ⟶ H^{∗}(BT_{X}; 𝔽_{p})^{G} .

The second arrow is a monomorphism, the composition is an isomorphism and so are both arrows. The action of W_{X} on H^{∗}(BT_{X}; 𝔽_{p}) is faithful because this is true for ℤ_{p}^{∧} as coefficients (Proposition 4:3) and because for odd primes the kernel of GL(n, ℤ_{p}^{∧}) ⟶ GL(n, 𝔽_{p}) is torsion free.

The ring R := H^{∗}(BT_{X}; 𝔽_{p}) is an integrally closed integral domain. Let F (R) denote the field of fractions. Then we have F (R^{G}) ≅ F (R)^{G} and F (R)^{G} ⊂ F (R) is a Galois extension with Galois group G. The same formula are true for the Weyl group W_{X} and F (R)^{G} ≅ F (R)^{W_{X}}. Thus, both groups have the same order and G = W_{X}. This proves the third part. □

Now we can start to study particular maps between classifying spaces of p– compact groups. The next lemma describes a well known trick in the theory of classifying spaces.

Lemma 4:5. Fix a prime p and a p-compact group X. Let G be a compact Lie group, let P ◁ G be a normal p-toral subgroup, and let κ : G ⟶ G/P be the projection. Then for any f : B(G/P) ⟶ BX,

(−∘Bκ) : map(B(G/P), BX)_{f} ⟶^{≃} map(BG, BX)_{f∘Bκ}  (1)

is a homotopy equivalence.

Proof. The map BP ⟶ map(BP, BX)_{const} is an equivalence. This follows from the Sullivan conjecture and is stated in [14, 9.7, 10.1]. Using this fact the statement might be found in [4, 7.3]. □

We are now ready to state our main proposition needed to describe certain mapping spaces between p-compact groups. We have to distinguish between centralizers in the (Lie) group theoretic sense and in the sense od p–compact groups. In the latter case we will use the notation of mapping spaces and keep the notation as centralizers for homomorphisms between compact Lie groups.

Proposition 4:6. Let G be a compact connected Lie group with maximal torus T ⊆ G and with central subgroup Z ⊆ Z(G) ⊆ G. Let i : T ↪ G and j : Z ↪ G be the inclusions, and let mult : G×Z ⟶ G denote multiplication (a homomorphism since Z is central). Let X be a connected p–compact group, and let f : BG ⟶ BX be a map such that

f′ : BG_{p}^{∧} ⟶^{≃} map(BZ, BX)_{f∘Bj},  (1)

adjoint to f ∘Bmult, is a homotopy equivalence. Then the following holds: (a) The composite BT ⟶^{Bi} BG ⟶^{f} BX is a maximal torus of X.

(b) Fix any abelian p–toral subgroup S ⊆ T with Z ⊆ S and assume that H := C_{G}(S) ⊆ G is connected. Then the maps

BZ(H)_{p}^{∧} ⟶^{β_{H}} map(BH, BG_{p}^{∧})_{incl} ⟶^{f∘−} map(BH, BX)_{f|BH},

and

β′_{H} : BH_{p}^{∧} ⟶ map(BS, BX_{p}^{∧})_{f|BS}

are homotopy equivalences. Here, β_{H} and β′_{H} are adjoint to the map induced by

Bmult : BZ(H)×BH ⟶ BG and BH×BS ⟶^{Bmult} BG ⟶^{f} BX. multiplication)

(c) Let S and H be as in (b) and assume that H is connected. Then for any map g : BH ⟶ BX, g ≃ f|BH iff H^{∗}(g; ℤ̂_{p})⊗ℚ = H^{∗}(f|BH; ℤ̂_{p})⊗ℚ iff g|BT ≃ f|BT.

Remark. The corresponding fact of (a) for Lie greoups is this: Let X be a compact connected Lie group. Because G is connected, Z is contained in a maximal torus for G and therefore in a maximal torus for X; consequently G ⟶ X is of maximal rank.

If S ⊂ T is a subtorus, then H is always connected [19; Proposition A.4].

Proof. Throughout the proof, Ad : BH_{p}^{∧} ⟶ map(BK, BG_{p}^{∧}) and Ad(f) : BH_{p}^{∧} ⟶ map(BK, BX) will denote the adjoint maps to B(mult) and f ∘B(mult), for any commuting pair of subgroups H, K ⊆ G and the multiplication mult : H×K ⟶ G. And similarly for any other map f′ : BG′ ⟶ BX, for any G′ ⊆ G. In particular, β_{H} is of the form Ad and β′_{H} of the form Ad(f).

For any H ⊆ G containing Z, we have

map(BH, BG_{p}^{∧})_{incl} ≃ map(BH, map(BZ, BX)_{f∘Bj})_{Ad(f)}

≃ map(BZ×BH, BX)_{f∘Bmult} ≃ map(BH, BX)_{f|BH}. (2)

The first equivalence holds by (1), the second by taking adjoints, and the third by Lemma 4:5 using the fibration BZ ⟶ BZ×BH ⟶^{Bmult} BH. Thus, the second map (f ∘−) in (b) is a homotopy equivalence. This even holds if we only assume that π_{0}(G) is a finite p group.

Now we assume that H is the centralizer in G of a p–toral subgroup S ⊆ T. In particular, we have H ⊇ T and π_{0}(H) is a finite p group (cf. [19, Proposition A.4]). Then by Proposition 4:1,

Ad : BH_{p}^{∧} ⟶^{≃} map(BS, BG_{p}^{∧})_{incl}  (3)

is a homotopy equivalence. Hence by (2) (after replacing G by H and BX by BG_{p}^{∧}), the map

map(BH, BH_{p}^{∧})_{Id} ⟶_{≃}^{(incl)∘−} map(BH, BG_{p}^{∧})_{incl}  (4)

is a homotopy equivalence. Furthermore,

map(BH, BH_{p}^{∧})_{Id} ≃ BZ(H)_{p}^{∧}.  (5)

For connected H, this follows from [19, Theorem 4.2] when H is simple, and from [20, Proposition 2.7] in the nonsimple case. For nonconnected H, this is proved in [14], but can also be proved in the context of compact Lie groups starting from the connected case. Points (2), (3), (4), and (5) now combine to prove that β_{H} as well as the composition (f ∘ − −)β_{H} are equivalences. The second part of (b) follows by applying (2) in the case of H = S and by (3) if we assume in addition that Z ⊆ S. Otherwise there is no multiplication Z × S ⟶ S.

For H = T, point (b) takes the form of a homotopy equivalence

BT_{p}^{∧} ⟶^{≃} map(BT, BX)_{f|BT}

(recall that T = C_{G}(T) [6, Theorem 2.3]). This proves part (a), namely that BT ↪ BG ⟶^{f} BX is a maximal torus (see the above definitions).

It remains to prove part (c). Let us assume that H be connected and let g : BH ⟶ BX be another map such that H^{∗}(g; ℤ̂_{p})⊗ℚ = H^{∗}(f|BH; ℤ̂_{p})⊗ℚ. By [13, Proposition 8.11], g|BT and f|BT lift to maps g′, f′ : BT ⟶ BT_{p}^{∧}. Since H^{∗}(g′; ℤ̂_{p})⊗ℚ and H^{∗}(f′; ℤ̂_{p})⊗ℚ agree on H^{∗}(BX; ℤ̂_{p})⊗ℚ ≅ (H^{∗}(BT; ℤ̂_{p})⊗ℚ)^{W_{X}} (Proposition 4:3), there exists w ∈ W_{X} such that g′ and w∘f′ induce the same map on H^{∗}(BT, ℤ̂_{p}) ⊗ ℚ and that g′ ≃ w∘f′. The first conclusion follows from [28, 7.1] and the second is obvious. Hence, we can assume that g|BT ≃ f|BT. We consider the following diagram

BT ⟶ map(BS, BT)_{Bincl}  map(BS, BT)_{Bincl}
↓  ↓  ↓^{(g|BT ∘−)≃(f|BT ∘−)}
BH ⟶^{Ad} map(BS, BH_{p}^{∧})_{Bincl_{p}^{∧}} ⟶_{f∘−}^{g∘−} {map(BS, BX)_{g|BS} \atop = map(BS, BX)_{f|BS}} ⟶^{ev} BX,

where incl always denotes the obvious inclusion. The composition in the lower row is nothing but g or f. The compositions (g∘−)Ad and (f ∘−)Ad ≃ (id_{BH})_{p}^{∧} induce the same map after restriction to BT and are therefore homotopic by [20, 2.5]. And so are f and g. □

Proof of Theorem 3.3. Theorem 3.3 is a special situation of (b) and (c) of Theorem 4.6. We only have to assume that X = G. To satisfy the extra condition Z ⊂ S we notice that we can replace S by the group S′ ⊂ T generated by S and Z, since H = C_{G}(S) = C_{G}(S′). Moreover, we notice that, for G a product of unitary groups, H = C_{G}(S) is always connected. □

Remark 4:7. In the case of X being the completion of a compact connected Lie group, the proof does not depend on the theory of p–compact groups. In the proof of part (b) and (c) we only have to replace some references, namely Proposition 4.3 by the fact that H^{∗}(BG; ℚ) ≃ H^{∗}(BT_{G}; ℚ)^{W_{G}} for any compact connected Lie group and [13, Proposition 8.11] by [20, Proposition 1.2]. In particular, this shows that Proposition 3.3, which is contained in (b) and (c), can be proved in the context of compact Lie groups.

The following proposition describes the principal situation in which Proposition 4:6 will be applied.

Proposition 4:8. Fix r|q|(p − 1) and n ≥ 2, and set G = G(q, r; n). Let BX be a p-complete space, together with an isomorphism

Φ : H^{∗}(BX; 𝔽_{p}) ⟶^{≅} 𝔽_{p}[x_{1}, …, x_{n}]^{G}

of algebras over the Steenrod algebra. Then there exists a map g : BU(n) ⟶ BX such that the square

\begin{matrix}H^{∗}(BX; 𝔽_{p}) & ⟶^{Φ} & 𝔽_{p}[x_{1}, …, x_{n}]^{G} \\ ^{g^{∗}}↓ & & ^{incl}↓ \\ H^{∗}(BU(n); 𝔽_{p}) & ⟶_{≅}^{Φ_{U}} & 𝔽_{p}[x_{1}, …, x_{n}]^{Σ_{n}} \end{matrix}  (1)

commutes (where Φ_{U} is the standard isomorphism). Furthermore, the following holds for any such map g : BU(n) ⟶ BX:

(a) For any maximal torus T ⊆ U(n), g|BT : BT ⟶ BX is a maximal torus of X.

(b) Let S ⊆ U(n) be any subtorus with Z(U(n)) ⊆ S, and set H = C_{U(n)}(S). Then the maps

BH_{p}^{∧} ⟶ map(BS, BX)_{g|BS} and BZ(H)_{p}^{∧} ⟶ map(BH, BX)_{g|BH} (2)

(adjoint to g∘Bmult.) are homotopy equivalences.

(c) Let S and H be as in (b). If f : BH ⟶ BX is any map such that H^{∗}(f; ℤ̂_{p})⊗ℚ = H^{∗}(g|BH; ℤ̂_{p})⊗ℚ, then f ≃ g|BH.

Proof. Let φ : H^{∗}(BX; 𝔽_{p}) ⟶ H^{∗}(Bℤ/p; 𝔽_{p}) be the composite

φ : H^{∗}(BX; 𝔽_{p}) ⟶_{≅}^{Φ} 𝔽_{p}[x_{1}, …, x_{n}]^{G} ⟶^{Φ_{U}^{−1}∘incl} H^{∗}(BU(n); 𝔽_{p})

⟶^{incl^{∗}} H^{∗}(Bℤ/p; 𝔽_{p}),

where the last map is the restriction to the central subgroup of order p. By [23, Théorème 0.4], there is a map g_{0} : Bℤ/p ⟶ BX such that H^{∗}(g_{0}; 𝔽_{p}) = φ. Let BT_{X} ⟶ BX be the maximal torus of X as in Proposition 4:4. Then a lift of g_{0} to BT_{X} is given by the imbedding k : Bℤ/p ⟶ BT_{X}. The isotropy group of k, i.e. the group of all elements of W_{X} such that w∘k ≃ k, is given by Σ_{n} ⊆ G = W_{X}.

Since H^{∗}(BX; 𝔽_{p}) ≅ 𝔽_{p}[x_{1}, ..., x_{n}]^{G}, an application of [29, 10.1, 10.2] calculates the mod–p cohomology of the mapping space. We get

H^{∗}(map(Bℤ/p, BX)_{g_{0}}; 𝔽_{p}) ≅ H^{∗}(BT_{X}, 𝔽_{p})^{Σ_{n}} ≅ H^{∗}(BU(n); 𝔽_{p}).

The uniqueness theorems in [29] yield that

map(Bℤ/p, BX)_{g_{0}} ≃ BU(n)_{p}^{∧}.  (3)

The composite

g : BU(n)_{p}^{∧} ≃ map(Bℤ/p, BX)_{g_{0}} ⟶^{eval} BX

makes diagram (1) commutative.

Now let g : BU(n) ⟶ BX be any map satisfying (1), and let g_{0} : Bℤ/p ⟶ BX be the restriction of g to the central subgroup of order p in U(n). Then using Proposition 4:1 and (3), we see that multiplication ℤ/p×U(n) ⟶ U(n) induces homotopy equivalences

BU(n)_{p}^{∧} ⟶^{≃} map(Bℤ/p, BU(n)_{p}^{∧})_{incl} ⟶^{≃} map(Bℤ/p, BX)_{g_{0}}.

Points (a), (b) and (c) now follow directly from Proposition 4:6. □

We end this section with some remarks we referred to in the introduction. Let f : BT_{p}^{∧} ⟶ BX be the realization of a pseudoreflection group W ⟶ GL(n, ℤ̂_{p}). We claimed that, for several coefficients, the map f induces an isomorphism between the cohomology of BX and the invariants of W acting on the cohomology of BT. This is stated in the next proposition.

Proposition 4:9. Let W ⊂ GL(n, ℤ̂_{p}) be a finite subgroup, T be a torus and BT_{p}^{∧} ⟶ BX be a map invariant with repect to the action of W. If H^{∗}(BX; 𝔽_{p}) ⟶ H^{∗}(BT; 𝔽_{p})^{W} is an isomorphism, then so are the maps H^{∗}(BX; ℤ̂_{p}) ⟶ H^{∗}(BT; ℤ̂_{p})^{W} and H^{∗}(BX; ℤ̂_{p})⊗ℚ ⟶ (H^{∗}(BT; ℤ̂_{p})⊗ℚ)^{W}.

Proof:. By assumption the mod–p cohomology of BX is concentrated in even degrees. Hence, the p–adic integral cohomology is torsionfree, and the map H^{∗}(BX, ℤ̂_{p})⊗𝔽_{p} ⟶ H^{∗}(BX; 𝔽_{p}) is an isomorphism. Therefore the isomorphism for mod–p cohomology can be written as the composition

H^{∗}(BX; 𝔽_{p}) ⟵^{≅} H^{∗}(BX; ℤ̂_{p})⊗𝔽_{p} ⟶ H^{∗}(BT; ℤ̂_{p})^{W} ⊗𝔽_{p} ⟶ H^{∗}(BT; 𝔽_{p})^{W} .

The right arrow is an epimorphism, because the composition is one. Passing to invariants is left exact. Thus this arrow is also a monomorphism and the middle arrow is an isomorphism. As a consequence of the Nakayama Lemma, this shows that H^{∗}(BX; ℤ̂_{p}) ⟶ H^{∗}(BT; ℤ̂_{p})^{W} is an isomorphism. The functor ⊗ℚ̂_{p} commutes with taking invariants. This establishes the second asserted isomorphism. □

Remark 4:10. Pseudo reflection groups appear as the Weyl group of connected p–compact groups (Theorem 4:3). Hence, one could say that a realization of a pseudoreflection group W ⟶ GL(n, ℤ̂_{p}) consists of a connected p–compact group X of rank n such that the action of the Weyl group W_{X} on the maximal torus T_{X} is described by the representation W ⟶ GL(n, ℤ̂_{p}). In this sense, every connected p–compact group is a realization of its Weyl group.

Because p–complete spaces with polynomial cohomology give rise to a connected p–compact group (see Proposition 4:4), this notion is a generalisation of the definition given in the introduction.

The work of several people [8] [36] [3] [14] and this paper (Theorem 1.4) show that every irreducible pseudoreflection group over ℚ̂_{p} (all primes) has a realization in this interpretation. The fact that every pseudoreflection group over ℚ̂_{p} has a realization in this sense follows immediately from the observation that such a group splits (canonically) as a product of irreducible pseudoreflection groups.

5. Homotopy uniqueness.

In this section, we show that the spaces BX(q, r; n) are determined by their mod-p cohomology: any p-complete space whose 𝔽_{p}-cohomology is isomorphic to that of BX(q, r; n) has the homotopy type of BX(q, r; n). The next theorem is nothing but Theorem 1:6.

Theorem 5:1. Fix r|q|(p − 1) (q > 1) and n > 1, set G = G(q, r; n) and BX = BX(q, r; n). Let BY be any p-complete space such that H^{∗}(BY; 𝔽_{p}) ≅ H^{∗}(BX; 𝔽_{p}) as algebras over the Steenrod algebra. Then BY ≃ BX.

Remark. We used the notation BY instead of Y, because p–complete spaces with the above properties can be thought of as the classifying space of a p–compact group (see Proposition 4:4).

Proof. Let

j_{Y} : BU(n) ⟶ BY

be the map constructed in Proposition 4:8, whose induced map in cohomology is the same as that of j : BU(n) ⟶ BX. For each space BU(π) in the decomposition of BX (Theorem 5:2), set f_{π} = j_{Y} |BU(π) : BU(π) ⟶ BY. We want to show that the f_{π} all fit nicely together to define a map

f̂: BX = (\underrightarrow{hocolim}_{O_{H}(G)}(BU(π)))_{p}^{∧} ⟶ BY,  (1)

which will then be a homotopy equivalence by the assumption on the cohomology of BY. Here, we take H = {Σ_{π} | π ∈ P_{p}(n)}.

We first want to show that for any morphism φ : G/Σ_{π_{1}} ⟶ G/Σ_{π_{2}} in O_{H}(G), the diagram

\begin{matrix}BU(π_{1}) & ⟶^{f_{π_{1}}} & BY \\ ^{φ}↓ & & ∥ \\ BU(π_{2}) & ⟶^{f_{π_{2}}} & BY \end{matrix}  (2)

commutes up to homotopy. Since the restriction of j_{Y} to BT (T ⊆ U(n) a maximal torus) gives a maximal torus of BY (Proposition 4:8(a)), and because j_{Y} ^{∗} : H^{∗}(BY; ℤ̂_{p}) ⟶ H^{∗}(BT; ℤ̂_{p}) is a monomorphism, the diagrams clearly commute in cohomology. Moreover, by Proposition 4:8(b) we have BU(π_{1}) ≃ map(BZ(U(π_{i})), BY)_{f_{π_{i}} |BZ(U(π_{i}))}. (Note that Z(U(π_{i})) = T^{Σ_{π_{i}}} is a torus containing the center Z(U(n)) of U(n).) Hence, Proposition 4:8(c) shows that the diagrams (2) commute up to homotopy.

The commutativity up to homotopy of the diagrams (2) shows that the maps f_{π} define a map from the 1-skeleton of the homotopy colimit into BY. The obstructions for extending this to a map on the full homotopy direct limit lie in the groups

\underleftarrow{lim}^{i+1}_{G/H∈O_{H}(G)} π_{i}(map(BU(π), BY)_{f_{π}})  (3)

[34] (for i ≥ 1), where

π_{i}(map(BU(π), BY)_{f_{π}}) ≅ π_{i}(BZ(U(π))_{p}^{∧})  (Prop. 4:8(b))

≅ \{ \begin{matrix}π_{2}(BT_{p}^{∧})^{Σ_{π}} & if i = 2 \\ 0 & if i ≠ 2. \end{matrix}

Lemma 3:1 now implies that the obstruction groups in (3) all vanish. Hence there exists a map f̂: BX ⟶ BY as in (1), which is a homotopy equivalence since BX and BY are p-complete and f̂ induces an isomorphism in 𝔽_{p}-cohomology. □

We also can offer a characterization of BX respectively X as a p–compact group in terms of the Weyl group data. We say that two p–compact groups X and Y have the same Weyl group type if the ranks are equal, say equal to n, and if the two representations W_{X}, W_{Y} ⟶ GL(n, ℤ̂_{p}), induced by the action on the maximal torus, are conjugate.

Theorem 5:2. Fix r|q|(p − 1) (q > 1) and n > 1 and set BX = BX(q, r; n). Let Y be a p–compact group, which has the same Weyl group type as X. Then, the p–compact groups X and Y are isomorphic.

Proof. Without loss of generality we can assume that T := T_{X} = T_{Y}, that W := W_{X} = W_{Y} and that the two representations W ⟶ GL(n; ℤ̂_{p}) are equal. Let j_{0} : ℤ/p ↪ T be the diagonal inclusion. Then, the centralizer C := C_{Y} (S^{1}) is a p–compact group of rank n with Weyl group given by the isotropy group of the inclusion S^{1} ↪ T. That is to say that W_{C} = Σ_{n} and that C has the same Weyl group type as U(n). Hence, by [27], the two p–compact groups C and U(n)_{p}^{∧} are isomorphic. The index of Σ_{n} ⊆ W is coprime to p. This implies that H^{∗}(BY; 𝔽_{p}) ⟶ H^{∗}(BU(n); 𝔽_{p}) is a monomorphism ([14] or [26]), that H^{∗}(BY; ℤ̂_{p}) is torsionfree, that H^{∗}(BY; ℤ_{p}^{∧}) ≅ H^{∗}(BT; ℤ_{p}^{∧})^{W} [27, 4.2] and that

H^{∗}(BY; 𝔽_{p}) ≅ H^{∗}(BT; ℤ_{p}^{∧})^{W} ⊗𝔽_{p} ≅ H^{∗}(BT; 𝔽_{p})^{W} ≅ H^{∗}(BX; 𝔽_{p}).

The second isomorphism follows from the proof of Proposition 4.9. Now, an application of Theorem 5:1 yields that BX ≃ BY, which is to say that X ≅ Y as p–compact groups. □

6. The normalizer of G(q, r; n).

In this section we calculate the quotient group N_{GL(n,ℚ̂_{p})}(G)/G, when G =

G(q, r; n) ⊆ GL(n, ℚ̂_{p}). This will be needed in the next section when describing the self maps of BX(q, r; n).

Set A_{max} = (μ_{q})^{n} = A(q, q; n), and set G_{max} = G(q, q; n). As usual, we write G = G(q, r; n) and A = A(q, r; n) for short. An easy calculation shows that the image of the homomorphism

H′ : A_{max} × ℚ̂_{p}^{∗} ⟶ GL(n, ℚ̂_{p}) ,

where H′((a_{1}, ..., a_{n}), k) = diag(ka_{1}, ..., ka_{n}), is contained in the normalizer N_{GL(n,ℚ̂_{p})}(G). Since H′ sends A×1 into G, and the subgroup μ_{q} ≅ {((k, …, k), k^{−1}) | k ∈ μ_{q}} ⊆ A_{max} × ℚ̂_{p} into 1, it factors through the quotient homomorphism

H : A_{max}/A ×_{μ_{q}} ℚ̂_{p}^{∗} ⟶ N_{GL(n,ℚ̂_{p})}(G)/G.

Lemma 6:1. Fix r|q|(p−1) and n > 1, and assume q > 2. Regard G = G(q, r; n) as a subgroup of GL(n, ℚ̂_{p}). Then, the homomorphism

H : A_{max}/A×_{μ_{q}}(ℚ̂_{p})^{∗} ⟶ N_{GL(n,ℚ̂_{p})}(G)/G

is an isomorphism except in the following two cases:

(a) G = G(4, 2; 2), and N_{GL(2,ℚ̂_{p})}(G)/G ≅ Σ_{3} × ((ℚ̂_{p})^{∗}/μ_{4})

(b) G = G(3, 1; 3), and N_{GL(3,ℚ̂_{p})}(G)/G ≅ A_{4}^{∗} ×_{(−1)} ((ℚ̂_{p})^{∗}/μ_{3}) (A_{4}^{∗} denotes the binary tetrahedral group). In both cases (a) and (b), the first factor is contained in GL(n, ℤ̂_{p}).

Proof. Step 1: We first check that H is a monomorphism. For any (a_{1}, …, a_{n}) ∈ A_{max} and any k ∈ (ℚ̂_{p})^{∗}, let [a_{1}, …, a_{n}; k] denote the class of ((a_{1}, …, a_{n}), k) in A_{max}/A×_{μ_{q}}(ℚ̂_{p})^{∗}. If [a_{1}, …, a_{n}; k] ∈ Ker(H), then diag(ka_{1}, …, ka_{n}) ∈ G, so k ∈ μ_{q} and (ka_{1}, …, ka_{n}) ∈ A(q, r; n), and

[a_{1}, …, a_{n}; k] = [ka_{1}, …, ka_{n}; 1] = 1  in  A_{max}/A×_{μ_{q}}(ℚ̂_{p})^{∗}.

Step 2: Fix an element M ∈ N_{GL(n,ℚ̂_{p})}(G), and let [M] denote its class mod G. We show here that [M] ∈ Im(H) if M normalizes A.

As an A-representation, (ℚ̂_{p})^{n} splits as a sum of n distinct 1-dimensional representations. So if M ∈ N_{GL(n,ℚ̂_{p})}(A), then M must be monomial (i.e., a product of a diagonal matrix and a permutation matrix). In particular, after multiplying M by an appropriate element in Σ_{n} ⊆ G, we can assume that M is diagonal.

Write M = diag(u_{1}, …, u_{n}), where u_{i} ∈ ℚ̂_{p}. For any σ ∈ Σ_{n},

MσM^{−1} = M·(σM^{−1}σ^{−1})·σ
= (u_{1}u_{σ^{−1}(1)}^{−1}, …, u_{n}u_{σ^{−1}(n)}^{−1})·σ ∈ G.

It follows that u_{1} ≡ u_{2} ≡ ⋯ ≡ u_{n} (mod μ_{q}), so

M = H′((1, u_{2}u_{1}^{−1}, …, u_{n}u_{1}^{−1}), u_{1});

and [M] ∈ Im(H).

Step 3: Again fix M ∈ N_{GL(n,ℚ̂_{p})}(G), and set ρ = (g ↦ MgM^{−1}) ∈ Aut(G). We now show that ρ(A) = A, except in the cases (a) and (b) listed above, by showing that no other subgroup of G is isomorphic to A. In fact, G contains no other abelian subgroup with the same order as that of A.

Assume first that for some prime p′|q, O_{p′}(Σ_{n}) = 1. Here, O_{p′}(−) denotes the intersection of all Sylow p′–subgroups. In this case, O_{p′}(G) ⊆ A and A = C_{G}(O_{p′}(G)), and so ρ(A) = A. Hence the only cases which remain to consider are those where q is a power of p′, and where (n, p′) = (2, 2), (3, 3), or (4, 2). (Recall that for n ≥ 5, the alternating group A_{n} ⊆ Σ_{n} is the only nontrivial normal subgroup.)

Now, if n = 3 and q is a power of 3, and if ρ(A) ≠ A, then ϕ(ρ(A)) must contain a 3-cycle, and so ρ(A) ∩ A is contained in the diagonal subgroup of order at most min(q, 3r). Thus, q^{2}r = |ρ(A)|| min(3q, 9r), and hence q = 3 and r = 1. If n = 2 or 4 and q is a power of 2, then a similar argument shows that (q, r; n) = (4, 1; 2) or (4, 2; 2) (recall that q > 2 by assumption). And finally, the group G(4, 1; 2) is dihedral of order 8, and A is its unique cyclic subgroup of order 4.

Step 4: If G = G(4, 2; 2), then it contains a total of three subgroups isomorphic to A. So by Step 2, Im(H) has index at most three in N_{GL(n,ℚ̂_{p})}(G)/G. Using this,

one checks that N_{GL(n,ℚ̂_{p})}(G)/G ≅ Σ_{3} ×(ℚ̂_{p})^{∗}/μ_{4}, where the second factor consists of multiples of the identity, and where the first factor is generated by the matrices

S_{1} = \frac{1}{1 + i} ( \begin{matrix}i & 1 \\ i & −1 \end{matrix} )  and  S_{2} = ( \begin{matrix}i & 0 \\ 0 & 1 \end{matrix} ) .

Note that 1 + i is a unit in ℤ̂_{p}, since 2 is invertible.

If G = G(3, 1; 3), then it contains a total of four subgroups isomorphic to A. Hence Im(H) has index at most four in N_{GL(n,ℚ̂_{p})}(G)/G. One now checks that

N_{GL(n,ℚ̂_{p})}(G)/G ≅ A_{4}^{∗} ×_{(−1)} (ℚ̂_{p})^{∗}/μ_{3}, where the second factor consists of multiples of the identity and the first is generated by the matrices

\begin{matrix}T_{i} = \frac{1}{\sqrt{−3}} ( \begin{matrix}1 & 1 & 1 \\ 1 & ζ & ζ^{2} \\ 1 & ζ^{2} & ζ \end{matrix} ) , T_{j} = \frac{1}{\sqrt{−3}} ( \begin{matrix}1 & ζ^{2} & ζ^{2} \\ ζ & ζ & ζ^{2} \\ ζ & ζ^{2} & ζ \end{matrix} ) , \\ T_{k} = \frac{1}{\sqrt{−3}} ( \begin{matrix}1 & ζ & ζ \\ ζ^{2} & ζ & ζ^{2} \\ ζ^{2} & ζ^{2} & ζ \end{matrix} ) , and T′ = diag(ζ, 1, 1) . \end{matrix}

Here, ζ denotes a primitive cube root of unity in ℤ̂_{p}. □

7. Classification of self maps.

Let BX = BX(q, r; n) be a generalized Grassmannian. In this section, we want to study the monoid [BX, BX] of homotopy classes of self maps of BX. As usual we assume that q|(p − 1), but throughout this section we also assume that q > 2. This restriction is necessary because we later use Lemma 6:1. When q = 2, BX ≃ BSO(2n)_{p}^{∧} or BSO(2n+1)_{p}^{∧}, and so [BX, BX] is described in [20, Theorem 2].

Proposition 7:1. Let p be an odd prime and let r|q|(p−1). Let BX = BX(q, r; n) be the p-complete space constructed in Theorem 3:2. Then, for any self map f : BX ⟶ BX which induces a nontrivial map in cohomology, the map H^{∗}(f; ℤ̂_{p}) extends to a map

H^{∗}(f_{T}) ∈ End(H^{∗}(BT; ℤ̂_{p})) ∩ Aut(H^{∗}(BT; ℤ̂_{p}) ⊗ ℚ) ≅ M_{n}(ℤ̂_{p}) ∩ GL(n, ℚ̂_{p}).

This extension H^{∗}(f_{T}) is contained in N(G) = N_{GL(n,ℚ̂_{p})}(G) and is unique modulo

G. If n ≥ p, then H^{∗}(f_{T}) ∈ GL(n, ℤ̂_{p}).

The proof of this statement also works for q = 2.

Proof. By [13, Proposition 8.11], for any map f : BX ⟶ BX, the map f|BT : BT ⟶ BX lifts to a self map f_{T} : BT_{p}^{∧} ⟶ BT_{p}^{∧} of the classifying space of the

maximal torus BT ⟶^{i} BX of X. And moreover, H^{∗}(f; ℤ̂_{p}) is nontrivial if and only if H^{∗}(f_{T}; ℤ̂_{p}) is nontrivial. For any w ∈ G = G(q, r; n), the two maps f_{T} ∘Bw and f_{T} are homotopic after composition with i. So by [28; 7.1], there is w′ ∈ G such that H^{∗}(f_{T} ∘Bw) = H^{∗}(Bw′∘f_{T}), and hence f_{T} ∘Bw ≃ Bw′∘f_{T}. In other words, there is a map of sets ρ : G ⟶ G such that f_{T}^{∗} = H^{∗}(f_{T}; ℤ̂_{p}) is equivariant with respect to ρ. An algebraic version, dealing with rational cohomology, and which is sufficient for our purpose, is already contained in [1] (see also [25, 0.2]).

In particular, this shows that the kernel of H^{2}(f_{T}; ℤ̂_{p}) is W_{X}-invariant. Also, H^{2}(BT; ℤ̂_{p})⊗ℚ is an irreducible representation of G = G(q, r; n) = W_{X}, since it splits as a sum of distinct 1–dimensional A = A(q, r; n)–representations which are permuted transitively by Σ_{n} = G/A. It follows that f_{T} induces either an isomorphism in rational cohomology and that ρ is an isomorphism of groups (given by conjugation with f_{T}^{∗} considered as a self map of H^{2}(BT; ℤ̂_{p})⊗ℚ), or that H^{∗}(f_{T}; ℤ̂_{p})⊗ℚ is the trivial map. We only have to consider the first case. Now the above argument says, that H^{∗}(f_{T}) ∈ N_{GL(n,ℚ̂_{p})}(G) (and clearly H^{∗}(f_{T}) ∈ M_{n}(ℤ̂_{p})). And by [28; 7.1] again, H^{∗}(f_{T}) is uniquely defined modulo G. This proves the first part of the statement.

Lemma 6:1 shows that, up to homotopy equivalences, the map f_{T} looks like an unstable Adams operation of degree p^{s}k with (p, k) = 1. Hence, we can assume that f_{T} = Ψ^{p^{s}k} is an unstable Adams operation.

Let g_{S} : BS^{1} = BZ(U(n)) ⟶ BX be the inclusion of the center. Then we have Ψ^{p^{s}k}g_{S} ≃ g_{S}Ψ^{p^{s}k} (Proposition 4:8), and Ψ^{p^{s}k} induces a map

BU(n)_{p}^{∧} ≃ map(BZ(U(n)), BX)_{g_{S}} ⟶ map(BS^{1}, BX)_{g_{S}} ≃ BU(n)_{p}^{∧},

which is again an unstable Adams operation of the same degree p^{s}k. By a result of Ishiguro [17], the degree p^{s}k is a p–adic unit, equal to 0 or n < p. Hence, for n ≥ p every non null homotopic map f : BX ⟶ BX gives rise to an element in GL(n, ℤ̂_{p}). This finishes the proof. □

The fact that the degree of an unstable Adams operations on BX is a p–adic unit when n ≥ p is also implicitly contained in a theorem of Møller [25, 4.5].

By Lemma 6:1 and Proposition 7:1, there are well defined homomorphisms of monoids

[BX, BX] ⟶^{D} {0}∐N_{GL(n,ℚ̂_{p})}(G)/G ⟵^{H} {0}∐((A_{max}/A)×_{μ_{q}}(ℤ̂_{p}∖0)),

where H is an isomorphism except in the cases BX(4, 2; 2) and BX(3, 1; 3). Using these maps, we can now formulate the main theorem of this section.

Theorem 7:2. Let p be an odd prime, r|q|(p−1) and q > 2. Let BX = BX(q, r; n) be the generalized Grassmannian constructed in Theorem 3:2. Then H^{−1}∘D induces an isomorphism

[BX, BX] ≅ {0}∐((A_{max}/A)×_{μ_{q}}(ℤ̂_{p})^{∗})

if n ≥ p, and an isomorphism

[BX, BX] ≅ {0}∐((A_{max}/A)×_{μ_{q}}(ℤ̂_{p}∖0))

if n < p and (q, r; n) ≠ (4, 2; 2) or (3, 1; 3). In the two exceptional cases, D induces isomorphisms

\begin{aligned}[BX(4, 2; 2), BX(4, 2; 2)] & ≅ \{0\}∐(M_{n}(ℤ̂_{p}) ∩ N_{GL(n,ℚ̂_{p})}(G)/G) \\ & ≅ \{0\}∐(Σ_{3} × (ℤ̂_{p}∖0)) \end{aligned}

and

\begin{aligned}[BX(3, 1; 3), BX(3, 1; 3)] & ≅ \{0\}∐(M_{n}(ℤ̂_{p}) ∩ N_{GL(n,ℚ̂_{p})}(G)/G) \\ & ≅ \{0\}∐(A_{4}^{∗} ×_{(−1)}(ℤ̂_{p}∖0)). \end{aligned}

The following corollary contains Theorem 1:7.

Corollary 7:3. Let BX be as in Theorem 7:2. Then the following conditions are equivalent for any pair of maps f, g : BX ⟶ BX.

(a) The maps f and g are homotopic.

(b) The restrictions f|BT, g|BT : BT ⟶ BX are homotopic.

(c) f^{∗}=g^{∗} : H^{∗}(BX; ℤ̂_{p}) ⟶ H^{∗}(BX; ℤ̂_{p}).

(d) f^{∗}⊗ℚ=g^{∗}⊗ℚ : H^{∗}(BX; ℤ̂_{p})⊗ℚ ⟶ H^{∗}(BX; ℤ̂_{p})⊗ℚ

Proof. Clearly, (a)⇒(b)⇒(c)⇒(d): the second implication since H^{∗}(BX; ℤ̂_{p})⊗ℚ injects into H^{∗}(BT; ℤ̂_{p})⊗ℚ (Proposition 4.3). And (d) implies (a) by Theorem 7:2 (the injectivity of D). □

Proof of Theorem 7:2. Set G = G(q, r; n). If n < p, then

BX ≃ (EG×_{G}BT_{p}^{∧})_{p}^{∧} ≃ (\underrightarrow{hocolim}_{G}(B)T_{p}^{∧})_{p}^{∧}

by [10, Theorem 1.2] (where T is the n-torus) and where we think of G as a category with one object and morphisms given by the elements of G. In this case, we will show that

[BX, BX] ⟶^{≅} {f_{T} ∈ [BT, BT] | equivariant w.r.t. some ρ : G ⟶ G.}/G

⟵^{≅} {0}∐(M_{n}(ℤ̂_{p}) ∩ N_{GL(n,ℚ̂_{p})}(G))/G.

The latter isomorphism is obvious. The action of G on the second set is induced from the action of G on the target space BT_{p}^{∧}. Because self maps of BT_{p}^{∧} are detected by rational cohomology, the arguments in the proof of Theorem 7:1 establish the first map. Now let f_{T} : BT_{p}^{∧} ⟶ BT_{p}^{∧} be a self map equivariant w.r.t. some

ρ : G ⟶ G. The obstructions for extending the composition BT_{p}^{∧} ⟶^{f_{T}} BT_{p}^{∧} ⟶ BX, also denoted by f_{T}, to a self map of BX lie in

\underleftarrow{lim}_{G}^{i+1}π_{i}(map(BT_{p}^{∧}, BX)_{f_{T}} ) ≅ \underleftarrow{lim}_{G}^{i+1}π_{i}(BT_{p}^{∧}) ≅ H^{i+1}(G; π_{i}(BT_{p}^{∧})).

Let f, f′ : BX ⟶ BX be two maps which, up to homotopy and the G–action, give rise to the same self map of BT_{p}^{∧}. The obstruction for f and f′ being homotopic lie in

\underleftarrow{lim}_{G}^{i}π_{i}(map(BT_{p}^{∧}, BX)_{f_{T}}) ≅ \underleftarrow{lim}_{G}^{i}π_{i}(BT_{p}^{∧}) ≅ H^{i}(G; π_{i}(BT_{p}^{∧})).

In both cases this follows from [34]. Because the order of G is coprime to p, all involved obstruction groups vanish, which proves the statement for n < p.

Now we assume that n ≥ p. In particular, by Proposition 7:1, any map f : BX ⟶ BX which does not induce the trivial map in rational cohomology induces an isomorphism on H^{∗}(BX; ℤ̂_{p}), and hence is a homotopy equivalence.

We first construct the unstable Adams operations. Fix k ∈ (ℤ̂_{p})^{∗}. Let g : BU(n) ⟶ BX be the inclusion of BU(n) into the homotopy direct limit constructed in Theorem 4:8. For each partition π ∈ P_{p}(n), set g_{π} = g|BU(π); and let f_{π} : BU(π) ⟶ BX be the composite

f_{π} : BU(π)_{p}^{∧} ⟶_{≃}^{ψ_{π}^{k}} BU(π)_{p}^{∧} ⟶^{g_{π}} BX,

where ψ_{π}^{k} denotes an unstable Adams operation of degree k on BU(π). Furthermore, by Proposition 4:8(c), the f_{π} commute up to homotopy with maps induced by morphisms φ : G/Σ_{π_{1}} ⟶ G/Σ_{π_{2}} (since they commute in cohomology). The f_{π} thus combine to define an element

f̂ ∈ \underleftarrow{lim}_{G/H∈O_{H}(G)} π_{0}(map(BU(π), BY)_{g_{π}}).

Here, H = {Σ_{π} | π ∈ P_{p}(n)} is the family of subgroups of G constructed in Section 5.

Since ψ_{π}^{k} is an equivalence (p ∤ k), Proposition 4:8(b) applies again to show that there are homotopy equivalences

\begin{aligned}BZ(U(π))_{p}^{∧} & ⟶^{≃} map(BU(π), BX)_{g_{π}} \\ & ⟶_{≃}^{−∘ψ_{π}^{k}} map(BU(π), BX)_{f_{π}} ⟵^{≃} BZ(U(π))_{p}^{∧}. \end{aligned}

These equivalences are compatible with all morphisms of the category O_{H}(G), and

π_{i}(BZ(U(π))_{p}^{∧}) ≅ \{ \begin{matrix}0 & if i ≠ 2 \\ π_{2}(BT_{p}^{∧})^{Σ_{π}} & if i = 2. \end{matrix}

This is a fixed point functor. So by Proposition 2:3,

\underleftarrow{lim}^{j}_{G/H∈O_{H}(G)} π_{2}(map(BU(π), BY)_{f_{π}}) = 0

for all j ≥ 1. Thus, the obstructions to extending the f_{π} to a map

BX ≅ \underrightarrow{hocolim}_{O_{H}(G)}(BU(π)) ⟶ BX

all vanish, and we get a map f : BX ⟶ BX which (since the cohomology of BX injects into the cohomology of BU(n)) is an unstable Adams operation of degree k.

Next we want to realize the elements of A_{max}/A as self maps of BX. Fix α ∈ A_{max}, and set

Φ = incl∘Bα_{X}^{∗} : H^{∗}(BX; 𝔽_{p}) ⟶ H^{∗}(BU(n); 𝔽_{p}).

By Proposition 4:8, there exists a map g_{α} : BU(n) ⟶ BX such that H^{∗}(g_{α}) = Φ. If we now set f_{π} = g_{α}|BU(π) for each π ∈ P_{p}(n), then the same arguments as those used to construct unstable Adams operations apply to show that the f_{π} extend to a map g : BX ⟶ BX. Then H^{∗}(f; ℤ̂_{p}) = α_{X}^{∗} (mod p), so D(g) = (α, k) for some k ≡ 1 (mod p).

It remains to show that D is injective. If D(f) = D(f′) and they are nontrivial in rational cohomology, then as noted above, f and f′ are both homotopy equivalences. Hence, upon replacing f by f ∘(f′)^{−1}, we can assume that H^{∗}(f; ℤ̂_{p}) is the identity, and must show that f ≃ Id_{BX}. By Proposition 4:8(b),

f ∘g ≃ g : BU(n) ⟶ BX,

where g : BU(n) ⟶ BX denotes the “canonical” map of Proposition 4:8. It follows that f and Id_{BX} have the same image under the restriction map

[BX, BX] ≅ [ \underrightarrow{hocolim}_{G/Σ_{π}∈O_{H}(G)}(BU(π)), BX] ⟶ \underleftarrow{lim}_{G/Σ_{π}∈O_{H}(G)}[BU(π), BX].

By [34] again, the obstructions to f being homotopic to Id_{BX} lie in the groups

\underleftarrow{lim}^{i}_{G/Σ_{π}∈O_{H}(G)} π_{i}(map(BU(π), BX)_{g_{π}}).

Because map(BU(π), BX)_{g_{π}} ≃ BZ(U(π))_{p}^{∧} by Proposition 4:8(b), these higher limits vanish by Proposition 2:3.

It remains to check that f : BX ⟶ BX is null homotopic if it induces the trivial map in rational cohomology. This is nothing but a special case of [25, 5.7]. □

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Mathematisches Institut, Bunsenstr. 3-5, 37073 Göttingen, Germany, email: notbohm@cfgauss.uni-math.gwdg.de