A UNIQUENESS RESULT FOR ORTHOGONAL

GROUPS AS 2-COMPACT GROUPS

D. Notbohm

Abstract. Two connected compact Lie groups are isomorphic if and only if their maximal torus normalizers are isomorphic. It is conjectured that this result generalizes to p-compact groups. Here, we prove the generalization for orthogonal groups O(n), the special orthogonal groups SO(2k + 1) and the spinor groups Spin(2k + 1) considered as 2-compact groups.

1. Introduction.

A p-compact group is the homotopy theoretic generalization of a compact Lie group. This concept was introduced by Dwyer and Wilkerson [8]. And it turned out that p-compact groups behave astonishingly similar as compact Lie groups; e.g. there exist maximal tori, Weyl groups and normalizer of maximal tori. Since their existence, the classification problem was one of the main questions. The classification scheme of connected p-compact groups is of the same form as for compact connected Lie groups. Every connected p-compact group has a finite covering which is a product of torus and a simply connected p-compact group [19], and every simply connected p-compact group splits into a product of finitely many simple simply connected p-compact groups [10] [23]. The classification boils down to the case of simple p-compact groups [15]. And in this situation, a case by case checking based on the Clark-Ewing list of rational pseudo reflection groups has solved the problem for odd primes ([2], [3], [4], [11], [17], [18], [20], [24]). For the prime 2, the known results in this direction concern unitary and special unitary groups [20], 2-compact groups with Weyl group isomorphic to an elementary abelian 2-group [11] and the case of the exceptional Lie group G_{2} [26].

In this work we consider products of orthogonal groups and characterize them as 2-compact groups by their associated normalizer of the maximal torus. Before we state our main results, we first have to recall some basic notions of the theory of p-compact groups from [8]. For further notation and definition as well as the theory of p-compact groups see the survey articles [14] or [22] or the references given there. Since all the basic notions are well discussed in these and other papers we keep explanations short and refer the interested reader to the mentioned papers for more details.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 55R35, 55P15.

Key words and phrases. orthogonal groups, compact Lie groups, p-compact groups, classification.

A loop space X is a pair (X, BX) of spaces, such that such that BX is pointed and such that the loop space ΩBX is equivalent to X. The loop space inherits properties from the space X, e.g. X is called connected if the space X is so and finite or ℤ-finite if the integral cohomology H^{∗}(X; ℤ) vanishes in large degrees and is in any degree a finitely generated abelian group. The loop space X is called a p-compact group, if X is 𝔽_{p}-finite and if X and BX are p-complete. Here, 𝔽_{p}-finite means that H^{∗}(X; 𝔽_{p}) is finite in each degree and vanishes in large degrees.

Every p-compact group X has a maximal torus T_{X} ⊂ X, a maximal torus normalizer N_{X} ⊂ X which is a finite extension of T_{X} and a Weyl group W_{X} which acts on the p-adic lattice L_{X} := π_{2}(BT_{X}). In this context, T_{X} denotes a completed torus and ⊂ signifies maps BT_{X} ⟶ BX and BN_{X} ⟶ BX whose homotopy fibers are 𝔽_{p}-finite. To ensure that T_{X} ⊂ X is maximal torus we also have to put some maximality condition on the map BT_{X} ⟶ BX (e.g. see [8], [14] or [22]). The maximal torus normalizer fits into a fibration BT_{X} ⟶ BN_{X} ⟶ BW_{X} which also determines the action of W_{X} on BT_{X} respectively on L_{X}. The map BN_{X} ⟶ BX is an extension of BT_{X} ⟶ BX.

The composition N_{X} ⟶ X ⟶ π_{0}(X) factors through a homomorphism W_{X} ⟶ π_{0}(X) and is always an epimorphism. The kernel of N_{X} ⟶ π_{0}(X) ≅ π_{1}(BX) respectively the fiber of the map BN_{X} ⟶ Bπ_{0}(X) is a maximal torus normalizer of the component X_{0} of the unit of X and the kernel of W_{X} ⟶ π_{0}(X) is the Weyl group of X_{0}. Examples of p-compact groups are given by the p-adic completion of compact Lie groups. If the group of components π_{0}(G) of a compact Lie group G is a finite p-group, then the pair (G_{p}^{∧}, BG_{p}^{∧}) gives rise to a p-compact group, also denoted by G. In particular, the orthogonal groups O(n) give rise to 2-compact groups.

1.1 Definition.

(i) Two loop spaces or p-compact groups X and Y are called isomorphic if the associated classifying spaces BX and BY are homotopy equivalent. (ii) Two p-compact groups X and Y are called N-isomorphic if the there exist isomorphisms N_{X} ⟶^{≅} N_{Y} and π_{0}(X) ⟶^{≅} π_{0}(Y) such that the diagram

BN_{X} ⟶ BN_{Y}
↓  ↓
Bπ_{0}(X) ⟶ Bπ_{0}(Y)

commutes up to homotopy.

In analogy to compact Lie groups, the classification conjecture reads as

1.2 Conjecture. Two p-compact groups X and Y are isomorphic if and only if they are N-isomorphic.

As already mentioned, for odd primes, this conjecture is known to be true. In this case, one even can proof a weaker version, namely, X and Y are isomorphic if and only if N_{X} and N_{Y} are isomorphic, and, if in addition X is connected, then it suffices to have isomorphic Weyl group data; i.e. there exists an abstract isomorphism W_{X} ≅ W_{Y} such that L_{X} and L_{Y} are isomorphic as W_{X}-modules. (see the above mentioned references). But, as a comparison of O(2) and SO(3) and of SO(2n + 1) and Sp(n) show, a weaker version cannot be true for the prime 2.

For p = 2, we want to prove the conjecture for finite products of orthogonal groups.

1.3 Theorem. Let G = ∏_{i} O(n_{i}) be a finite product of orthogonal groups and let X be a 2-compact group. Then, G and X are isomorphic if and only if they are N-isomorphic.

The proof of this theorem is based on homotopy uniqueness results for products of orthogonal groups [21] in terms of mod-2 cohomology and

1.4 Theorem. Let G = ∏_{i} O(n_{i}) be a finite product of orthogonal groups and let X be a 2-compact group. If G and X are N-isomorphic, then there exists an isomorphism H^{∗}(BX) ≅ H^{∗}(BG) of algebras over the Steenrod algebra.

Proof of Theorem 1.3. As shown in [21], every 2-complete space with the same mod-2 cohomology as BG is homotopy equivalent to the 2-adic completion BG_{2}^{∧}. □

As a corollary we will also see that the groups Spin(2k + 1) and SO(2k + 1) satisfy the above conjecture.

1.5 Corollary. Let G = Spin(2k + 1) or SO(2k + 1) and let X be a 2-compact group. Then X and G are isomorphic if and only if they are N-isomorphic.

Proof. Let X be a 2-compact group which is N-isomorphic to SO(2k + 1). Since N_{O(2k+1)} ≅ N_{SO(2k+1)} × ℤ/2, the 2-compact group X × ℤ/2 is N-isomorphic to O(2k + 1) and therefore, by Theorem 1.3, isomorphic to O(2k + 1). The projection BX × Bℤ/2 ⟶ Bℤ/2 on the second factor is given by the classifying map of the non trivial 1-dimensional mod-2 cohomology class. The fiber of this map is BX. the same holds for BO(2k + 1)_{2}^{∧}. This shows that BX and BSO(2k + 1)_{2}^{∧} are equivalent and that X and SO(2k + 1) are isomorphic as 2-compact groups.

Let X be a 2-compact group, which is N-isomorphic to Spin(2k + 1). The center of X can be read off from the maximal torus normalizer [9]. Hence, X and Spin(2k + 1) have the same center equal to ℤ/2 and the quotients \overline{X} := X/ℤ/2 and SO(2k + 1) are N-isomorphic and therefore isomorphic. Since X is the simply connected cover of \overline{X} and Spin(2k + 1) the simply connected cover of SO(2k + 1), the 2-compact groups X and Spin(2k + 1) are isomorphic. □

Starting from Corollary 1.5, we plan in a further paper to prove uniqueness results for the 2-compact group DI(4) constructed in [7] in terms of the maximal torus normalizer as well as in terms of the mod-2 cohomology of BDI(4).

The paper is organized as follows. In Section 2 we provide some facts for orthogonal groups as 2-compact groups and fix some notation. Section 3, the last section, contains the tedious proof of Theorem 1.4. We also note that cohomology is always taken with ℤ/2-coefficients; i.e. H^{∗}(−) = H^{∗}(−; 𝔽_{2}).

We will use the language and theory of p-compact groups all over the places. For references, about p-compact groups we refer the reader to the survey articles [14] and [22] or the references mentioned there.

We thank the Max-Planck-Institut für Mathematik in Bonn for the hospitality when this work was done. We also thank the referee who pointed out to us an argument which simplified parts of our original proof of Proposition 3.1 (see Claim 3.9).

2. Products of orthogonal type.

In this section we provide necessary facts about products of orthogonal groups and fix some notation.

2.1 Notation and Remarks. Let n = 2k + θ with θ ∈ {0, 1}. We use Θ to denote the group ℤ/2^{θ}. The maximal torus normalizer of O(n) is isomorphic to O(2) ≀ Σ_{k} × Θ. The group O(2) contains a maximal elementary abelian 2-subgroup E_{O(2)} of rank 2. We denote by (ℤ/2)^{k} ≅ t_{O(n)} ⊂ T_{O(n)} the maximal elementary abelian 2-subgroup of the maximal torus of O(n). We have a sequence

t_{O(n)} ⊂ (ℤ/2)^{n} ≅ E_{O(n)} := E_{O(2)}^{k} × Θ ⊂ N_{O(n)} ⊂ O(n)

of inclusions.

Every elementary abelian 2-subgroup of O(n) is contained in E_{O(n)}. The subgroup E_{O(n)} is self centralizing, i.e. the inclusion E_{O(n)} ⊂ C_{O(n)}(E_{O(n))} induces an isomorphism of E_{O(n)} with the centralizer of E_{O(n)}. The Weyl group \overline{W}_{O(n)} := W_{O(n)}(E_{O(n)}) is isomorphic to Σ_{n} which acts by permutations on E_{O(n)}. Actually, the normalizer N_{O(n)}(E_{O(n)}) of E_{O(n)} is isomorphic to ℤ/2 ≀ Σ_{n} ⊂ O(n). All this follows from classical representation theory. Moreover, H^{∗}(BO(n); ℤ/2) ≅ H^{∗}(BE_{O(n)}; ℤ/2)^{Σ_{n}}.

For G = ∏_{i} O(n_{i}), taking products establishes the same type of groups which fit into the diagram of inclusions

t_{G} ⟶ T_{G}
↓  ↓
E_{G} ⟶ N_{G} ⟶ N_{G} ⟶ G

and satisfy the same properties. Among other things we collect these facts in the next proposition. And \overline{W}_{G} := W_{G}(E_{G}) denotes the Weyl group of E_{G} in G.

By SG we denote the component of the unit of G. Then SG = ∏_{i} SO(n_{i}) is a product of special orthogonal groups. We will also use the notation SX for the component of the unit of a p-compact group X.

2.2 Proposition. Let G ≅ ∏_{i} O(n_{i}) be a finite product of orthogonal groups. Then, the following holds:

(i) E_{G} is self centralizing. (ii) The mod-2 cohomology of BN_{G} is detected by elementary abelian 2-subgroups and so is the mod-2 cohomology of BG. (iii) For any monomorphism ρ : E_{G} ⟶ N_{G}, the image is subconjugated to E_{G} ⊂ N_{G}. (iv) Up to conjugation, every elementary abelian subgroup of G is contained in E_{G}.

(v) \overline{W}_{G} := ∏_{i} \overline{W}_{O(n_{i})} ≅ ∏_{i} Σ_{n_{i}}. (vi) H^{∗}(BG; ℤ/2) ≅ H^{∗}(BE_{G}; ℤ/2)^{\overline{W}^{G}}. (vii) For any elementary abelian 2-group E, the inclusion E_{G} ⟶ G induces a bijection [BE, BE_{G}]/\overline{W}_{G} ≅ [BE, BG]. (viii) For any elementary abelian 2-subgroup E ⊂ G, the centralizer C_{G}(E) is again a finite product of orthogonal groups.

Proof. : Most of the proof is obvious and/or a straight forward application of real representation theory. Only (ii) and (iii) need a more detailed argument.

As a product of Θ and a wreath product of O(2) with a symmetric group, the mod-2 cohomology of BN_{O(n)} is detected by elementary abelian 2-subgroups [12] and so is the mod-2 cohomology of BN_{G}. This proves part (ii).

Part (iii) is a straightforward consequence of the lemma below, which contains the claim in a particular case. □

2.3 Lemma. Let N := O(2) ≀ Σ_{n}. The 2-rank rk_{2}(N)of N equals 2n and all elementary abelian 2-subgroups isomorphic to (ℤ/2)^{2n} are conjugated.

Proof. Let (ℤ/2)^{2n} ≅ Δ_{n} ⊂ (O(2)^{n} be the elementary abelian 2-subgroup given by the product of all diagonal matrices. Since Δ_{n} ≅ (ℤ/2)^{2n}, rk_{2}(N) ≥ 2n.

The rest of the statement we prove by induction. For n=1 there is nothing to show. For abbreviation we set N_{k} := O(2) ≀ Σ_{k}. Let E ⊂ N_{n} be an elementary abelian 2-subgroup of dimension 2n and let U ⊂ Σ_{n} be the image of E under the projection N ⟶ Σ_{n} and E′ the kernel of E ⟶ U. Then, E′ ⊂ (O(2)^{n})^{U}.

The group U acts on the set \underline{n} := \{1, ..., n\}. If this action has more than one orbit, U is subconjugated to the subgroup Σ_{k} × Σ_{n−k}, where k < n, and E is subconjugated to N_{k} × N_{n−k}. By induction hypothesis, rk_{2}(N_{k}) = 2k and rk_{2}(N_{n−k} = 2(n − k). Hence, the image of E ≅ E_{k} × E_{n−k} where E_{K} ⊂ N_{K} and E_{n−k} ⊂ N_{n−k} are elementary abelian subgroups of maximal rank. Again by induction hypothesis, E_{k} is conjugated to Δ_{k} and E_{n−k} to Δ_{n−k}, which proves the statement in this case.

If the action of U on \underline{n} produces exactly one orbit, then E′ ⊂ (O(2)^{n})^{U}) ≅ O(2) and E′ has dimension ≤ 2. Since rk_{2}(Σ_{n}) ≤ n/2, this implies that 2 + n/2 ≥ 2n and hence that n = 1, which also proves the statement. □

3. Proof of Theorem 1.4.

Let G ≅ ∏_{i} O(n_{i}) be a finite product of orthogonal groups and let X be a 2compact group which is N-isomorphic to G. In this section we want to prove a version of Proposition 2.2 for X. It will include Theorem 1.4.

Since X and G are N-isomorphic, we have the following diagram of inclusions

t_{G} ⟶ T_{G}
↓  ↓
E_{G} ⟶ N_{G} ⟶ X

We denote by i_{X} : E_{G} ⟶ X the inclusion given by the bottom line. As already mentioned, SX denotes the component of the unit of X.

3.1 Proposition.

(i) The subgroup E_{G} ⊂ X is self centralizing; i.e. C_{X}(E_{G}) = E_{G}. (ii) The mod-2 cohomology of BX is detected by elementary abelian 2-subgroups. (iii) Every elementary abelian 2-subgroup is subconjugated to E_{G}. (iv) \overline{W}_{X} = \overline{W}_{G}.

(v) H^{∗}(BX) ≅ H^{∗}(BE_{G})^{\overline{W}^{G}}.

The proof of this proposition splits into several parts and is worked out in the rest of this section. In particular, the parts (iii) and (iv) are proved via inductions, which themselves split into several steps.

3.2 Remark. According to the splitting of G = ∏_{i} O(n_{i}), the maximal torus normalizer N_{G} = ∏ N_{O(n_{i})} splits into a product. By [23], this implies a splitting X ≅ ∏_{i} X_{i} such that N_{O(n_{i})} ⊂ X_{i} is a maximal torus normalizer and such that the diagram

∏_{i} N_{O(n_{i})} ⟶^{≅} N_{G}
↓  ↓
∏_{i} X_{i}  ⟶^{≅} X

commutes. In particular, X_{i} and O(n_{i}) are N-isomorphic. Moreover, all statements of Proposition 3.1 also split in the same manner. Hence, on the way of proving the proposition, we always can assume that G = O(n) is an orthogonal group. This even is possible for the inductions necessary for the proofs of (iii) and (iv).

3.3 Proof of (i) and (ii): Part (ii) follows from Proposition 2.2(ii), since N_{G} is the maximal torus normalizer of X and since H^{∗}(BX) ⟶ H^{∗}(BN_{G}) is a monomorphism [8].

Since E_{G} is self centralizing in G, we have C_{N_{G}}(E_{G}) = E_{G}, too. By Proposition 2(iii), every monomorphism of E_{G} ⟶ N_{G} takes image in E_{G} ⊂ N_{G}. By [16], there exist a lift E_{G} ⟶ N_{G} of i_{X} : E_{G} ⟶ X such that C_{N}(E_{G}) ⊂ C_{X}(E_{G}) is the maximal torus normalizer. Hence, the maximal torus normalizer of C_{X}(E_{G}) is E_{G}. This implies that C_{X}(E_{G}) is discrete, therefore its own maximal torus normalizer, and congruent to E_{G}. This proves the first part. □

Before we continue with the proof we digress for the following lemma:

3.4 Lemma. Let E ⊂ T_{G} be an elementary abelian 2-subgroup. Then C_{G}(E) and C_{X}(E) are N-isomorphic.

Proof. By [9, 7.6], both C_{G} := C_{G}(E) and C_{X} := C_{X}(E) have the same Weyl groups and therefore the same maximal torus normalizer, namely C_{N} := C_{N_{G}}(E). We only have to show that the components of the unit are N-isomorphic.

Since E is a toral subgroup and therefore E ⊂ SX and since SC_{X}(E) is contained in SX, we have SC_{X}(E) = SC_{SX}(E). But the Weyl group of SC_{SX}(E) is completely determined by the Weyl group action of W_{SX} on the maximal torus T_{G} = T_{X} [9, 7.6]. Since SG and SX are N-isomorphic, and since the same holds for G, both centralizers, C_{G} and C_{X}, have N-isomorphic components of the unit. □

3.5 Claim. Proposition 3.1 (iii) is true for E ≅ ℤ/2.

Proof. We prove this via an induction over the order of the Weyl group of G. Because of Remark 3.2 we can assume that G = O(n). If G ≅ O(2), then X = N_{G} = G. And for O(2) all of Proposition 3.1 is true. This is the beginning of the induction.

Up to conjugation E ⊂ N_{G}. Since N_{G} = O(2) ≀ Σ_{k} × Θ, the generator of E is given by an element x = ((A_{1}, ..., A_{k}; σ), a) ∈ N_{G} with A_{i} ∈ O(2), σ ∈ Σ_{k} and a ∈ Θ. Since E ≅ ℤ/2, the permutation σ is of order two. Without loss of generality we can assume that σ = ∏_{j=1}^{r} τ_{2j−1,2j} is a product of commuting transpositions. Therefore x ∈ (O(2) ≀ ℤ/2)^{r} × O(2)^{k−2r} × Θ. If r = 0, then E obviously is subconjugated to E_{G}. If r ≥ 1 then T_{G}^{E} ≠ T_{G}^{Σ_{n}} and there exists a ℤ/2 ≅ E′ ⊂ (T_{G})^{E} which is neither central in N_{G} nor in G or in X. By Lemma 3.4, C_{G}(E′) and C_{X}(E′) are N-isomorphic. And by Proposition 2.8(viii), C_{G}(E′) is a product of orthogonal groups. Moreover, since E′ ⊂ T_{G} is not central in G, this centralizer has a smaller Weyl group than G. Since E_{G} and E are contained in C_{N_{G}}(E′) ⊂ X, E is subconjugated to E_{G} by induction hypothesis. □

3.6 Claim. Proposition 3.1 (iii) is true for any general elementary abelian 2-group E.

Proof. This time we make an induction over the order of E. The starting point of the induction is given by Claim 3.5. Again, we can assume that G = O(n). If the dimension of E is greater than 1, the composition E ⊂ X ⟶ π_{0}(X) has a non trivial kernel and we can split E ≅ E_{0} ⊕ E_{1} such that ℤ/2 ≅ E_{0} ⊂ SX. Since E_{0} is cyclic, it is subconjugated to T_{G}, in fact to t_{G}. The centralizers C_{X}(E_{0}) and C_{G}(E_{0}) are N-isomorphic (Lemma 3.4), and E_{G} ⊂ C_{X}(E_{0}). By induction hypothesis, E_{1} is subconjugated to E_{G} inside of C_{X}(E_{0}), which shows that E is subconjugated to E_{G} inside X. □

We make the map BE_{G} ⟶ BX into a fibration and denote the fiber by X/E_{G}. The inclusion i_{X} : E_{G} ⟶ X establishes a proxy action of E_{G} on X/E_{G} (see [8]) and the homotopy fixed point set (X/E_{G})^{E_{G}} fits into a fibration

(X/E_{G})^{E_{G}} ⟶ map(BE_{G}, BE_{G})_{lifts} ⟶ map(BE_{G}, BX)_{Bi_{X}} .

The total space is given by all self maps of BE_{G} which are lifts of Bi_{X}. Since E_{G} ⊂ X is self centralizing (Proposition 3.1 (i)) the fiber is homotopically discrete and the set of components \overline{W}_{X} := π_{0}((X/E_{G})^{E_{G}}) is a group, the Weyl group W_{X}(E_{G}) of E_{G} ⊂ X. Since π_{0}(map(BE_{G}, BE_{G}) = Hom(E_{G}, E_{G}), every element of \overline{W}_{X} is represented by an automorphism of E_{G}. This establishes an action of \overline{W}_{X} on E_{G} as well as on BE_{G}.

3.7 Claim. For G = O(n) the following holds:

(i) H^{1}(BX) ≅ ℤ/2. (ii) Let 0 ≠ d ∈ H^{1}(BX) represented by the map d : BX ⟶ Bℤ/2. Then, the composition BN_{G} ⟶ BX ⟶ Bℤ/2 is homotopic to the map induced by the determinant N_{G} ⊂ O(n) ⟶^{det} ℤ/2 where det denotes the determinat.

Proof. Since X and O(n) are N-isomorphic, there exists a commutative diagram

\xymatrix{ & π_{1}(BN_{G}) \ar[dr] \ar[dl] \\ π_{1}(BX) \ar[rr] &  & π_{1}(BO(n)) & .}

The Hurewicz map and universal coefficient theorems establish a commutative diagram

\xymatrix{ & H^{1}(BN_{G}) \ar[dr] \ar[dl] \\ H^{1}(BX) \ar[rr]^{≅} &  & H^{1}(BO(n)) & .}

This proves part (i). And part (ii) is a straightforward consequence of this diagram. □

3.8 Claim. \overline{W}_{G} ⊂ \overline{W}_{X}. Moreover there exist a chain of inclusions H^{∗}(BX) ⊂ H^{∗}(BE_{G})^{\overline{W}^{X}} ⊂ H^{∗}(BE_{G})^{\overline{W}^{G}}

Proof. Again we can assume that G = O(n). That is that \overline{W}_{G} = Σ_{n} For n ≥ 3 the two subgroups Σ_{2} × Σ_{n−2} and Σ_{n−2} × Σ_{2} generate Σ_{n}. The inclusions of both subgroups are induced by the Weyl group inclusion \overline{W}_{C_{G}(E)} ⊂ \overline{W}_{G} for an elementary abelian 2-subgroup ℤ/2 ≅ E ⊂ t_{G}. By Lemma 3.4, C_{G}(E) and C_{X}(E) are N-isomorphic. Now, the first part of the claim follows by an induction over the order of the Weyl group. And the last is then a consequence of part (ii) and (iii) of Proposition 3.1. □

The next claim is nothing but part (iv) of Proposition 3.1. The argument in the proof was pointed out to us by the referee. It simplifies and shortens our original proof.

3.9 Claim. \overline{W}_{X} = \overline{W}_{G}.

Proof. As usual we can assume that G = O(n). To prove the statement we only have to compare the orders of both groups. We consider the element x := (−1, −1, 1, ..., 1) ∈ (ℤ/2)^{n} ≅ E_{G}. Actually, this element is contained in T_{G} and generates the subgroup E in the proof of Claim 3.8. By induction, C_{X}(E) ≅ O(2) × O(n − 2). On the other hand we can calculate the mod-2 cohomology of BC_{X}(E) with the help of the Lannes’ T -functor [8]. Let α denote any of the compositions

H^{∗}(BX) ⟶ H^{∗}(BE_{G})^{\overline{W}^{X}} ⟶ H^{∗}(BE_{G})^{\overline{W}^{X}} ⟶ H^{∗}(BE_{G}) ⟶ H^{∗}(BE)

with target H^{∗}(BE). Let T_{α}^{E}( ) denote the componet of the T -functor associated to the morphism α [13]. And let I_{X} respectively I_{G} denote the isotropy group of E with respect to the action of \overline{W}_{X} or \overline{W}_{G} on E_{G}. Since the T -functor is exact and commutes, for finite groups, with taking invariants (e.g. see [25]) we get a sequence of monomorphisms

H^{∗}(BC_{X}(E)) ≅ T_{α}^{E}(H^{∗}(BX)) ⟶ H^{∗}(BE_{G})^{I_{X}} ⟶ H^{∗}(BE_{G})^{I_{G}}.

By induction hypothesis, C_{G}(E) ≅ O(2) × O(n − 2) as 2-compact groups. Since I_{G} = Σ_{2} × Σ_{n−2} = W_{O(2)×O(n−2)}, all monomorphism are isomorphism and Σ_{2} × Σ_{n−2} = I_{G} = I_{X}.

Now we look at the orbit of x under \overline{W}_{X}. Let x′ be an element in the orbit which has k cordinates equal to −1. Since x and x′ are conjugate in X both have the same determinant. This implies that k is even. Hence, since Σ_{n} ⊂ \overline{W}_{X}, x′ is conjugate to an element in T_{G} with the same number of −1’s. We can assume that x′ ∈ T_{G}. Since both are conjugate in X they are already conjugate in N_{G}. Therefore, there exist an element in W_{X} = W_{G} ⊂ Σ_{n} which transports x into x′. this implies that k = 2, that the \overline{W}_{X}-orbit of x is the same as the Σ_{n}-orbit, that both groups have the same order and that they are equal. □

Finally we have to prove part (v) of Proposition 3.1. We denote by A_{p} the Steenrod algebra and by K and U the categories of unstable algebras and unstable modules over A_{p}. For an object A^{∗} in K we denote by F (A^{∗}) the field of fractions. The action of A_{p} on A^{∗} can be extended to an action on F (A^{∗}) which, in general, is not unstable. We denote Un(F (A^{∗})) the unstable part of F (A^{∗}). This is again an object of K.

3.10 Claim. H^{∗}(BX) ≅ H^{∗}(BG).

Proof. By Proposition 3.1 (ii) and (iii), H^{∗}(BX) ⟶ H^{∗}(BE_{G}) is the Adams-Wilkerson embedding [1]. Let R^{∗} := H^{∗}(BX), A^{∗} := H^{∗}(BE_{G}) and D^{∗} ⊂ A^{∗} be the smallest Hopf-subalgebra containing R^{∗} which is also an algebra over the Steenrod algebra. By [6, 3.6], D^{∗} ≅ T_{i}^{E_{G}}(R^{∗}) ≅ H^{∗}(BC_{X}(E_{G}). Since E_{G} ≅ C_{X}(E_{G}) (part (i)), we have D^{∗} = A^{∗}.

Since A^{∗} is a finitely generated as an R^{∗}-module F (R^{∗}) ⊂ F (A^{∗}) is an algebraic extension. We can apply [27, Theorem II]. In our situation, it tells us that there exists a ”Galois group” W ⊂ Gl(E_{G}) such that R^{∗} ⊂ Un(F (R^{∗}) = (A^{∗})^{W}. Since the homotopy classes of maps BE_{G} ⟶ BX are in a one to one relation with the algebraic maps of algebras over the Steenrod algebra H^{∗}(BX) ⟶ H^{∗}(BE_{G}), the Galois group W can be identified with \overline{W}_{X} = \overline{W}_{G}. This establishes a chain of monomorphism

H^{∗}(BX) ≅ R^{∗} ⟶^{j} H^{∗}(BG) ≅ (A^{∗})^{W_{G}} ⟶ H^{∗}(BN_{G}) .

Since the homotopy fiber of BN_{G} ⟶ BX is 𝔽_{p}-finite and has Euler characteristique equal to 1 [8], there exists a transfer tr : H^{∗}(BN_{G}) ⟶ H^{∗}(BX) which is a U-map H^{∗}(BX)-linear. Moreover, tr j = id [5]. Let x ∈ H^{∗}(BG). Then, there exist classes a, b ∈ H^{∗}(BX) such that j(a) = x j(b). Applying the transfer gives tr(x) b = a and applying j yields j(tr(x)) j(b) = j(a). Since H^{∗}(BG) is a polynomial ring, this shows that j tr(x) = x and that j is an isomorphism. This finishes the proof of the claim and of Proposition 3.1. □

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Max-Planck-Institut für Mathematik, P.O.Box: 7280, D-53072 Bonn, Germany. e-mail : notbohm at mpim-bonn.mpg.de