ON THE 2-COMPACT GROUP DI(4)

D. Notbohm

Abstract. Besides the simple connected compact Lie groups there exists one further simple connected 2-compact group, constructed by Dwyer and Wilkerson, the group DI(4). The mod-2 cohomology of the associated classifying space BDI(4) realizes the rank 4 mod-2 Dickson invariants. We show that mod-2 cohomology determines the homotopy type of the space BDI(4) and that the maximal torus normalizer determines the isomorphism type of DI(4) as 2-compact group. We also calculate the set of homotopy classes of self maps of BDI(4).

1. Introduction.

As introduced by Dwyer and Wilkerson in their influential paper [9], a p-compact group is a pair (X, BX) of spaces such that X is 𝔽_{p}-finite, i.e. H^{∗}(X; 𝔽_{p}) is a finite vector space, BX is pointed, both spaces are p-complete (in the sense of [2]) and the loop space ΩBX and X are homotopy equivalent. This is a homotopy theoretic generalization of the concept of compact Lie groups. For a connected compact Lie group G, the pair (G_{p}^{∧}, BG_{p}^{∧}) together with the natural equivalence ΩBG_{p}^{∧} ≃ G_{p}^{∧} is a p-compact group, which, by abuse of notation, we also denote by G. For odd primes, there exist many connected p-compact groups which do not come from compact Lie groups, e.g. Sullivan showed that, if n divides p − 1, the sphere S^{2n−1} gives rise to a connected p-compact group ((S^{2n−1})_{p}^{∧}, B(S^{2n−1})_{p}^{∧}) [27]. At the prime 2, Dwyer and Wilkerson constructed the only known example DI(4) of a connected 2-compact group, which is not a Lie group [8]. The classifying space BDI(4) realizes the rank 4 mod-2 Dickson invariants, namely the ring of invariants of the natural action of Gl(4, 𝔽_{2}) on the rank 4 polynomial algebra H^{∗}((Bℤ/2)^{4}; 𝔽_{2}). In fact, they conjectured that every connected 2-compact group is a product of a connected compact Lie group and copies of DI(4). In this work we will approach this conjecture by showing that the 2-compact group DI(4) satisfies some uniqueness results. We also show that this conjecture is at least rationally true (see Theorem 1.5)

p-compact groups behave astonishingly similar as compact Lie groups. Every p-compact group X has a maximal torus T_{X} ⟶ X, a Weyl group W_{X}, a maximal torus normalizer N_{X} ⟶ X and a center Z(X) ⟶ X, all with basically the same properties as in the case of compact Lie groups. And two p-compact groups X and

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 55R35, 55P15.

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

Date: January 17, 2001.

Y are isomorphic if the classifying spaces BX and BY are homotopy equivalent. For further details about p-compact groups see the survey article [16] or the references mentioned there.

For p-compact groups, there are two classification conjectures around. A map between algebras over the Steenrod algebra is called a K-map.

1.1 Conjecture.

(i) Let X be a simply connected or centerfree p-compact group. A p-complete space A is equivalent to BX if and only if there exists a K-isomorphism H^{∗}(A; 𝔽_{p}) ≅ H^{∗}(BX; 𝔽_{p}). (ii) Two connected p-compact groups X and Y are isomorphic, i.e. the classifying spaces BX and BY are equivalent, if and only if the maximal torus normalizer N_{X} and N_{Y} are isomorphic.

At the prime two, both conjectures are proved for special unitary groups [5] [6]

[18] [21], for the exceptional Lie group G_{2} [30] and for the special orthogonal groups SO(2n + 1) [19] [24] [26]. For the spinor groups Spin(2n + 1) [24], for Sp(n) [28] and the exceptional Lie group F_{4} [29] only the second part is known.

We will show that both conjectures are also valid for the Dwyer-Wilkerson example DI(4).

1.2 Theorem. A 2-complete space A is homotopy equivalent to BDI(4) if and only if there exists a K-isomorphism H^{∗}(A; 𝔽_{2}) ≅ H^{∗}(BDI(4); 𝔽_{2}).

1.3 Theorem. A connected 2-compact group X is isomorphic two DI(4) if and only if the maximal torus normalizer N_{X} and N_{DI(4)} are isomorphic.

If H^{∗}(A; 𝔽_{2}) ≅ H^{∗}((Bℤ/2)^{4}; 𝔽_{2})^{Gl(4,𝔽_{2})}, standard methods show that ΩA is 𝔽_{2}-finite (see [8]). The pair (ΩA, A) gives rise to a 2-compact group. The next proposition shows that Theorem 1.2 and Theorem 1.3 are equivalent statements.

1.4 Proposition. For a 2-compact group X, the maximal torus normalizer N_{X} and N_{DI(4)} are isomorphic if and only if there exists a K-isomorphism H^{∗}(BX; 𝔽_{2}) ≅ H^{∗}(BDI(4); 𝔽_{2}).

Every p-compact group X has an associated p-adic lattice L_{X} := π_{1}(T_{X}) which carries an action of the Weyl group W_{X}. If X is connected, the representation W_{X} ⟶ Gl(L_{X} ⊗ ℚ) represents W_{X} as a pseudo reflection group, i.e. W_{X} is finite and generated by elements fixing a hyperplane of codimension 1. The above representation gives the rational Weyl group data of X. We say that two p-compact groups X and Y have the same rational Weyl group data if the two representations W_{X} ⟶ Gl(L_{X} ⊗ ℚ) and W_{Y} ⟶ Gl(L_{Y} × ℚ) are weakly isomorphic. That is there exists abstract isomorphism W_{X} ⟶^{≅} W_{Y} such that L_{X} ⊗ ℚ and L_{Y} ⊗ ℚ are isomorphic as ℤ_{p}^{∧}[W_{X}]-modules. The next theorem shows that the above mentioned conjecture of Dwyer and Wilkerson is rationally true.

1.5 Theorem. Every connected 2-compact group X splits into a product X ≅ X_{1} × X_{2} such that X_{1} is isomorphic to some copies of DI(4) and X_{2} has the same rational Weyl group data as a suitable connected compact Lie group G.

Finally we characterize the homotopy classes of self maps of BDI(4).

1.6 Theorem. Let i : T_{DI(4)} ⊂ DI(4) be a maximal torus of DI(4). Then, the following conditions are equivalent.

(i) f and g are homotopic. (ii) The compositions f ∘ Bi and g ∘ Bi are homotopic. (iii) H^{∗}(f; ℤ_{2}^{∧}) ⊗ ℚ = H^{∗}(g; ℤ_{2}^{∧}) ⊗ ℚ.

In fact, we will show that every self f : BDI(4) ⟶ BDI(4) is an Adams operation (see Section 6). That is there exists a k_{f} ∈ ℤ_{2}^{∧} such that f induces on H^{2i}(BDI(4); ℤ_{2}^{∧}) ⊗ ℚ multiplication by (k_{f})^{i}. The 2-adic integer k_{f} is called the degree of f. It will turn out that k_{f} is either 0 or a 2-adic unit, that all these values can be realized and that k_{f} determines f up to homotopy (Theorem 3.5).

The strategy for the proofs of Theorem 1.2 and Theorem 1.3 is clear and known to experts. It uses the construction of the classifying space BDI(4) of Dwyer and Wilkerson. They constructed BDi(4) as a homotopy colimit of a diagram based on the Rector category of H^{∗}((Bℤ/2)^{4}; 𝔽_{2})^{Gl(4,𝔽_{2})}, which is equivalent to the Quillen category of DI(4), namely the category of elementary abelian 2-subgroups of DI(4). One of the spaces in that diagram is the classifying space BSpin(7) of the 2-compact group Spin(7), which plays a major role in the construction. The missing gap for completing the proof is a uniqueness result for Spin(7) which is provided in [24]. Having this at hand we construct a map from the homotopy colimit approximating BDI(4) to the classifying space BX of a p-compact group satisfying Proposition 1.4. And this map will turn to be an equivalence after completion.

Remark: As Wilkerson pointed out to us, one can get away with a weaker assumptions in Theorem 1.3. If the Weyl groups of a 2-compact group X and DI(4) are isomorphic as abstract groups, then both have isomorphic normalizers as well as isomorphic mod-2 cohomology. In fact, this applies to several 2-compact groups. A detailed proof will appear in future work by Dwyer and Wilkerson.

There is also some overlap with work by Dwyer and Wilkerson. Proposition 1.4 and Theorem 1.6 are also known to them and were presented by Wilkerson at conference talks in Seattle and Orlando [32].

The paper is organized as follows. In Section 2 we recall the construction of BDI(4) of Dwyer and Wilkerson. In Section 3 we discuss self maps of DI(4), Section 4 contains the proof of Proposition 1.4, Section 5 the proofs of the theorems 1.2 and 1.3, and the last section the proof of Theorem 1.5.

We will use the language of p-compact groups all over the places. For references we refer the reader to the survey article [16] or the references mentioned there.

Mostly, cohomology is taken with 𝔽_{2} as coefficients. And H_{ℚ_{2}^{∧}}^{∗}(−) denotes the cohomology groups H^{∗}(−; ℤ_{2}^{∧}) ⊗ ℚ.

We thank the Max-Planck-Institut für Mathematik in Bonn for their hospitality when the main part of this work was done.

2. The construction of DI(4).

In this section we recall the construction of DI(4) and other material from [8]. The general linear group Gl(4, 𝔽_{2}) acts on the 4-dimensional 𝔽_{2}-vector space E_{4} = (𝔽_{2})^{4} and therefore on the algebra of formal polynomial functions 𝔽_{2}[E_{4}] on E_{4} with coefficients in 𝔽_{2}. That is the symmetric part of the tensor algebra of the dual of E. We can identify this algebra with the algebra H^{∗}(BE_{4}; 𝔽_{2}). The ring of invariants H^{∗}(BE_{4})^{Gl(4,𝔽_{2})} ≅ 𝔽_{2}[c_{8}, c_{12}, c_{14}, c_{15}] carries an action of the Steenrod algebra A_{2}. The indices denote the degree of the classes. The action is determined by Sq^{4}(c_{8}) = c_{12}, Sq^{2}(c_{12}) = c_{14} and Sq^{1}(c_{14}) = c_{15}.

The Bockstein spectral sequence collapses at the E_{2}-page and shows that H_{ℚ}^{∗}(BDI(4); ℤ_{p}^{∧}) ≅ ℚ_{p}^{∧}[x_{8}, x_{12}, x_{28}]. This is also true for any space A such that H^{∗}(A) ≅ H^{∗}(BE)^{Gl(4,𝔽_{2})}.

DI(4) is a 2-compact group of rank 3 with maximal torus T_{DI(4)} ≃ (S^{1})^{3} ⟶ DI(4). The Weyl group W_{DI(4)} ≅ ℤ/2 × Gl(3, 𝔽_{2}) arises as the pseudo reflection group No. 24 in the Clark-Ewing list of irreducible rational pseudo reflection groups [3]. The representation

W_{DI(4)} ≅ ℤ/2 × Gl(3, 𝔽_{2}) ⟶ Gl(L_{DI(4)}) ≅ Gl(3, ℤ_{2}^{∧})

maps the generator of ℤ/2 on −I where I is the identity. The composition W_{DI(4)} ⟶ Gl(3, ℤ_{2}^{∧}) ⟶ Gl(3, 𝔽_{2}) is the projection on the second factor [8, Theorem 5.1]. In particular L_{DI(4)}/2 := L_{DI(4)} ⊗ 𝔽_{2} is an irreducible W_{DI(4)}-module.

2.1 Lemma. Every W_{DI(4)}-lattice of L_{DI(4)} ⊗ ℚ is isomorphic to L_{DI(4)} as ℤ_{p}^{∧}[W_{DI(4)}]-module.

Proof. Let L ⊂ L_{DI(4)} ⊗ ℚ be a W_{DI(4)}-lattice. There exists a W_{DI(4)}-equivariant monomorphism L ⟶ L_{DI(4)} such that the cokernel K is a torsion group and such that rank of K/2 := K ⊗ 𝔽_{2} is smaller than 3. Tensoring with ℤ/2 yields an exact sequence

0 ⟶ T or(K, ℤ/2) ⟶ L/2 ⟶ L_{DI(4)}/2 ⟶ K/2 ⟶ 0

of W_{DI(4)}-modules. Since L_{DI(4)}/2 is irreducible, K/2 = 0, hence K = 0 and L ≅ L_{DI(4)}. □

Next we describe the Quillen category of DI(4). Since H^{∗}(BDI(4); 𝔽_{2}) ≅ H^{∗}(BE_{4}; 𝔽_{2})^{Gl(4,𝔽_{2})} every elementary abelian 2-subgroup E ⊂ DI(4) is subconjugated to E_{4} and every pair of subgroups E, E′ ⊂ E_{4} of the same dimension are conjugate in DI(4). Essentially, for i = 1, .., 4, there is exactly one elementary abelian 2-subgroup E_{i} ⊂ DI(4) of dimension i. The Quillen category is equivalent to the full subcategory of the following shape

[figure: {1}; Gl(2,𝔽_{2}); Gl(3,𝔽_{2}); Gl(4,𝔽_{2}); E_{1}; E_{2}; E_{3}; E_{4}]

where the groups above the objects denote the self maps and where ⟹ denotes the set of morphisms from E_{i} to E_{i+1}.

We have C_{DI(4)}(E_{4}) = E_{4}. And C_{3} := C_{DI(4)}(E_{3}) ≅ T_{DI(4)} ⋊ ℤ/2 where ℤ/2 acts on T_{DI(4)} via an Adams operation ψ^{−1} of degree −1, i.e. ψ^{−1} induces on π_{2}(BT_{DI(4)}) multiplication by −1. The group E_{3} is contained in T_{DI(4)} and therefore toral in DI(4) as well as E_{2} and E_{1}. The induced action of Gl(3, 𝔽_{2}) on T_{DI(4)} comes from the representation Gl(3, 𝔽_{2}) ⟶ Gl(3, ℤ_{2}^{∧}) given by restricting the 2-adic Weyl group representation W_{DI(4)} ⟶ Gl(3, ℤ_{2}^{∧}) to the second component. Moreover, C_{2} := C_{DI(4)}(E_{2}) ≅ SU(2)^{3}/Δ, where Δ is the image of the diagonal embedding of ℤ/2 ⟶ SU(2)^{3}. And C_{1} := C_{DI(4)}(E_{1}) ≅ Spin(7).

The decomposition diagram of BDI(4) given by the centralizers of the elementary abelian subgroups of DI(4) [7] [10] has the form

[figure: {1}; Gl(2,𝔽_{2}); Gl(3,𝔽_{2}); Gl(4,𝔽_{2}); BC_{1}; BC_{2}; BC_{3}; E_{4}; .]

Let Top denote the category of topological spaces and let Θ : A_{p}(DI(4)) ⟶ Top be the functor given by the above diagram. Then, evaluation at basepoints establishes a map

hocolim_{A_{p}(DI(4))} Θ ⟶ BDI(4)

2.2 Theorem. ([7] [8]) The map hocolim_{A_{p}(DI(4))} Θ ⟶ BDI(4) is a mod-2 equivalence.

The 2-compact groups C_{1} and Spin(7) are isomorphic. We will switch between them. For example, for the above decomposition diagram it is more appropriate to work with C_{1} since we have to think of it a s a centralizer, and for describing self maps of BDI(4) we prefer to work with Spin(7).

2.3 Remark. We do need some more detailed information about the subgroup Spin(7) ⊂ DI(4). Let E_{1} ⟶ E_{4} be the inclusion of the first coordinate. The isotropy subgroup Γ := Iso(E_{1}) ⊂ Gl(4, 𝔽_{2}) is the subgroup of all matrices of the

form ( \begin{matrix}1 & ∗ \\ 0 & A \end{matrix} ) such that A ∈ Gl(3, 𝔽_{2}). The classifying space BC_{DI(4)}(E_{1}) ≃

BSpin(7)_{2}^{∧} has mod-2 cohomology

H^{∗}(BSpin(7); 𝔽_{2}) ≅ H^{∗}(BC_{DI(4)}(E_{1}); 𝔽_{2}) ≅ H^{∗}(BE_{4}; 𝔽_{2})^{Γ} .

This ring of invarinats is a polynomial algebra and H^{∗}(BSpin(7); 𝔽_{2}) ≅ 𝔽_{2}[d_{4}, d_{6}, d_{7}, d_{8}] where subindices denote degrees. In particular, H^{∗}(BDI(4); 𝔽_{2}) ⟶ H^{∗}(BSpin(7); 𝔽_{2}) is a monomorphism. Moreover, since E_{1} is subconjugated to T_{DI(4)}, Spin(7) ⊂ DI(4) is a subgroup of maximal rank and W_{Spin(7)} ⊂ W_{DI(4)}. The Euler characteristique χ(DI(4)/Spin(7)) of the homotopy fiber of BSpin(7) ⟶ BDI(4) is the index of [W_{DI(4)} : W_{Spin(7)}] of the Weyl groups. The order of W_{DI(4)} is 336, the order of W_{Spin(7)} is 48 and χ(DI(4)/Spin(7)) = 7. We explained this calculation in detail since, unfortunately, there is is misprint in [8, Theorem 1.8]. This calculation always works for subgroups of p-compact groups of maximal rank [10] [17].

The maximal torus normalizer N_{DI(4)} contains the subgroup T_{DI(4)} ⋊ ℤ/2 ≅ C_{DI(4)}(E_{3}). And therefore a unique subgroup E_{4} ⊂ T_{DI(4)} ⋊ ℤ/2 ⊂ N_{DI(4)}. Since Spin(7) ≅ C_{DI(4)}(E_{1}) and since E_{1} ⊂ T_{DI(4)}, the Weyl group W_{Spin(7)} ⊂ W_{DI(4)} is the isotropy subgroup of E_{1} for the W_{DI(4)}-action on T_{DI(4)} [10].

We finish this section with a some statements necessary for later purpose.

2.4 Proposition. For any elementary abelian 2-group E, the map [BE, BE_{4}] ⟶ [BE, BDI(4)] factors through the bijection [BE, BE_{4}]/Gl(4, 𝔽_{2}) ⟶ [BE, BDI(4)].

Proof. Since H^{∗}(BDI(4); 𝔽_{2}) ≅ H^{∗}(BE_{4}; 𝔽_{2})^{Gl(4,𝔽_{2})}, this statement is well known. It follows from facts about Lannes’ T -functor, namely that H^{∗}(map(BE, BX); 𝔽_{2}) ≅ T^{E}(H^{∗}(BDI(4); 𝔽_{2})) and that

T^{E}(H^{∗}(BE_{4}; 𝔽_{2})^{Gl(4,𝔽_{2})} ≅ (T^{E}(H^{∗}(BE_{4}; 𝔽_{2})))^{Gl(4,𝔽_{2})} ≅ H^{∗}(map(BE, BE_{4}); 𝔽_{2})^{Gl(4,𝔽_{2})}

[14] [9] (for the latter sequence of isomorphisms see also [8]). □

2.5 Lemma. There exist exactly one monomorphis R^{∗} := H^{∗}(BDI(4); 𝔽_{2}) ⟶ S^{∗} := H^{∗}(BSpin(7); 𝔽_{2}) of algebras over the Steenrod algebra, such that S becomes a finitely generated R-module.

Proof. Let A^{∗} := H^{∗}(BE_{4}; 𝔽_{2}) and E_{4} ⊂ Spin(7) the standard inclusion. This inclusion induces a monomorphism S^{∗} ⟶ A^{∗}. Therefore it suffices to show the statement if we replace S^{∗} by A^{∗}. Let α : R^{∗} ⟶ A^{∗} be the map induces by the inclusion BE_{4} ⟶ BDI(4). Let β : R^{∗} ⟶ A^{∗} be any map which makes A^{∗} to a finitely generated module over R^{∗}. In particular, β is a monomorphism. Then there exists an automorphism γ : A^{∗} ⟶ A^{∗} such that βγ = α. Since Gl(4, 𝔽_{2}) is the Galois group of the integral extension R^{∗} ⟶ A^{∗}, the maps α and β have the same image. □

3. Self maps of BDI(4).

In this section we want to prove Theorem 1.6. We use the same notation as in Section 2.

3.1 Remark. Every elementary abelian 2-subgroup E ⊂ DI(4) of rank k, 1 ≤ k ≤ 4, is conjugate to the inclusion E_{k} ⊂ E_{4} ⊂ DI(4). We denote by i_{k} : BC_{k} := BC_{DI(4)}(E_{k}) ⟶ BDI(4) the inclusion of the centralizer of E_{k}. Let ϕ : C_{k} ⟶ DI(4) be a monomorphism. Since E_{4} ⊂ C_{k}, there exists a homomorphism ρ : E_{4} ⟶ E_{4} such that the diagram

E_{4} ⟶^{⊂} C_{k}
ρ↓  ^{ϕ}↓
E_{4} ⟶^{⊂} DI(4)

commutes up to conjugation (Proposition 2.4). Actually, ρ is a monomorphism, because ϕ is one [9], and therefore an isomorphism. Since the Weyl group of E_{4} ⊂ DI(4) is Gl(4; 𝔽_{2}) we can assume that ρ = id and that the restriction ϕ|_{E_{k}} is the inclusion E_{k} ⊂ DI(4). Passing to centralizers establishes a monomorphism ϕ′ : C_{k} ≅ C_{C_{k}}(E_{k}) ⟶ C_{DI(4)}(E_{k}) = C_{k}. In fact, this is an isomorphism [17].

Because E_{k} ⊂ C_{k} is central there exists a homomorphism μ : C_{k} × E_{k} ⟶ C_{k}. By construction, Bϕ′ : BC_{k} ⟶ map(BE_{k}, BDI(4))_{Bϕ|_{BE_{k}}} ≃ BC_{k} is the adjoint of

the composition BC_{k} × BE_{k} ⟶^{Bμ} BC_{k} ⟶^{Bϕ} BDI(4) and an equivalence. If i = 1 or 2, then C_{k} is connected and E_{k} ⊂ C_{k} is subconjugated to the maximal torus of C_{k}.

All this together establishes the assumptions made in Proposition 4.6 of [23]. The conclusion we can draw is the following. If Bψ : BC_{k} ⟶ BDI(4) is another map such that H_{ℚ}^{∗}(Bϕ) = H_{ℚ}^{∗}(Bψ), then Bϕ and Bψ are homotopic.

3.2 Lemma. A map g : BSpin(7)_{2}^{∧} ⟶ BDI(4) is either homotopic to the constant map or there exists a self equivalence h : BSpin(7)_{2}^{∧} ⟶ BSpin(7)_{2}^{∧} such that Bi_{1} ∘ h ≃ g.

Proof. Let ϕ : Spin(7) ⟶ DI(4) be the underlying homomorphism such that g ≃ Bϕ. The homomorphism ϕ is either conjugate to the constant map or the kernel of ϕ is a central subgroup of Spin(7) [20], hence a subgroup of ℤ/2. If the kernel is the trivial group, then ϕ is a monomorphism. If the kernel equals ℤ/2, then ϕ factors through a monomorphism \overline{ϕ} : SO(7) = Spin(7)/(ℤ/2) ⟶ DI(4) [20]. This gives a contradiction, since SO(7) contains an elementary abelian 2-group of rank 6 and DI(4) does not. Therefore, ϕ is either the constant map or a monomorphism. In the latter case, the existence of h follows from Remark 3.1. □

3.3 Proposition. Let f : BDI(4) ⟶ BDI(4) be a self map. Then the following holds:

(i) The map f is either homotopic to the constant map or an equivalence. (ii) There exists a self map h : BSpin(7)_{2}^{∧} ⟶ BSpin(7)_{2}^{∧} such that the diagram

BSpin(7)_{2}^{∧} ⟶^{h} BSpin(7)_{2}^{∧}
Bi_{1}↓  ^{Bi_{1}}↓
BDI(4) ⟶^{f} BDI(4)

commutes up to homotopy.

Proof. If f ∘ Bi_{1} is nullhomotopic we can choose the constant map for h. Hence, the second part is a consequence of Lemma 3.2.

The map h is either null homotopic or an equivalence [13]. If h is an equivalence, then f induces a monomorphism in mod-2 cohomology, therefore an isomorphism, and is an equivalence.

Since the Weyl group index [W_{DI(4)} : W_{Spin(7)}] is odd, Spin(7) and DI(4) have the same 2-Sylow subgroup P ⊂ Spin(7) ⊂ DI(4). thus, if h is null homotopic, the restriction f|_{BP} is null homotopic and so is f itself [22] □

For a p-compact group X, we call a self map f : BX ⟶ BX an Adams operation of degree k ∈ ℤ_{p}^{∧} if, for any class x ∈ H_{ℚ_{p}^{∧}}^{2i}(BX), f^{∗}(x) = k^{i}x. By ℤ_{2}^{∧}^{∗} we denote the units of ℤ_{2}^{∧}.

3.4 Corollary. Every self equivalence of BDI(4) is an Adams operation whose degree is a 2-adic unit.

Proof. . Every self equivalence of BSpin(7)_{2}^{∧} has this property [13]. Since H_{ℚ_{2}^{∧}}^{∗}(BDI(4)) ⟶ H_{ℚ_{2}^{∧}}^{∗}(BSpin(7)) is a monomorphism, the statement is a consequence of Proposition 3.3. □

Considering the constant map as an Adams operation of degree 0, the above corollary establishes a map

D : [BDI(4), BDI(4)] ⟶ {0} ∐ ℤ_{2}^{∧}^{∗} .

3.5 Theorem. The map D : [BDI(4), BDI(4)] ⟶ {0} ∐ ℤ_{2}^{∧}^{∗} is a bijection.

We postpone the proof of Theorem 3.5 to the end of this section.

Proof of Theorem 1.6. If, for two maps f, g : BDI(4) ⟶ BDI(4), the restrictions to BT_{DI(4)} are homotopic, then both induce the same map in rational cohomology, are Adams operations of the same degree and therefore homotopic by Theorem 3.5. □

As in Section 2 we denote by A := A_{p}(DI(4)) the Quillen category of DI(4) and by Θ := Θ_{DI(4)} : A ⟶ Top the functor describing the Dwyer-Wilkerson decomposition. Since hocolim_{A} Θ ⟶ BDI(4) induces an equivalence after 2-adic completion (Theorem 2.2) we get an equivalence

map(BDI(4), BDI(4)) ≃ holim_{A} map(Θ, BDI(4)) .

The equivalence establishes a map

[BDI(4), BDI(4)] ⟶ lim_{\overleftarrow{A}} [Θ, BDI(4)] .

For ϕ ∈ lim_{\overleftarrow{A}} [Θ, BDI(4)] and an object E ⊂ X of A we denote by

map(BC(E), BDI(4))_{ϕ} the component determined by ϕ and by

map(BDI(4), BDI(4))_{ϕ} := holim_{A} map(Θ, BDI(4))_{ϕ}

the union of all components whose image is ϕ.

The homotopy groups of the last space can be calculated with the Bousfield-Kan spectral sequence [2]. The E^{2}-term has the form

E_{i,j}^{2} ≅ lim_{\overleftarrow{A}}^{i} π_{j}(map(B̃C, BDI(4))_{ϕ} =: lim_{\overleftarrow{A}}^{i} Π_{j}(ϕ) .

If ϕ is the image of the constant map, then, for all E_{k},

map(BC_{DI(4)}(E_{k}), BDI(4))_{ϕ} ≃ BDI(4)

and the functor Π_{j}(ϕ) is the constant functor. If ϕ is the image of a self equivalence f : BDI(4) ⟶ BDI(4), then composition with f induces a natural equivalence

Π_{j} := Π_{j}(id) ⟶^{≅} Π_{j}(ϕ) .

Since for any abelian p-compact group A and any p-compact group X

BZ(C_{X}(A)) ≃ map(BC_{X}(A), BC_{X}(A))_{id} ≃ map(BC_{X}(A), BX)_{Bi}

where i denotes the inclusion of the centralizer [10] [23], the functor Π_{j} is naturally equivalent to the inclusion functor J : A ⟶ Ab into the category of abelian groups.

In both cases the higher derived limits all vanish [8], the Bousfield spectral sequence collapses and is concentrated in a singular line. This allows to prove the following proposition:

3.7 Proposition.

(i) The map [BDI(4), BDI(4)] ⟶ lim_{\overleftarrow{A}} [Θ, BDI(4)] is a bijection

(ii) For all non trivial self maps f : BDI(4) ⟶ BDI(4), the mapping space map(BDI(4), BDI(4))_{f} is contractible, and map(BDI(4), BDI(4))_{const} ≃ BDI(4).

Proof. The first part follows from the above considerations. They also show that, for all non trivial self maps f : BDI(4) ⟶ BDI(4), map(BDI(4), BDI(4))_{id} ≃ map(BDI(4), BDI(4))_{f}. The first mapping space is equivalent to the classifying space of the center of DI(4) [10], which is trivial [8]. The component of the constant map is calculated in [10]. □

Proof of Theorem 3.5. Let A′ ⊂ A denote the full subcategory whose objects are elementary abelian 2-subgroups of DI(4) of rank less or equal to 2. Then, the results of [25] say that the second arrow of

[BDI(4), BDI(4)] ⟶^{≅} lim_{\overleftarrow{A}} [Θ, BDI(4)] ⟶ lim_{\overleftarrow{A′}} [Θ, BDI(4)]

is an isomorphism. The first isomorphism follows from Proposition 3.7. We denote by i_{E} : C_{DI(4)}(E) ⟶ DI(4) the inclusion of the centralizer of E.

We first show that D is an injection. The degree of a self map of BDI(4) is determined by the the induced map in rational cohomology. Let f, g : BDI(4) ⟶ BDI(4) be two self maps. Since a self map is null homotopic if and only if it induces the trivial map in rational cohomology [15], the map f is null homotopic if and only if g is. Moreover, if f is not null homotopic it is an equivalence as well as g. The compositions f ∘ Bi_{E} and g ∘ Bi_{E} induce the same map in rational cohomology. If the rank of E is less or equal to 2, then Remark 3.1 shows that both compositions are actually homotopic. By Proposition 3.6, this implies that f ≃ g. Hence, D is an injection.

The degree 0 is realized by the constant map. For k ∈ ℤ_{2}^{∧}^{∗} and E ⊂ DI(4) of rank ≤ 2, the centralizer C_{DI(4)}(E) is the completion of a connected compact Lie group. There exists an Adams operation f′_{E} : BC_{DI(4)}(E) ⟶ BC_{DI(4)}(E) of degree k [13]. Let f_{E} := Bi_{E} ∘ f′_{E} : BC_{DI(4)}(E) ⟶ BDI(4) denote the composition. We want to show that the tuple (f_{E})_{E∈A′} gives an element of lim_{\overleftarrow{A′}} [Θ, BDI(4)]. That is that

for every morphism E_{0} ⟶ E_{1} the composition BC(E_{1}) ⟶ BC(E_{0}) ⟶^{f_{E_{0}}} BDI(4) is homotopic to f_{E_{1}}. By construction, the composition and f_{E_{1}} induce the same map in rational cohomology. Remark 3.1 implies that both maps are homotopic. Therefore, (f_{E})_{E∈A′} ∈ lim_{\overleftarrow{A′}} [Θ, BDI(4)] and D is surjective (Proposition 3.7), which

finishes the proof. □

4. The proof of Proposition 1.4.

First we assume that X is a 2-compact group, such that

H^{∗}(BX; 𝔽_{2}) ≅ H^{∗}(BDI(4); 𝔽_{2}) ≅ H^{∗}(BE_{4}; 𝔽_{2})^{Gl(4,𝔽_{2})} .

4.1 Lemma.

(i) Every elementary abelian 2-subgroup E ⊂ X is subconjugated to E_{4} ⊂ X. (ii) The 2-compact group X is connected, centerfree and simple. In particular, the rational Weyl group representation W_{X} ⟶ Gl(L_{X} ⊗ ℚ) is irreducible. (iii) X and DI(4) have the same rational Weyl group data.

Proof. The K-map H^{∗}(BX; 𝔽_{2}) ⟶ H^{∗}(BE_{4}; 𝔽_{2}) can be realized by a topological map BE_{4} ⟶ BX [14] which makes E_{4} ⊂ X to a subgroup of X. Moreover, since H^{∗}(BE; 𝔽_{2}) is an injective object in the category K of algebras over the Steenrod algebra, every map BE ⟶ BX factors through BE_{4}. This proves (i).

For p-compact groups, the T -functor calculates the cohomology of classifying spaces of centralizers of elementary abelian p-subgroups [9]. Since the T -functor commutes with taking invariants, a central subgroup E ⊂ E_{4} ⊂ X has to be fixed under the action of Gl(4, 𝔽_{2}). There are no nontrivial fixed points, and therefore X is centerfree.

As a connected centerfree 2-compact group, X splits into a product of simple centerfree 2-compact groups [11]. A non trivial splitting of X would establish a non trivial splitting of H^{∗}(BX; 𝔽_{2}) into a tensor product. But such a splitting does not exist for H^{∗}(BE_{4}; 𝔽_{2})^{Gl(4,𝔽_{2})}. Hence, X is simple. Since, for p-compact groups, simple just means that the associated rational Weyl group is irreducible [11] [22], the last fact of (ii) is obvious.

As already mentioned in Section 2, the A_{2}-action on H^{∗}(BX; 𝔽_{2}) implies that H_{ℚ_{2}^{∧}}^{∗}(BX) ≅ ℚ_{2}^{∧}[y_{8}, y_{12}, y_{28}] ≅ H_{ℚ_{2}^{∧}}^{∗}(BDI(4)). Since H_{ℚ_{2}^{∧}}^{∗}(BX) ≅ H_{ℚ_{2}^{∧}}^{∗}(BT_{X})^{W_{X}} [9], the 2-compact groups X and DI(4) have the same rank and the Weyl groups W_{X} and W_{DI(4)} the same degrees. Looking at the classification of all irreducible rational pseudo reflection groups [3] shows that both Weyl groups are weakly isomorphic. That is that X and DI(4) have the same rational Weyl group data. □

Proof of Proposition 1.4 (first half): Let X be a 2-compact group, such that H^{∗}(X; 𝔽_{2}) ≅ H^{∗}(BDI(4); 𝔽_{2}). We want to show that both have isomorphic maximal torus normalizer. By Lemma 4.1, we can identify W_{X} and W_{DI(4)} via the rational Weyl group data. Then L_{X} and L_{DI(4)} are W_{DI(4)}-lattices of L_{DI(4)} ⊗ ℚ. By Lemma 2.1, both lattices are isomorphic. That is we can identify BT_{X} and BT_{DI(4)} as W_{DI(4)}-spaces. Since for every connected centerfree p-compact group the maximal torus normalizer splits [11] we have BN_{X} ≃ B(T_{X}⋊W_{X}) ≃ B(T_{DI(4)}⋊ W_{DI(4)}) ≃ BN_{DI(4)} which proves the claim. □

4.3 Remark. The last arguments in the above proof show, that the maximal torus normalizer of a 2-compact group X and DI(4) are isomorphic if and only if they have the same rational Weyl group data.

Now we prove the other half of Proposition 1.4, which is nothing but Part (vi) of the following proposition.

4.4 Proposition. For a 2-compact group X, such that N_{X} and N_{DI(4)} are isomorphic, the following holds:

(i) The 2-compact group X is connected. (ii) There exists a monomorphism Spin(7) ⟶ X such that Spin(7) is a subgroup of maximal rank. (iii) The Euler characteristique χ(X/Spin(7)) is odd. (iv) The map j^{∗} : H^{∗}(BX; 𝔽_{2}) ⟶ H^{∗}(BSpin(7); 𝔽_{2}) is a monomorphism.

(v) The subgroup E_{4} ⊂ N_{DI(4)} ⊂ X is self centralizing and

H^{∗}(BX; 𝔽_{2}) ≅ H^{∗}(BE_{4}; 𝔽_{2})^{\overline{W}^{X}} ,

where \overline{W}_{X} is the Weyl group of E_{4} ⊂ X. (vi) H^{∗}(BX; 𝔽_{2}) ≅ H^{∗}(BDI(4); 𝔽_{2}) as algebras over the Steenrod algebra.

Here, the Weyl group \overline{W}_{X} is the group of all elements w ∈ Gl(4, 𝔽_{2}) such that the inclusion i_{X} : E_{4} ⟶ X and the composition i_{x}w are conjugate in X.

Proof. By [9], for any p-compact group Y, the map W_{Y} = π_{0}(N_{Y}) ⟶ π_{0}(Y) is an epimorphism and the kernel is the Weyl group of the component Y_{0} of the unit. Hence, there exists an epimorphism W_{X} = ℤ/2 × Gl(3, 𝔽_{2}) ⟶ π_{0}(X). The target is a 2-group and the kernel W_{0} is the Weyl group of X_{0}. In particular, since X and X_{0} have both rank 3, the rational representation, W_{0} ⟶ Gl(3, ℚ_{2}^{∧}) represents W_{0} as a pseudo reflection group. The order of W_{0} divides 336 = 2^{4} · 3 · 7 which is the order of W_{DI(4)}. We have to figure out the pseudo reflection groups such that the product of the degrees is divisible by 3 and 7. Only the group No. 24 satisfies this property, which is the W_{DI(4)}. Hence, π_{0}(X) is trivial and X connected. This proves part (i).

The maximal torus normalizer N_{X} ≅ N_{DI(4)} contains a unique subgroup E_{4} ⊂ N_{DI(4)} (Remark 2.3). Let E_{1} ⊂ E_{4} be the inclusion of the first coordinate. Since the Weyl group of C_{X}(E_{1}) and of the component of the unit C_{X}(E_{1})_{0} can be read off from the action of W_{X} on T_{X} [10, 7.6], the centralizers C_{X}(E_{1}) and C_{DI(4)}(E_{1}) have isomorphic maximal torus normalizer as well as C_{X}(E_{1})_{0} and C_{DI(4)}(E_{1})_{0}. Since Spin(7) ≅ C_{DI(4)}(E_{1}) is connected, C_{X}(E_{1}) and C_{X}(E)_{0} have isomorphic Weyl groups. Hence, C_{X}(E_{1}) is connected and has the same maximal torus normalizer as Spin(7). By [24], this implies that both are isomorphic, which proves part (ii).

The subgroup Spin(7) ⊂ X is of maximal rank. As explained in Remark 2.3, the Euler characteristique of X/Spin(7) equals the Weyl group index [W_{X} : W_{Spin(7)}] = [W_{DI(4)} : W_{Spin(7)}] which is odd (Remark 2.3). This proves Part (iii).

There exists a transfer tr : H^{∗}(BSpin(7); 𝔽_{2}) ⟶ H^{∗}(BX; 𝔽_{2}) such that the composition with the restriction map j^{∗} : H^{∗}(BX; 𝔽_{2}) ⟶ H^{∗}(BSpin(7); 𝔽_{2}) is multiplication by χ(X/Spin(7) [4]. Since this is an odd number, the composition is the identity and the map j^{∗} of (iv) a monomorphism as desired. We also notice that the transfer is a linear map of H^{∗}(BX)-modules.

Since E_{4} centralizes E_{1}, we have E_{4} ⊂ Spin(7). And since E_{1} ⊂ E_{4}, we have C_{X}(E_{4}) ≅ C_{Spin(7)}(E_{4}) ≅ E_{4}. The last identity follows because E_{4} ⊂ Spin(7) is self centralizing.

Let R^{∗} := H^{∗}(BX; 𝔽_{2}), S^{∗} := H^{∗}(BSpin(7); 𝔽_{2}) and A^{∗} := H^{∗}(BE_{4}; 𝔽_{2}). Then the composition i^{∗} : R^{∗} ⟶ S^{∗} ⟶ A^{∗} is a monomorphism and the Adams-Wilkerson embedding. Let D^{∗} ⊂ A^{∗} b the smallest Hopf algebra containing R^{∗} which is also an algebra over the Steenrod algebra. By [6, 3.6], we have D^{∗} = T_{i^{∗}}^{E_{4}}(R^{∗}) = H^{∗}(BC_{X}(E_{4}); 𝔽_{2}) = A^{∗}. The last identity follows since E_{4} ⊂ X is self centralizing.

Since A^{∗} is a finitely generated R^{∗}-algebra, the fields of fractions F (R^{∗}) ⊂ F (A^{∗}) establish an algebraic extension. We can apply [31, Theorem II]. In or situation it tells us that there exists a ‘Galois’ group W ⊂ Gl(E_{4}) such that R^{∗} ⊂ Un(F (R^{∗}) = (A^{∗})^{W} ⊂ F (R^{∗}), where Un(F (R^{∗}) denotes the unstable part of F (R^{∗}). Since the homotopy classe of maps BE_{4} ⟶ BX can be identified with the maps H^{∗}(BX; 𝔽_{2}) ⟶ H^{∗}(BE_{4}; 𝔽_{2}) of algebras over the Steenrod algebra, W = \overline{W}_{X} is the Weyl group of E_{4} ⊂ X. This establishes a chain of monomorphism

R^{∗} ⟶ (A^{∗})^{\overline{W}^{X}} ⟶ (A^{∗})^{Γ} = S^{∗} ,

where Γ, as explained in Section 2, denotes the Weyl group of E_{4} ⊂ Spin(7). Let x ∈ (A^{∗})^{\overline{W}^{X}}. Then, x = a/b with a, b ∈ R^{∗} and b · tr(x) = tr(bx) = tr(a) = a. since we are doing calculations in a subring of a polynomial ring this shows that tr(x) ∈ R^{∗} is divisible by b, that R^{∗} ⟶ (A^{∗})^{\overline{W}^{X}} is an epimorphism and that R^{∗} ≅ (A^{∗})^{\overline{W}^{X}}. This proves part (v).

Finally we have to prove part (vi). By construction, Γ ⊂ \overline{W}_{X} ⊂ Gl(4, 𝔽_{2}). The first inclusion is proper. Otherwise, the map BSpin(7)_{2}^{∧} ⟶ BX induces an isomorphism in mod-2 cohomology and is an equivalence. This is a contradiction, since Spin(7) and X have non isomorphic maximal torus normalizer, and are therefore non isomorphic [17].

Lemma 4.5 below shows that Γ ⊂ Gl(4, 𝔽_{2}) is a maximal subgroup. This implies that \overline{W}_{X} = Gl(4, 𝔽_{2}) and that H^{∗}(BX; 𝔽_{2}) ≅ H^{∗}(BDI(4); 𝔽_{2}). This proves the last part and finishes the proof. □

The stabilizer subgroup of a subspace U′ ⊂ U := (𝔽_{p})^{n} is the subgroup of Gl(n, 𝔽_{p}) of all elements which map U′ onto itself. For p = 2 and for a 1-dimensional subspace, the isotropy subgroup and the stabilizer subgroup are equal.

4.5 Lemma. For any prime p, the stabilizer subgroup Γ ⊂ Gl(n, 𝔽_{p}) of any 1dimensional subspace of U is a maximal subgroup.

Proof. We can assume that Γ is the stabilizer subgroup of the subspace generated by the first standard basis vector e_{1}. We write every element of B ∈ Gl(n, 𝔽_{p}) in

the form B = ( \begin{matrix}μ & c \\ b & M \end{matrix} ), where μ ∈ 𝔽_{p}, b and c have n − 1 components and where

M is a (n − 1) × (n − 1) matrix. For elements of Γ, we have μ ≠ 0, b = 0 and M ∈ Gl(n − 1, 𝔽_{p}).

Let Γ ⫋ Γ′ ⫅ Gl(n, 𝔽_{p}). We have to show that Γ′ = Gl(n, 𝔽_{p}). Since Γ \ Gl(n, 𝔽_{p}) ≅ U′ := U \ {0}, the quotient Γ′ \ Gl(n, 𝔽_{p}) is a quotient of U′. Therefore, it is sufficient to show that Γ′ acts transitively on U′.

Let B = ( \begin{matrix}μ & c \\ b & M \end{matrix} ) ∉ Γ; i.e. b ≠ 0. Multiplying B by elements of Γ from the

left shows that, for every vector u ∈ U, there exists a matrix B′ ∈ Γ′ such that the first column (μ′, b′) equals u. Thus, the Γ′-orbit of e_{1} contains all vectors of U′ and Γ′ acts transitively on U′. □

5. Proof of the theorems 1.2 and 1.3.

Because of Proposition 1.4, both theorems are a consequence of the following statement.

5.1 Theorem. Let X be a 2-compact group, , such that X and DI(4) have isomorphic maximal torus normalizer and such that H^{∗}(BX; 𝔽_{2}) ≅ H^{∗}(BDI(4); 𝔽_{2}) (as algebras over the Steenrod algebra). Then, X and DI(4) are isomorphic as 2-compact groups.

The rest of this section is devoted to the proof of Theorem 5.1. And X will be always denote a 2-compact group satisfying the assumptions of the theorem. Before we start with the proof we fix some notation, which partly was already introduced in Section 2 and Section 3. Any compact Lie groups G appearing in the decomposition diagram of BDI(4) is the centralizer C_{DI(4)}(E_{k}) =: C_{k} of a k-dimensional elementary abelian 2-subgroup of DI(4). Actually, for all k, C_{k} is already contained in C_{1} ≅ Spin(7). We denote by j_{k} : BC_{k} ⟶ BC_{1} and j_{k} : BC_{k} ⟶ BDI(4)4 the maps associated to these inclusions. In particular, i_{1} : BC_{1} ⟶ BDI(4) comes from the standard inclusion C_{1} ⊂ DI(4). We denote by T the maximal torus of C_{1} and denote by j_{T} : BT ⟶ BC_{1} the asoociated map of the inclusion. The composition T ⊂ C_{1} ⊂ DI(4) is also a maximal torus and establishes the map i_{T} : BT ⟶ BDI(4). By W := W_{DI(4)} we denote the Weyl group of DI(4) and by L_{T} := π_{1}(T) the ‘dual weight lattice’.

5.2 Lemma.

(i) There exist a monomorphism f_{1} : BC_{1} ⟶ BX and an isomorphism H^{∗}(BX; 𝔽_{2}) ≅ H^{∗}(BDI(4); 𝔽_{2}) such that the diagram

\xymatrix{ & H^{∗}(BC_{1}; ℤ/2) \\ H^{∗}(BX; 𝔽_{2}) \ar[rr]^{≅} \ar[ur]^{f_{1}^{∗}} &  & H^{∗}(BDI(4); 𝔽_{2}) \ar[ul]_{i_{1}^{∗}}}

commutes. (ii) The composition T ⟶^{j_{T}} BC_{1} ⟶^{f_{1}} X is a maximal torus and the associated Weyl group representation W_{X} ⟶ Gl(L_{T}) is weakly isomorphic to the Weyl group representation W ⟶ Gl(L_{T}).

Proof. The composition of inclusions E_{1} ⊂ E_{4} ⊂ T ⋊ ℤ/2 ≅ C_{2} ⊂ N_{DI(4)}4(T) ⊂ X factors through the maximal torus T. By Proposition 4.3, C_{X}(E_{1}) ≅ Spin(7) ≅ C_{1} which establishes the monomorphism f_{1} : BC_{1} ⟶ BX. This map induces a monomorphism f_{1}^{∗} : H^{∗}(BX; 𝔽_{2}) ⟶ H^{∗}(BC_{1}; 𝔽_{2}) in mod-2 cohomology (Proposition 4.3) and makes the target to a finitely generated module over the source. By Lemma 2.5, both maps, f_{1}^{∗} and i_{1}^{∗} have the same image. Therefore the desired isomorphism H^{∗}(BX; 𝔽_{2}) ≅ H^{∗}(BDI(4); 𝔽_{2}) exists. This proves the first part.

The composition T ⊂ C_{1} ⊂ X is obviously a maximal torus. The weak isomorphism between the two Weyl group representations follows from Lemma 2.1. □

Using Part (ii) of this lemma we can identify the maximal tori and Weyl groups of DI(4) and X. The inclusion T ⊂ X establishes a map f_{T} : BT ⟶ BX and f_{T} = f_{1}j_{T}.

Let A_{p}(DI(4)) denote the Quillen category of the DI(4) and let

Θ : A_{p}(DI(4)) ⟶ Top

be the functor described in Section 2, which gives the Dwyer-Wilkerson decomposition for DI(4). This functor maps E_{k} on the space BC_{k} := BC_{DI(4)}(E_{k}). For each space BC_{k} we use the inclusion j_{k}BC_{k} ⟶ BC_{1} (see Section 2). to define f_{k} : BC_{k} ⟶ BX by the composition f_{k} := f_{1}j_{k}.

5.3 Proposition. Let α : E_{l} ⟶ E_{k} be any morphism of A_{p}(DI(4)). Then, f_{l}Θ(α) and f_{k} are homotopic maps.

Proof. Since i_{k} = i_{1}j_{k}, we can assume that l = 1. By assumption, i_{1}Θ(α) and i_{k} are homotopic. Now we check the statement for each value of i.

For k = 1 there is nothing to show since the auotomorphism group of E_{1} is trivial.

Let E_{4} ⊂ C_{1} be the standard inclusion. Then, BC_{4} ≃ BE_{4} and, by Lemma 5.2, the triangle

\xymatrix{ & H^{∗}(BE_{4}; ℤ/2) \\ H^{∗}(BX; 𝔽_{2}) \ar[rr]^{≅} \ar[ur]^{j_{E_{4}}^{∗}} &  & H^{∗}(BDI(4); 𝔽_{2}) \ar[ul]_{i_{E_{4}}^{∗}}}

commutes. Since homotopy classes of maps BE_{4} ⟶ BX are determined by the induced map in mod-2 cohomology, the statement also follows for k = 4.

For k = 2, 3 we first notice that the maximal torus T ⊂ C_{1} also establishes a maximal torus for C_{k}.

Claim: The restrictions f_{1}Θ(α)|_{BT} and f_{k}|_{BT} are homotopic.

Proof. The restriction Θ(α)|_{BT} factors through a self map w : BT ⟶ BT. Since i_{1}Θ(alpha) ≃ i_{1}, we have i_{T} w ≃ i_{T} which shows that w ∈ W. Hence, f_{1}Θ(α)|_{BT} ≃ f_{1}j_{T} w ≃ f_{T} w ≃ f_{T} ≃ f_{1}j_{T} ≃ f_{k}|_{BT}, which proves the claim. □

Now we consider the cases k = 2 and k = 3 separately. The centralizer C_{2} comes from a compact connected Lie group with center E_{2} ⊂ T subsetC_{2} and the inclusion C_{2} ⊂ X induces a homotopy equivalence BC_{2} ≃ BC_{X}(E_{2}). Therefore all assumptions of Proposition 4.6 of [23] are satisfied (for X = DI(4) this is explained in more detail in Remark 3.1). Part (c) of that proposition and the above claim imply that f_{1}Θ(α) ≃ f_{2}. This proves the statement for k = 2.

The final case k = 3 is a consequence of the above claim and the next Proposition. □

5.4 Proposition. Let g, h : BC_{3} ⟶ BX be two maps such that the restrictions g|_{BT} and h|_{BT} are homotopic and monomorphisms. Then g and h are homotopic.

Proof. The 2-compact group C_{2} establishes a fibration BT ⟶ BC_{2} ⟶ Bℤ/2. The group action of ℤ/2 on L := π_{2}(BT) is given by multiplication by −1. Let g′ := g|_{BT} and let M_{g′} denote the union of all components of map(BC_{2}, BX) whose restriction to BT is homotopic to g′. We make BT ⟶ BC_{2} into a fibration respectively into a principal ℤ/2-bundle. Then M_{g′} can be described as the homotopy fixed-point set (map(BT, BX)_{g′})^{hℤ/2} [12]. We only have to show that this homotopy fixed-point set is connected. Since g′ is a monomorphism, it can serve as a maximal torus and map(BT, BX)′_{g} ≃ BT. Hence, the Borel construction establishes a fibration BT ⟶ E ⟶ Bℤ/2 and the space of sections is homotopy equivalent to the homotopy fixed-point set. In this case, up to homotopy, the sections are classified by the obstruction group H^{2}(Bℤ/2, L where ℤ/2 acts on L as desribed above. Since H^{2}(ℤ/2; ℤ_{2}^{∧}) = 0 for the nontrivial action of ℤ/2 on ℤ_{2}^{∧}, this obstruction group vanishes and the space of sections of the above fibration is connected as well as the homotopy fixed-point set. This proves the statement. □

Let K := hocolim_{A_{p}(DI(4))}Θ. This is the realization of a simplicial space. By K^{(n)} we denote the realization of the n-skeleton of this simplicial space. By the definition of the homotopy colimit, Proposition 5.3 establishes a map

ϕ_{1} : K^{(1)} ⟶ BX .

Proof of Theorem 5.1 We want to extend the map ϕ_{1} : K^{(1)} ⟶ BX to K by induction over the skeleton. There exists an obstruction theory for such extension problems [33], whose obstruction groups we desribe next.

Let Π_{i} : A_{p}(DI(4)) ⟶ Ab be the functor taking values in the category of abelian groups, which is defined by Π_{i}(E_{k}) := π_{k}(map(BC_{k}, BX)_{f_{k}}). The above mentioned obstruction groups are given by higher derived limits of this functor. Since BE_{k} ≃ map(BC_{k}, BX)_{f_{k}}, the functor Π_{i} is trivial for i ≠ 1 and nothing but the inclusion functor J : A_{p}(DI(4)) ⟶ Ab for i = 1. Here, J is defined by J(E_{k}) := E_{K}. The higher derived limits of J are calculated in [8] and shown to vanish. Therefore, we can solve the extension problems and there exists a map ϕ : K ⟶ BX. By construction K ⟶ BDI(4) is a mod-2 equivalence. And this establishes a homotopy equivalence BDI(4) ≃ BX and finishes the proof of Theorem 5.1. □

6. Proof of Theorem 1.5.

We start directly with the proof. For any connected 2-compact group X there exists a short exact sequence Z ⟶ X ⟶ \overline{X} of 2-compact groups respectively a fibration BZ ⟶ BX ⟶ B\overline{X} for the associated classifying spaces, such that Z ⊂ X is a central subgroup, in particular a product of a finite abelian 2-group and a torus, and such that \overline{X} is centerfree [17]. In fact, the fibration is principal and has a classifying map B\overline{X} ⟶ B^{2}Z. By [11] \overline{X} ≅ ∏_{i} Y_{i} is congruent to a finite product of simple centerfree 2-compact groups Y_{i}. In particular, for all i, the rational Weyl group representation W_{Y_{i}} ⟶ Gl(L_{Y_{i}} ⊗ ℚ) is irreducible. (Actually, all this is true for any prime.) If Y_{i} has the same rational Weyl group data as DI(4), then Y_{i} and DI(4) have isomorphic maximal torus normalizer (Remark 4.3) and are isomorphic by Theorem 1.3.

Let X_{1} be the product of all factors Y_{i} with the same rational Weyl group data as DI(4) and \overline{X}_{2} the product of all the others. At the prime 2, the rational Weyl group representation of DI(4) is the only irreducible pseudo reflection group over ℚ_{2}^{∧} which is not weakly isomorphic to the 2-adic rational Weyl group representation of a simple connected compact Lie group. This follows by checking the Clark-Ewing list [3]. Therefore \overline{X}_{2} has the same rational Weyl group data as a suitable connected compact Lie group, and X_{1} is congruent to a product of copies of DI(4).

Since BDI(4) is 7-connected, the composition BX_{1} ⟶ BX_{1} × B\overline{X}_{2} ⟶ B^{2}Z is null homotopic and BX ≃ BX_{1} × BX_{2} where BX_{2} is the fiber of the composition B\overline{X}_{2} ⟶ BX_{1}×B\overline{X}_{2} ⟶ B^{2}Z. Since BX_{2} fits into a fibration BZ ⟶ BX_{2} ⟶ B\overline{X}_{2}, we can think of it as the classifying space of a 2-compact group X_{2} . And, up to some trivial summands, BX_{2} has the same rational Weyl group data as B\overline{X}. In particular, BX_{2} has the same rational Weyl group data as a suitable connected compact Lie group. This proves Theorem 1.5. □

References.

[1] J.F. Adams & C.W. Wilkerson, Finite H–spaces and algebras over the Steenrod algebra, Ann. Math. 111 (1980), 95-143.

[2] A. Bousfield & D. Kan, Homotopy limits, completion and localisations, SLNM 304, Springer Verlag, 1972.

[3] A. Clark & J. Ewing, The realization of polynomial algebras as cohomology rings, Pacific J. Math. 50 (1974), 425-434.

[4] W.G. Dwyer, Transfer maps for fibrations, Math. Proc. Camb. Philos. Soc. 120 (1996), 221-235.

[5] W.G. Dwyer, H.Miller, & C.W. Wilkerson, The homotopy uniqueness of BS^{3}, Algebraic Topology, Barcelona 1986,, SLNM 1298, Springer Verlag, 1987, pp. 90-105.

[6] W.G. Dwyer, H.Miller, & C.W. Wilkerson, The homotopical uniqueness of classifying spaces, Topology 31 (1992), 29-45.

[7] W.G. Dwyer & C.W. Wilkerson, A cohomology decomposition theorem, Topology 31 (1992), 433-443.

[8] W.G. Dwyer & C.W. Wilkerson, A new finite loop space at the prime two, J. AMS 6 (1993), 37-63.

[9] W.G. Dwyer & C.W. Wilkerson, Homotopy fixed–point methods for Lie groups and finite loop spaces, Ann. Math. (2) 139 (1994), 395-442.

[10] W.G. Dwyer & C.W. Wilkerson, The center of a p-compact group, The Cech Contennial. Contemporary Math. 181 (1995), 119-157.

[11] W.G. Dwyer & C.W. Wilkerson, Product splittings of p–compact groups, Fund. Math. 147 (1995), 279-300.

[12] W.G. Dwyer and A. Zabrodsky, in ”Algebraic Topology, Barcelona 1986”, SLNM 1298, pp. 106-119.

[13] S. Jackowski, J. McClure & R. Oliver, Self homotopy equivalences of classifying spaces of connected compact Lie groups, Fund. Math. 147 (1995), 99-126.

[14] J. Lannes, Sur les espaces fonctionelles dont la source est la classifiant d’un p–groupe abélien élemementaires, IHES 75 (1992), 135-244.

[15] J.M. Møller, Rational isomorphism of p-compact groups, Topology 35 (1996), 201-225.

[16] J.M. Møller, Homotopy Lie groups, Bull. Amer. Math. Soc. (N.S.) 32 (1995), 413-428.

[17] J.M. Møller & D. Notbohm, Centers and finite coverings of finite loop spaces, J. reine u. angew. Math. 456 (1994), 99-133.

[18] J.M. Møller & D. Notbohm, Connected finite loop spaces with maximal tori, Trans. AMS 350 (1998), 3483-3504.

[19] Homotopieeindeutigkeit von Produkten orthogonaler Gruppen, Thesis, Göttingen 1996 or ftp://hopf.math.purdue.edu/pub/morgenroth/BOn.dvi.

[20] D. Notbohm, Kernels of maps between classifying spaces, Isreal J. Math. 87 (1994), 243-256.

[21] D. Notbohm, Homotopy uniqueness of classifying spaces of compact connected Lie groups at primes dividing the order of the Weyl group, Topology 33 (1994), 271-330.

[22] D. Notbohm, Unstable splittings of classifying spaces of p–compact groups, Quart. J. Math. Oxford 51 (2000), 237-266.

[23] D. Notbohm, Topological realzation of a family of pseudo reflection groups, Fundam. Math. 155 (1998), 1-31.

[24] A uniqueness result for orthogonal groups as 2-compact groups, Preprint.

[25] R. Oliver, Higher limits via Steinberg representations, Comm. in Algebra 22 (1994), 1381–1402.

[26] K. Premadosa, Uniqueness of SO(7), unpublished work.

[27] D. Sullivan, Geometric Topology, Part I: Localisations, periodicity and Galois theory, Mimeographed notes, MIT (1970).

[28] Simplectic groups are N-determined p-compact groups, in preparation.

[29] A. Vavpetic and A. Viruel, Homotopy uniqueness of BF_{4}, in preparation.

[30] A. Viruel, Homotopy uniqueness of BG_{2}, Manuscr. Math. 95 (1998), 471-497.

[31] Rings of invariants and insepaprable forms of algebras over the Steenrod algebra, Preprint.

[32] C.W. Wilkerson, private communication.

[33] Z. Wojtkowiak, On maps from holim F to Z, Algebraic Topology, Barcelona 1986, SLNM 1298, Springer Verlag, 1987, pp. 227-236.

Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester, LE1 7RH, UK. e-mail : dn8 at mcs.le.ac.uk