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 [10], a pcompact 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 [9]. 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: April 26, 2000.

Y are isomorphic if the classifying spaces BX and BY are homotopy equivalent. For further details about p-compact groups see the survey article [17] 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 [6] [22] [19], for the exceptional Lie group G_{2} [28] and for the special orthogonal groups SO(2n + 1) [20] [25]. For the spinor groups Spin(2n + 1) only the second part is known [25].

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 H^{∗}(A; 𝔽_{2}) ≅ H^{∗}(BDI(4); 𝔽_{2}) as algebras over the Steenrod algebra.

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 [9]). 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 6.5).

The strategy for the proofs of Theorem 1.2 and Theorem 1.3 is clear and known to the 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)_{2}^{∧} of the 2compact 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 [25]. Having this at hand the idea of the proof is basically straightforward. We show that a 2-compact group X satisfying Proposition 1.4 has the same decomposition diagram as DI(4); i.e. BX is the completion of the homotopy colimit of a diagram which, as a diagram of spaces, is equivalent to the diagram used for the construction of BDI(4).

The paper is organized as follows. In Section 2 we recall the construction of BDI(4) of Dwyer and Wilkerson. Section 3 contains the proof of Proposition 1.4, Section 4 the proof of the theorems 1.2 and 1.3, Section 5 the proof of theorem 1.5 and the last section the proof of Theorem 1.6.

We will use the language of p-compact groups all over the places. For references we refer the reader to the survey article [17] 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 this work was done.

2. The construction of DI(4).

In this section we recall the construction of DI(4) and other material from [9]. 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 [9, Theorem 4.1]. 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 as well as 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 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_{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)} is given by the restriction of the representation Gl(3, 𝔽_{2}) ⟶ Gl(3, ℤ_{2}^{∧}) given by the action of the 2-adic Weyl group representation W_{DI(4)} ⟶ Gl(3, ℤ_{2}^{∧}) on the second component. Moreover, C_{DI(4)}(E_{2}) ≅ SU(2)^{3}/Δ =: Q, where Δ is the image of the diagonal embedding of ℤ/2 ⟶ SU(2)^{3}. And 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) [8] [11] has the form

[figure: {1}; Gl(2,𝔽_{2}); Gl(3,𝔽_{2}); Gl(4,𝔽_{2}); BSpin(7); BQ; B(T DI(4) ⋊ ℤ/2); E_{4}; .]

2.2 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} .

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 characteristic 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 [9, Theorem 1.8]. This calculation always works for subgroups of p-compact groups of maximal rank [11] [18].

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)} [11].

We finish this section with a proposition necessary for later purpose.

2.3 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})}

[15] [10] (for the latter sequence of isomorphisms see also [9]). □

3. 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})} .

3.1 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 [15] 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 [10]. 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 [12]. 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 [12] [23], 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}} [10], 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 3.1, 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 [10] we have BN_{X} ≃ B(T_{X}⋊W_{X}) ≃ B(T_{DI(4)}⋊ W_{DI(4)}) ≃ BN_{DI(4)} which is the claim. □

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

3.3 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 Q χ(X/Spin(7)) is odd. (iv) The map H^{∗}(BSpin(7); 𝔽_{2}) ⟶ H^{∗}(BX; 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 [10], 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′ 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 2.2). 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} [11, 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 [25], this implies that both are isomorphic, which proves part (ii).

The subgroup Spin(7) ⊂ X is of maximal rank. As explained in Remark 2.2, the Euler characteristic of X/Spin(7) equals the Weyl group index [W_{X} : W_{Spin(7)}] = [W_{DI(4)} : W_{Spin(7)}] which is odd (Remark 2.2). This proves Part (iii).

There exists a transfer H^{∗}(BSpin(7); 𝔽_{2}) ⟶ H^{∗}(BX; 𝔽_{2}) such that the composition with the restriction map 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 K-map of (iv) is a monomorphism as desired.

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. Since H^{∗}(BX; 𝔽_{2}) ⟶ H^{∗}(BE_{4}; 𝔽_{2}) is the Adams-Wilkerson embedding [1], a combination of [7, 3.6], [29, Theorem II] and of properties of the T -functor [15] shows that H^{∗}(BX; 𝔽_{2}) ≅ H^{∗}(BE_{4}, 𝔽_{2})^{\overline{W}^{X}}, which is Part (v).

Finally we have to prove part (vi). By construction Γ ⊂ \overline{W}_{X} ⊂ Gl(4, 𝔽_{2}) where, as explained in Section 2, Γ is the Weyl group of E_{4} ⊂ Spin(7). 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

Lemma 3.4 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 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.

3.4 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′. □

4. Proof of the theorems 1.2 and 1.3.

The Quillen category of a p-compact group X is given by purely algebraic data and depends only on the mod-p cohomology of the classifying space BX. For compact lie groups this is shown in [8] and for p-compact groups the same proof works. This shows that two p-compact groups X and Y with H^{∗}(BX; 𝔽_{p}) ≅ H^{∗}(BY; 𝔽_{p}) have equivalent Quillen categories. And this equivalence only depends on the chosen K-isomorphism between the mod-p cohomology algebras of the classifying spaces. Moreover, for any elementary abelian p-subgroup E ⊂ X the mod-p cohomology H^{∗}(BC_{X}(E); 𝔽_{p}) can be calculated with the help of Lannes’ T -functor and also depends only on the algebraic object H^{∗}(BX; 𝔽_{p}).

Let Top denote the category of topological spaces. Let A_{p}(X) denote the Quillen category of the p-compact group X and let

B̃C_{X} : A_{p}(X) ⟶ Top

denote the functor which maps E on the space BC_{X}(E) and describes the Dwyer-Wilkerson decomposition of BX [8] [11]. We can collect the above considerations in the following proposition.

4.1 Proposition. Let X and Y be two p-compact groups. Let ϕ : H^{∗}(BY; 𝔽_{p}) ⟶^{≅} H^{∗}(BX; 𝔽_{p}) be a K-isomorphism. Then the following holds:

(i) There exists an equivalence ϕ_{A} : A_{p}(X) ⟶ A_{p}(Y) between the Quillen categories of X and Y only depending on ϕ. (ii) For each elementary abelian subgroup E ⊂ X there exists an K-isomorphism ψ_{E} : H^{∗}(BC_{Y} (ϕ_{A}(E)); 𝔽_{p}) ⟶^{≅} H^{∗}(BC_{X}(E); 𝔽_{p}) which fit together to a natural transformation Ψ : H^{∗}(B̃C_{Y} ∘ ϕ_{A}; 𝔽_{p}) ⟶ H^{∗}({ \atop BC_{X}}; 𝔽_{p}).

If we identify the Quillen categories of X and Y, then the proposition says that the Dwyer-Wilkerson decompositions of both p-compact groups are identical in mod-p cohomology. In fact, for Y = DI(4), we will show that these decomposition diagrams are equivalent on the level of diagrams of topological spaces. This will be done in three steps. First we show that B̃C_{X} is an H^{∗}-clone of B̃C_{DI(4)} in the sense of [13] (see below). In the second step we show that both diagrams are equal inside the homotopy category of spaces and in the last step we show that, up to homotopy of diagrams, there is only one lift to the category Top. The last two steps are lifting problems. In our situation, for both lifting problems, there exists an obstruction theory, in the first case due to Dwyer and Wilkerson [13] and in the latter case due to Dwyer and Kan [5]. Fortunately, in [9] is shown that all these obstruction groups vanish. In particular, only the first step needs some work. And this is the place where we do need a uniqueness result for Spin(7).

We have to recall what H^{∗}-clones are. Let Φ, Ψ : C ⟶ HoTop be two functors from a discrete category into the homotopy category of Top. Following [13] we say that Ψ is a H^{∗}-clone of ϕ, if for each object c of C there exists an equivalence s_{c} : ϕ(c) ⟶ Ψ(c) such that the isomorphisms H^{∗}(s_{c}; 𝔽_{p}) establish a natural equivalence H^{∗}(Ψ; 𝔽_{p}) ⟶ H^{∗}(Φ; 𝔽_{p}). A lift of Ψ is a functor Ψ̂ : C ⟶ HoTop such that H^{∗}(Ψ̂; 𝔽_{p}) is naturally equivalent to Ψ. And a lift of Ψ̂ to Top is a functor Ψ̃ : C ⟶ Top such that composition of Ψ̃ with Top ⟶ HoTop is naturally equivalent to Ψ̂.

In the rest of this section, we will prove the following theorem which, because of Proposition 1.4, implies Theorem 1.2 as well as Theorem 1.3.

4.3 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}). Then, X and DI(4) are isomorphic as 2-compact groups.

The proof of this theorem will occupy the rest of this section. From now on and throughout this section X denotes a 2-compact group which satisfies the assumption of the theorem. By Proposition 4.1, we can and we will identify the Quillen categories A_{p}(X) ≅ A_{p}(DI(4)) =: A. We can assume that X is connected (Lemma 3.1). The composition H^{∗}(BX; 𝔽_{2}) ≅ H^{∗}(BE_{4}; 𝔽_{2})^{Gl(4,𝔽_{2})} ⟶ H^{∗}(BE_{4}; 𝔽_{2}) establishes a map BE_{4} ⟶ BX [15] which gives rise to a monomorphism i_{X} : E_{4} ⟶ X of 2-compact groups.

4.4 Lemma. There exists a monomorphism Spin(7) ⟶ X such that the diagram

\xymatrix{H^{∗}(BX; 𝔽_{2}) \ar[rr]^{≅} \ar[dr] &  & H^{∗}(BDI(4); 𝔽_{2}) \ar[dl] \\  & H^{∗}(BSpin(7); 𝔽_{2})}

commutes.

Proof. Let T := T_{DI(4)} denote the maximal torus and N := N_{DI(4)} the maximal torus normalizer of DI(4). We have a chain E_{4} ⊂ T ⋊ ℤ/2 ⊂ N ⊂ X of subgroups and E_{1} ⊂ E_{4} is actually contained in T. By Proposition 3.3, C_{X}(E_{1}) ≅ Spin(7) which establishes the monomorphism f : Spin(7) ⟶ X. And f induces a monomorphism f^{∗} : H^{∗}(BX; 𝔽_{2}) ⟶ H^{∗}(BSpin(7); 𝔽_{2}) in mod-2 cohomology (Proposition 3.3).

Let i_{Spin} : E_{4} ⊂ Spin(7) be the standard embedding and let j_{X} := f ∘ i_{Spin(7)} : E_{4} ⟶ X be the composition. Then, Bj_{X}^{∗} : H^{∗}(BX; 𝔽_{2}) ⟶ H^{∗}(BE_{4}; 𝔽_{2}) is a monomorphism and there exists an isomorphism ρ : E_{4} ⟶ E_{4} such that the diagram

\xymatrix{ & H^{∗}(BX; 𝔽_{2}) \ar[dl]_{Bi_{X}^{∗}} \ar[dr]^{Bj_{X}^{∗}} \\ H^{∗}(BE_{4}; 𝔽_{2}) \ar[rr]^{Bρ^{∗}} &  & H^{∗}(BE_{4}; 𝔽_{2})}

commutes.

Since H^{∗}(BX; 𝔽_{2}) is isomorphic to the Dickson invariants, the map Bj_{X}^{∗} also maps H^{∗}(BX; 𝔽_{2}) onto the Dickson invariants. The same is true for the compo-

sition H^{∗}(BDI(4); 𝔽_{2}) ⟶ H^{∗}(BSpin(7); 𝔽_{2}) ⟶^{Bi_{Spin}^{∗}} H^{∗}(BE_{4}; 𝔽_{2}). Because of the degrees of the polynomial generators, H^{∗}(BDI(4); 𝔽_{2}) has only one automorphism and there exists only one isomorphism between the mod-2 cohomology algebras of BX and BDI(4). This shows that there exists a diagram as desired. □

4.5 Proposition. The two functors B̃C_{X}, B̃C_{DI(4)} : A ⟶ Top are H^{∗}-clones.

Proof. For the elementary abelian 2-subgroups E_{i} ⊂ E_{4} ⊂ Spin(7) ⊂ X, i = 1, .., 4, we have to ways to calculate the mod-2 cohomology of BC_{X}(E_{i}). As explained in the beginning of this section we can use Lannes’ T -functor [15]. Then, H^{∗}(BC_{X}(E_{i}); 𝔽_{2}) = T^{E_{i}}(H^{∗}(BX; 𝔽_{2}); ϕ_{i}), where ϕ_{i} denotes the map induced by the composition BE_{i} ⟶ BE_{4} ⟶ BX. On the other hand, since E_{1} ⊂ E_{i} for all i, the inclusion C_{Spin(7)}(E_{i}) ⟶ C_{X}(E_{i}) is an isomorphism of 2-compact groups. The same holds if we replace by X by DI(4).

Now, for each E_{i}, we define s_{E_{i}} to be the composition

BC_{DI(4)}(E_{i}) ≅ BC_{Spin(7)}(E_{i}) ≅ BC_{X}(E_{i}) .

By construction, the collection of these maps establishes a natural transformation H^{∗}(B̃C_{X}; 𝔽_{2}) ⟶ H^{∗}(B̃C_{DI(4)}; 𝔽_{2}) which shows that B̃C_{X} and B̃C_{DI(4)} are H^{∗}- clones. □

4.6. Proof of Theorem 4.3 By the last lemma we know that B̃C_{X} and B̃C_{DI(4)} are H^{∗}-clones. We have to analyze, up to equivalence, how many lifts to HoTop do exist for H^{∗}(B̃C_{DI(4)}). Since the diagram is given as the diagram of classifying spaces of centralizers of elementary abelian subgroups, the diagram is centric in the sense of [13] and [5]. Basically this means that

map(BC_{DI(4)}(E_{i}), BC_{DI(4)}(E_{i}))_{id} ⟶ map(BC_{DI(4)}(E_{i}), BC_{DI(4)}(E_{j}))_{Bi}

is an equivalence for any E_{j} ⊂ E_{i} where i : C_{DI(4)}(E_{i}) ⟶ C_{DI(4)}(E_{j}) denotes the induced inclusion [13]. As shown in [13], there exists an obstruction theory for the above question . The obstruction groups are given by the first higher derived functor of the inverse limit functor of particular functors Φ_{i} : A ⟶ Ab with values in the category of abelian groups. On objects, Φ_{i}(E_{i}) = π_{i}(BZ(C_{DI(4)}(E_{i})) where Z(C_{DI(4)}(E_{i})) denotes the center of C_{DI(4)}(E_{i}) [13].

For the number of lifts of the composition A ⟶^{_{BC}_{DI(4)}} Top ⟶ HoTop to Top up to natural equivalences, there also exists an obstruction theory. Again, the obstruction groups are given by the higher derived functors of the inverse limit functor of the same functors Φ_{i} [5].

In [9] all these higher derived functors are proven to vanish. This shows that B̃C_{X} and B̃C_{DI(4)} are equivalent as functors taking values in HoTop as well as as functors taking values in Top. In particular, both functors B̃C_{X} and B̃C_{DI(4)} have homotopy equivalent homotopy colimits. And, since the homotopy colimits are mod-2 equivalent to BX respectively to BDI(4), this proves that X and DI(4) are isomorphic as 2-compact groups. □

5. Proof of Theorem 1.5.

We start directly with th 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 2-group and a torus, and such that \overline{X} is centerfree [18]. In fact, the fibration is principal and has a classifying map B\overline{X} ⟶ B^{2}Z. By [12] \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, as the proof of Proposition 1.4 (first half) shows (see Section 3), Y_{i} and DI(4) have isomorphic maximal torus normalizer 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 3-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

12

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. □

6. Self maps of BDI(4).

In this section we want to prove Theorem 1.6. We use the notation of the previous chapters.

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

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

commutes up to conjugation (Proposition 2.3). Actually, ρ is a monomorphism, because ϕ is one [10], 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_{i}} is the inclusion E_{i} ⊂ DI(4). Passing to centralizers establishes a monomorphism ϕ′ : C_{i} ≅ C_{C_{i}}(E_{i}) ⟶ C_{DI(4)}(E_{i}) = C_{i}. In fact, this is an isomorphism [18].

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

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

All this together establishes the assumptions made in Proposition 4.6 of [24]. The conclusion we can draw is the following. If Bψ : BC_{i} ⟶ BDI(4) is another map such that H_{ℚ}^{∗}(Bϕ) = H_{ℚ}^{∗}(Bψ), then Bϕ and Bψ are homotopic.

6.2 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 Bj ∘ 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) [21], 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) [21]. 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 6.1. □

6.3 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}^{∧}
Bj↓  ^{Bj}↓
BDI(4) ⟶^{f} BDI(4)

commutes up to homotopy.

Proof. If f ∘ Bj is null homotopic we can choose the constant map fro h. Hence, the second part is a consequence of Lemma 6.2.

The map h is either null homotopic or an equivalence [14]. 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 [23] □

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}^{∧}.

6.4 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 [14]. Since H_{ℚ_{2}^{∧}}^{∗}(BDI(4)) ⟶ H_{ℚ_{2}^{∧}}^{∗}(BSpin(7)) is a monomorphism, the statement is a consequence of Proposition 6.3. □

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}^{∧}^{∗} .

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

We postpone the proof of Theorem 6.5 to the end of this section.

6.6. 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 6.5. □

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

map(BDI(4), BDI(4)) ≃ holim map(B̃C, BDI(4)) .

The equivalence establishes a map

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

For ϕ ∈ lim_{\overleftarrow{A}} [B̃C, 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 map(B̃C, 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_{i},

map(BC_{DI(4)}(E_{i}), 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 [11] [24], the functor Π_{j} is naturally equivalent to the functor Φ_{j} considered in Section 4.

In both cases the higher derived limits all vanish [9], the Bousfield spectral sequence collapses and is concentrated in a singular line. This allows to prove the following proposition:

6.7 Proposition.

(i) The map [BDI(4), BDI(4)] ⟶ lim_{\overleftarrow{A}} [B̃C, 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) [11], which is trivial [9]. The component of the constant map is calculated in [11]. □

Proof of Theorem 6.5. 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 [26] say that the second arrow of

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

is an isomorphism. The first isomorphism follows from Proposition 6.7. We denote by j_{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 [16], 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 ∘ Bj_{E} and g ∘ Bj_{E} induce the same map in rational cohomology. If the rank of E is less or equal to 2, then Remark 6.1 shows that both compositions are actually homotopic. By Proposition 6.6, 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 [14]. Let f_{E} := Bj_{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′}} [B̃C, 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 6.1 implies that both maps are homotopic. Therefore, (f_{E})_{E∈A′} ∈ lim_{\overleftarrow{A′}} [B̃C, BDI(4)] and D is surjective (Proposition 6.7),

which finishes the proof. □

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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