HOMOTOPY UNIQUENESS OF BG_{2}

ANTONIO VIRUEL

Abstract. In this work, we prove that the homotopy type of the classifying space of the exceptional Lie group G_{2} is determined by its mod 2 type as well as by its Weyl group type at the prime 2. The proof is based on the mod p approximation of the classifying space of a compact Lie group via its p-stubborn subgroups developed by Jackowski, McClure and Oliver. The construction of the mod 2 equivalence follows a slightly different version of the programmeme developed by Notbohm in the case of p-convenient Lie groups.

1. Introduction.

A loop space L := (L, BL, e) consist of a pair of spaces L and BL, BL pointed, and a homotopy equivalence e : ΩBL ≃ L defining a loop structure on L. The space BL is called the classifying space of L.

In [7], Dwyer and Wilkerson introduced the concept of p-compact group, a homotopy theoretic generalization of compact Lie group. A loop space X := (X, BX, e) is said to be a p-compact group if the following hold: C1) X is 𝔽_{p}-finite, C2) X is p-complete, C3) π_{0}X is a finite p-group. Note that conditions C2) and C3) are equivalents to the following one:

C2’) BX, is p-complete.

The main examples of p-compact groups are given by the p-completion of compact Lie groups, but the triple (G_{p}^{∧}, BG_{p}^{∧}, e) is only a p-compact group if π_{0}G is finite p-group. Futher examples are given by the realization of polynomial algebras, i.e., by pairs (ΩBX, BX), where BX has polynomial mod p cohomology.

Then p-compact groups are shown to admit maximal tori in the sense of Rector ([22], [7]) and the Weyl group of X, W_{X}, is defined as π_{0}𝕎_{X}, where 𝕎_{X} is the space of self-maps of BT_{X} over BX, BT_{X} → BX being the maximal torus of X. The p-adic representation of the Weyl group as a pseudo-reflection group is therefore obtained as W_{X} → Aut(H_{2}(BT_{X}, ℤ_{p}^{∧})). Two p-compact groups are said to have the same Weyl group type if both have the same rank and the p-adic representation of the Weyl groups are conjugate.

0The author is partially supported by DGICYT grant PB94-0725.

Given a p-compact group X, two different kind of homotopy uniqueness problems arise: (P1) Let Y be a p-complete space having the mod p type of BX, i.e. there is an

isomorphism H^{∗}(Y; 𝔽_{p}) ≅ H^{∗}(BX; 𝔽_{p}) of algebras over the Steenrod algebra. Does Y have the homotopy type of BX?

(P2) Let X be a connected p-compact group with the same Weyl group type as

Y. Does BX have the homotopy type of BY? Both problems have been partially or totally solved in the torsion free case ([20],

[17], [6]) but not much is known in the torsion case ([3]). In this work we consider the case of two spaces having only 2-torsion.

Let SO(4) be the group of special unitary real matrices of order four. According to [20], the homotopy type of BSO(4) is determined, up to completion, by its mod p cohomology at primes different from 2, for in that case BSO(4) ≃ BSU(2). We solve problem (P1) in this case, proving the following result for p = 2.

Theorem 1.1. Let X be a 2-complete space such that H^{∗}(X; 𝔽_{2}) ≅ H^{∗}(BSO(4); 𝔽_{2}) as algebras over the mod 2 Steenrod algebra. Then X is homotopy equivalent to BSO(4) up to 2-completion.

The solution of problem (P2) for SO(4) has been anounced by Dwyer and Wilkerson.

Let G_{2} be the exceptional Lie group of rank two. Again, according to [20], the homotopy type of BG_{2} is determined, up to completion, by its mod p cohomology at primes different from 2. At the prime 2, BG_{2} has torsion and it is well known that

H^{∗}(BG_{2}; 𝔽_{2}) ≅ 𝔽_{2}[y_{4}, y_{6}, y_{7}],

where Sq^{2}y_{4} = y_{6} and Sq^{1}y_{6} = y_{7}. We solve the problem (P1) in this case, proving the following result.

Theorem 1.2. Let X be a 2-complete space such that H^{∗}(X; 𝔽_{2}) ≅ H^{∗}(BG_{2}; 𝔽_{2}) as algebras over the mod 2 Steenrod algebra. Then X is homotopy equivalent to BG_{2} up to 2-completion.

The problem (P2) also has a positive answer in the case of G_{2} at primes different from 2, [17]. We consider again the case p = 2.

Theorem 1.3. Let X be a connected 2-compact group with the same Weyl group type as G_{2}. Then BX and BG_{2} are homotopy equivalent up to 2-completion.

This result allows us to determine the integral homotopy type of BG_{2} in the following sense:

Corollary 1.4. Let L be a connected finite loop space with maximal torus and Weyl group W_{L} such that the integral representation of W_{L} is conjugate to that of G_{2}. Then BL is homotopy equivalent to BG_{2}.

In fact, both theorems 1.2 and 1.3 can be easily obtained from the following one:

Theorem 1.5. Let X be a connected 2-compact group with the same Weyl group type as G_{2} and such that H^{∗}(BX; 𝔽_{2}) ≅ H^{∗}(BG_{2}; 𝔽_{2}) as algebras over the mod 2 Steenrod algebra. Then BX is homotopy equivalent to BG_{2} up to 2-completion.

Organization of the paper. The paper is organized as follows. In section 2, we prove theorem 1.1. In section 3 we prove theorem 1.5. The gaps left open in that section are filled in the following ones. In section 4 we deal with the maximal torus, the Weyl group and some centralizers of 2-elements in BX. In section 5 we consider the theory relating to the 2-stubborn subgroups of G_{2}. In section 6 we construct a map from the normalizer of the torus in G_{2} to a space X with the same mod 2 cohomology as BG_{2}, and this map looks cohomologically like the standard inclusion in BG_{2}. In section 7, we contruct the mod 2 homotopy equivalence. In the last section we prove theorems 1.2 and 1.3 and corollary 1.4.

Notation. Here A_{2} is the mod 2 Steenrod algebra, all spaces are are assumed to have the homotopy type of CW-complexes, and completion means Bousfield-Kan completion. Given a space Y, we write H^{∗}Y for H^{∗}(Y; 𝔽_{2}), H_{ℚ_{2}}^{∗}(Y) for H^{∗}(Y; ℤ_{2}^{∧})⊗ ℚ and Y_{p}^{∧} for the Bousfield-Kan (ℤ/p)_{∞}-completion of the space Y. We write “Bss” for the Bockstein spectral sequence, “EMss” for the Eilenberg-Moore spectral sequence and “Sss” for the Serre spectral sequence. The symbol Λ_{𝔽} is used to denote an exterior algebra with field of coefficients 𝔽. Given a group A and an A-module M, we denote by H^{∗}(A; M) the cohomology with twisted coefficients.

Acknowledgements. To finish the introduction, it is a pleasure to thank M. Crossley, S. Jackowski, R. Levi, H. Morgenroth, F. Neumann and R. Oliver for useful conversations, and SFB 170 at Göttingen for hospitality and economical support while this work was being done. Finally, the author wants to thank specially C. Broto and D. Notbohm for “uncountably” many interesting suggestions and conversations.

2. Homotopy uniqueness of BSO(4).

In this section we prove theorem 1.1 following the idea in [3]. First, we recall the structure of H^{∗}BSO(4) and determine the Steenrod algebra action. Then we obtain a fibration Bℤ/2 → Y → X for a 2-complete space X with the same cohomology as BSO(4) and compute the cohomology of Y as an algebra over the Steenrod algebra. Finally we prove that Y ≃ B(SU(2)^{2})_{2}^{∧} and therefore X ≃ BSO(4).

The mod 2 cohomology of the classifying space BSO(4) is the polynomial algebra

H^{∗}BSO(4) = 𝔽_{2}[w_{2}, w_{3}, w_{4}]

where the w_{i} are the Stiefel-Whitney classes and A_{2} operates in the following way, [13]

Sq^{1}w_{2} = w_{3}, Sq^{1}w_{3} = Sq^{1}w_{4} = 0,
Sq^{2}w_{3} = w_{2}w_{3}, and Sq^{2}w_{4} = w_{2}w_{4}.

Note that the mod 2 cohomology of the classifying space of P U(2) × SU(2) is isomorphic to H^{∗}BSO(4) as an algebra but not over the Steenrod algebra because Sq^{2} acts in a differently.

Let X be a 2-complete space such that H^{∗}X is isomorphic to H^{∗}BSO(4) over the Steenrod algebra, and let X → B^{2}ℤ/2 the natural map induced by w_{2}, the generator of H^{∗}X in dimension 2. Let Y be the homotopy fibre of that map. This map classifies a principal fibration:

Bℤ/2 ⟶^{h} Y ⟶^{j}X. (1)

We compute H^{∗}Y as algebra over the Steenrod algebra.

Proposition 2.1. The mod 2 Sss associated to the fibration (1) collapses at the E_{4} term and:

E_{∞}^{∗,∗} ≅ E_{4}^{∗,∗} ≅ 𝔽_{2}[x_{4}, z_{4}].

Proof. Let us denote H^{∗}Bℤ/2 = 𝔽_{2}[ι] where |ι| = 1 and let (E_{∗}^{∗,∗}, d_{∗}) be the mod 2 Sss asociated to the fibration (1). The spectral sequence starts at:

E_{2}^{∗,∗} = H^{∗}(X; H^{∗}Bℤ/2) ≅ H^{∗}X ⊗ H^{∗}Bℤ/2.

The first non trivial differential is the transgression of ι, which is by construction, w_{2}; that is, d_{2}(ι) = w_{2}. This determines d_{2} and:

E_{3}^{∗,∗} = 𝔽_{2}[ι^{2}, w_{3}, w_{4}].

Since ι transgresses to w_{2}, Sq^{1}ι = ι^{2} must be transgressive too, and d_{3}(ι^{2}) = Sq^{1}w_{2} = w_{3}. This determines d_{3} and:

E_{4}^{∗,∗} = 𝔽_{2}[ι^{4}, w_{4}].

Now, dimensional arguments show that all the following differentials vanish, so:

E_{∞}^{∗,∗} ≅ E_{6}^{∗,∗} ≅ 𝔽_{2}[x_{4}, z_{4}],

where x_{4} is represented by ι^{4} and z_{4} is represented by w_{4}.

There is only one possible action of A_{2} on H^{∗}Y:

Corollary 2.2. H^{∗}Y ≅ 𝔽_{2}[x_{4}, z_{4}] ≅ H^{∗}BSU(2)^{2} as algebras over the Steenrod algebra.

Moreover, we know how the maps j and h look in cohomology.

Corollary 2.3. The maps,

H^{∗}X ⟶^{j^{∗}} H^{∗}Y ⟶^{h^{∗}} H^{∗}Bℤ/2

induced by the fibration (1) are trivial for all the generators except j^{∗}(w_{4}) = z_{4} and h^{∗}(x_{4}) = ι^{4}.

Proof. It follows from the edge homorphisms of the Sss asociated to the fibration.

Now we have to prove the homotopy uniqueness of B(SU(2)^{2}) because it does not appear in the literature although it follows easily from the work of Dwyer and Wilkerson [8].

Proposition 2.4. Let Y a 2-complete space such that H^{∗}Y ≅ H^{∗}(BSU(2)^{2}) as A_{2}-algebras. Then Y ≃ B(SU(2)^{2})_{2}^{∧}.

Proof. By the EMss we see that H^{∗}ΩY is finite, so Y can be considered as the classifying space of a 2-compact group. Denote by W the Weyl group of ΩY and by BT its maximal torus. Now, by Dwyer and Wilkerson, [7], we know that the map

H_{ℚ_{2}}^{∗}Y → (H_{ℚ_{2}}^{∗}BT)^{W}

is an isomorphism, so (H_{ℚ_{2}}^{∗}BT)^{W} ≅ ℚ_{2}[a_{4}, b_{4}], where the subscript indicates the degree of the element. Checking the Shepard-Todd list, [23], or more precisely the Clark-Ewing one, [5], we know that W ≅ ℤ/2 × ℤ/2.

Because ΩY is simply connected, by [8] we get that Y splits as a product of simple simply connected 2-compact groups, Y ≃ BX_{1} × BX_{2}. The algebra structure of H^{∗}Y forces H^{∗}BX_{i} ≅ 𝔽_{2}[a_{2}] ≅ H^{∗}BSU(2), and by [6] BX_{i} has to be homotopy equivalent to BSU(2)_{2}^{∧}, which completes the proof.

Now we are in a position to prove Theorem 1.1.

Proof of theorem 1.1. Let X be a 2-complete space such that H^{∗}X ≅_{A_{2}} H^{∗}BSO(4). Hence, for such a space X, we have obtained a fibration

Bℤ/2 → Y → X,

where Y ≃ B(SU(2)^{2})_{2}^{∧}. Moreover, according to Dwyer-Zabrodsky ([9]) we have

map(Bℤ/2, B(SU(2) ×_{C_{2}} SU(2))_{2}^{∧})_{ct} ≃ B(SU(2) ×_{C_{2}} SU(2))_{2}^{∧},

where ct stands for the constant map. Then we can apply Zabrodsky’s Lemma ([25],[14]) and obtain that the composition of Y ≃ B(SU(2)^{2})_{2}^{∧} with the 2-completion of the standard B(SU(2)^{2}) → B(SU(2)×_{C_{2}}SU(2)) factors up to homotopy through X, completing the following diagram:

Bℤ/2 = = = = = = = = = = = Bℤ/2

Y^{∨}  \overline{>} B(SU(2)^{2})_{2}^{∧}^{∨}

X^{∨}  \overline{>} B(SU(2) ×_{C_{2}} SU(2))_{2}^{∧}^{∨}

Finally, we have to check that the map X → B(SU(2) ×_{C_{2}} SU(2))_{2}^{∧} induces an isomorphism in cohomology (remember that SU(2) ×_{C_{2}} SU(2) ≅ SO(4)).

This holds because of the natural morphism between the mod 2 Sss associated to each vertical fibration. The classes w_{2} and w_{3} are detected because they are the image of d_{2}(ι) and d_{3}(ι^{2}). The class w_{4} is detected because the commutativity of the diagram.

Finally, note that the same arguments allow us to prove the homotopy uniqueness of B(P U(2) × SU(2)), although this is not relevant to our aims.

3. Proof of theorem 1.5.

In what follows in this section, X is a connected 2-compact group with the same Weyl group type as G_{2} and such that H^{∗}BX ≅ H^{∗}BG_{2} as algebras over the mod 2 Steenrod algebra. The construction of the mod 2 equivalence f : BG_{2} → BX is based on the mod p approximation of the classifying space BG of a compact Lie group G via its p-stubborn subgroups that appears in [10]. To apply this result, we follow a slightly version of the programme developed by Notbohm in [20].

Let R_{p}(G) ⊂ O(G) be the full subcategory of the orbit category O(G), whose objects are the homogenous spaces G/P, where P is p-stubborn. Let EG denote an acyclic free G-CW-complex.

Theorem 3.1. [10] Let G be a connected Lie group. The canonical map

holim_{R_{p}(G)} EG ×_{G} G/P → BG

is a p-local equivalence.

In our case, we have

map(BG_{2}, BX) ≃ map(holim_{R_{p}(G)} EG ×_{G} G/P, BX)
= holim_{R_{p}(G)} map(EG ×_{G} G/P, BX)

hence to get the desired map f : BG_{2} → BX, we construct a collection of maps EG ×_{G} G/P ≃ BP ⟶^{f_{P}} BX such that they produce a map between the diagrams

{BP }_{R_{2}(G_{2})} → BX,

where {BP }_{R_{2}(G_{2})} denotes the diagram given by the Borel construction applied to the category R_{2}(G_{2}), that commutes up to homotopy and the associated obtruction groups to extend the map to the total colimit vanish.

We do it in the following steps.

Write ϕ^{∗} : H^{∗}BX → H^{∗}BG_{2} for the given isomorphism of A_{2}-algebras. We suppose that it is induced by a map ϕ : (BG_{2})_{2}^{∧} ^{>} BX, but use a dotted arrow, since the map may not exist. Then,

Proposition 3.2. Let BT_{2}^{∧} a 2-compact torus of rank 2. There exist maps Bi and f_{T} such that

BT_{2}^{∧}
[figure: Bi; f_{T}]
(BG_{2})_{2}^{∧}  ϕ  ^{>} BX

is W_{G_{2}}-equivariant, commutes in mod 2 cohomology and the maps Bi and f_{T} look like the standard inclusions of the maximal torus in (BG_{2})_{2}^{∧} and BX.

Proof. See section 4.

The next step is to extend the inclusion of maximal torus BT_{2}^{∧} ⟶^{f_{T}} X to a map BNT_{2}^{∧} ⟶^{f_{N}} X, where BNT is the normalizer of the torus in G_{2}, whose cohomological induced map looks like that induced by the standard inclusion of BNT in BG_{2}. We do this in the next proposition.

Proposition 3.3. There exists an extension BNT_{2}^{∧} ⟶^{f_{N}} X of BT_{2}^{∧} ⟶^{f_{T}} BX such that it fits in the following commutative diagram (also in cohomology):

BT_{2}^{∧}

[figure: Bi; f_{T}]
BNT_{2}^{∧}^{∨}
[figure: f_{N}]
(BG_{2})_{2}^{∧}  ϕ  ^{>} BX

Proof. See section 6.

Now, note that the standard map

{BP }_{R_{2}(G_{2})} → BG_{2}

factor through BNT, so we have

{BP }_{R_{2}(G_{2})} ⟶^{h} BNT → BG_{2},

but it does not imply the commutativity of the diagram

{BP }_{R_{2}(G_{2})} ⟶^{h} BNT.

We prove that the composition with the map f_{N} constructed in proposition 3.3 commutes again up to homotopy. Note that the diagram commutes in cohomology by construction.

Proposition 3.4. The diagram

{BP }_{R_{2}(G_{2})} ⟶^{h} BNT ⟶^{f_{N}} BX (2)

commutes up to homotopy.

Proof. See section 7.

The homotopy commutativity of the diagram 2 induces a map from the 1-skeleton of the homotopy colimit to BX. The obtruction groups to extending this map to the total homotopy colimit are

lim_{R_{2}(G_{2})^{←}}^{i} π_{j}(map(BP, BX)_{f_{P}}),

where lim^{i} is the i-th derived functor of the inverse functor [2], [24].

To calculate those groups, we compare the functors

Π_{j}(X), Π_{j}(G_{2}) : R_{2}(G_{2}) → Ab

defined as

Π_{j}(X)(G/P) := π_{j}(map(BP, BX)_{f_{P}}),
Π_{j}(G_{2})(G/P) := π_{j}(map(BP, (BG_{2})_{2}^{∧})_{Bi_{P}}).

The category Ab is the category of abelian groups. These functors are well defined for j ≥ 2 and note that according to [10], Π_{1}(G_{2})(G/P) = π_{1}(BZ(P)) which is an abelian group and therefore the functor is well defined in this case as well. The case of Π_{1}(X) is similar (see section 7).

The relation between those functors is given by the following proposition.

Proposition 3.5. There exists a natural transformation

T : Π_{j}(G_{2}) → Π_{j}(X)

which is an equivalence.

Proof. See section 7.

So we have that:

lim_{R_{2}(G_{2})^{←}}^{i} π_{j}(map(BP, BX)_{f_{P}}) ≅ lim_{R_{2}(G_{2})^{←}}^{i} π_{j}(map(BP, (BG_{2})_{2}^{∧})_{Bi_{P}}),

and by [10]:

lim_{R_{2}(G_{2})^{←}}^{i} π_{j}(map(BP, (BG_{2})_{2}^{∧})_{Bi_{P}}) = 0.

Hence we have a map f : BG → BX. To finish the proof we have to check that the map induced by f on cohomology is an isomorphism. This follows from the commutative diagram (see section 5 for the definition of N_{2}(T))

BN_{2}(T)
[figure: Bi_{N_{2}T}; f_{N_{2}T}]
BG_{2}  \overline{f  _{>}} BX

because both diagonal maps induce injective maps in cohomology, so H^{∗}f is injective too, and because H^{∗}BX and H^{∗}BG_{2} have the same Poincaré series, which implies that H^{∗}f is an isomorphism.

4. The maximal tori, centralizers and Weyl group.

In this section, we consider the Weyl group of BX, its centralizers and related ideas. First we prove that the Weyl group type is determined by cohomology in the case of G_{2}.

Proposition 4.1. Let X be a 2-compact group such that H^{∗}BX ≅_{A_{2}} H^{∗}BG_{2}. Then the Weyl group of X, W_{X}, has 2-adic representation conjugate to that of the Weyl group of G_{2}.

Proof. Because H_{ℚ_{2}}^{∗}BX = 𝔽_{2}[x_{4}, x_{12}], checking the list of Clark-Ewing, [5], we show that the Weyl group of X has to be isomorphic to ℤ/2 × ℤ/6 or D_{12} ≅ W_{G_{2}}. But ℤ/2 × ℤ/6 is not generated by 2-reflections, so W_{X} has to be isomorphic to W_{G_{2}}.

Consider L, the lattice associated to the 2-adic representation of W_{X} and Q the lattice associated to that of W_{G_{2}}. We have to prove that L ≅ Q in order to prove that these two representations are conjugate.

We know that L ⊗ ℚ ≅ Q ⊗ ℚ, so we can construct a short exact sequence of D_{12} modules

L ⟶^{α} Q ⟶Q/L,

where Q/L is finite and rk(Q/L) < rk(Q) = rk(L) = 2.

The mod 2 reduction produces a long exact sequence

0 ⟶Tor(Q/L, 𝔽_{2}) ⟶L/2 ⟶^{α/2} Q/2 ⟶Q/L ⊗ 𝔽_{2} ⟶0.

Now we know that Q/2 is irreducible, so α/2 has to be trivial or an isomorphism. But rk(Q/L) < rk(Q), so α/2 has to be an isomorphism and therefore Q/L⊗𝔽_{2} = 0. Hence Q/L = 0, which implies that L and Q are isomorphic lattices.

Now, we construct a maximal 2-compact torus for BX such that the natural map

BT_{2}^{∧} ⟶^{f_{T}} BX

looks in cohomology like the natural inclusion

BT ⟶^{Bi_{T}} BG_{2}

and is W_{G_{2}} ≅ W_{X} equivariant.

In order to do that, first we contruct a map BSO(4)_{2}^{∧} → BX which looks in cohomology like the natural inclusion BSO(4) → BG_{2}.

Proposition 4.2. Let BX be a 2-complete space such that H^{∗}BX ≅_{A_{2}} H^{∗}BG_{2}, then there exists a map BSO(4)_{2}^{∧} → BX such that the following diagram commutes:

BSO(4)_{2}^{∧}
[figure]
(BG_{2})_{2}^{∧}  ϕ  ^{>} BX

Proof. By [11], there is an unique conjugacy class of elements of order 2 in G_{2} and the centralizer of any of those elements is SO(4).

Because H^{∗}BX ≅_{A_{2}} H^{∗}BG_{2}, Lannes’ theory provides a map Bℤ/2 ⟶^{f} BX such that the induced homomorphism in cohomology looks like the homomorphism induced by the natural inclusion of the center of BSO(4) in G_{2}, hence we have a commutative diagram in cohomology

Bℤ/2
[figure: Bi_{SO(4)}|_{ℤ/2}; f]
(BG_{2})_{2}^{∧}  ϕ  ^{>} BX

Because H^{∗}BSO(4) is free in degree less or equal than two, we can apply 3.2.4 in [12] and the Lannes T functor shows that H^{∗} map(Bℤ/2, BX)_{f} ≅_{A_{2}} H^{∗}BSO(4). Now, using theorem 1.1, we get that map(Bℤ/2, BX)_{f} ≃ BSO(4)_{2}^{∧}. Moreover, the evaluation map induces in cohomology the augmentation map M → T M, so we get the desired map.

Now the proof of proposition 3.2 is trivial.

Proof of proposition 3.2. The desired map appears as the composition BT_{2}^{∧}

[figure: Bi; f_{T}]

BSO(4)_{2}^{∧}^{∨}

[figure]

(BG_{2})_{2}^{∧}  ϕ  ^{>} BX

We finish this section proving a technical result about the Weyl group of the centralizers of p-compact groups.

Proposition 4.3. Let X be a p-compact group with maximal torus BT → BX and Weyl group W_{X}. Let P be a p-compact toral group. Given a monomorphism

Bi : BP → BT → BX

the centralizer of Bi, BC := map(BP, BX)_{Bi}, is a p-compact group with the same rank as X and Weyl group W_{C} = Iso(Bi) := {w ∈ W_{X} : w ∘ Bi ≃ Bi}.

Proof. It is a p-compact group by [7], and it has the same rank as X, because the monomorphism BT → BX induces a monomorphism

map(BP, BT)_{Bi} ⟶map(BP, BX)_{Bi}

and, because BT is abelian, map(BP, BT)_{Bi} ≃ BT.

Note that the monomorphism induced by the evaluation

BT ⟶map(BP, BX)_{Bi} ⟶^{ev} BX

induces a natural map

W_{C} ⟶W_{X} ⟶π_{2}BT.

On the one hand, because the representation of W_{C} in π_{2}BT is faithfull, that natural map has to be injective. On the other hand, it can be easily checked that every element in Iso(Bi) is an element in W_{C}, which finishes the proof.

5. The 2-stubborn subgroups of G_{2}.

In this section we deal with the theory relating to the 2-stubborn subgroups of G_{2}. First some technical lemmas.

Lemma 5.1. The cohomology group H^{2}(ℤ/2; (LT_{P U(2)})_{2}^{∧}) is trivial.

Proof. In [20], proof of proposition 6.7, we see that H^{2}(ℤ/2; (LT_{U(2)})_{2}^{∧}) = 0. To finish the proof, we consider the exact sequence of ℤ/2-modules:

0 → ℤ_{2}^{∧} → (LT_{U(2)})_{2}^{∧} → (LT_{P U(2)})_{2}^{∧} → 0.

Lemma 5.2. Let P ⊂ P U(2) be the 2-normalizer of T, the torus of P U(2). Then, given an inclusion Bi : BT → BP U(2), there exists only one extension to a map Bi : BP → BP U(2) up to homotopy, i.e. there exists only one extension i : P → P U(2) up to conjugation.

Proof. The number of those extensions are calculated by the obstruction groups

H^{∗}(P/T; π_{∗} map(BT; BP U(2))_{Bi} ≅ H^{2}(ℤ/2; (LT_{P U(2)})_{2}^{∧})

and by lemma 5.1 this group is trivial, hence we are finished.

In the following proposition, we describe the 2-stubborn subgroups of G_{2}. They are calculated in [11].

Proposition 5.3. G_{2} contains exactly 6 conjugacy classes of 2-stubborn subgroups, with representatives P_{1}, …, P_{6} as listed below. They are all presented as subgroups of Sp(1) ×_{C_{2}} Sp(1) ⊂ G_{2}. Also, Q ⊂ N ⊂ Sp(1) are the subgroups Q =< i, j > (the quaternion group of order 8), and N =< S^{1}, j >.

\begin{matrix}\underline{P} & \underline{N(P)/P} \\ P_{1} = N ×_{C_{2}} N = N_{2}(T) & 1 \\ P_{2} = N ×_{C_{2}} Q & 1 × Σ_{3} \\ P_{3} = Q ×_{C_{2}} N & Σ_{3} × 1 \\ P_{4} = Q ×_{C_{2}} Q & Σ_{3} × Σ_{3} \\ P_{5} =< S^{1} ×_{C_{2}} S^{1}, (j, j) >≅ T ⋊ C_{2} & Σ_{3} \\ P_{6} =< (i, i), (j, j), (1, −1) >≅ (C_{2})^{3} & GL_{3}(𝔽_{2}) \end{matrix}

Futhermore, all morphisms in R_{2}(G_{2}) between the G_{2}/P_{i} listed here are composites of the automorphisms of the orbits, and of the maps induced by the given inclusions.

In the next proposition, we describe some properties of the 2-stubborn subgroups of G_{2} that we need. Note that Sp(1) ×_{C_{2}} Sp(1) ≅ SO(4).

Proposition 5.4. The representatives i : P ↪ N_{G_{2}}(T) ↪ G_{2} defined in proposition 5.3 satisfy the following conditions:

(1) P_{T} := P ∩ T is a 2-toral group.

(2) The short exact sequence P_{T} ⟶P ⟶^{π} P/P_{T} has a section.

(3) Z(SO(4)) ⊂ P_{T} .

(4) C_{G_{2}}(P_{iT}) = T for all i ≠ 6. C_{G_{2}}(P_{6T}) = P_{5}.

(5) The canonical map

π_{0}(map(BP, (BG_{2})_{2}^{∧})_{Bα|BP_{t}=Bi}) → hom(H^{∗}BG_{2}, H^{∗}BP)

is an injection.

Remark 5.5. Note that by (3), every element in C_{G_{2}}(P_{T}) centralizes Z(SO(4)), so the stament (4) can be proved by calculating the centralizers in SO(4).

Remark 5.6. By map(BP, (BG_{2})_{2}^{∧})_{Bα|_{BP_{T}} =Bi}) we denote the components of the mapping space map(BP, (BG_{2})_{2}^{∧}) given by maps Bα : BP → (BG_{2})_{2}^{∧}, such that Bα|_{BP_{T}} ≃ Bi. Hence (5) can be reformulated as “the extensions of the canonical inclusion Bi|_{BP_{T}} : BP_{T} → (BG_{2})_{2}^{∧} to maps Bα : BP → (BG_{2})_{2}^{∧} are controlled by cohomology”.

Proof. The proof of the statements (1) to (4) is a straight foward calculation.

Let α, β : P → G_{2} be two homomorphisms, such that H^{∗}Bα ≡ H^{∗}Bβ and Bα|_{BP_{T}} ≃ Bβ|_{P_{T}}, we see that Bα ≃ Bβ. That is trivial in the case of P_{6} by Lannes’ theory, but the others cases are more complicated and will be done in several steps.

Step 1. Both maps factor through BSO(4)_{2}^{∧}.

Consider the induced map between mapping spaces:

Bα, Bβ : map(BZ(P), BP)_{Bi} → map(BZ(P), (BG_{2})_{2}^{∧})_{Bi}.

According to [9] and [18], and by (2), we get that map(BZ(P), BP)_{Bi} ≃ BP_{2}^{∧} and map(BZ(P), (BG_{2})_{2}^{∧})_{Bi} ≃ BSO(4)_{2}^{∧} and the evaluation map allows us to reconstruct the original map. Hence we have two maps:

Bα, Bβ : BP → BSO(4)_{2}^{∧},

which when composed with the standard inclusion of BSO(4) in BG_{2}, give the original ones. Moreover, by Lannes’ T functor, we get that H^{∗}Bα ≡ H^{∗}Bβ.

Step 2. β(x)(α(x))^{−1} ∈ Z(SO(4)) for all x ∈ P.

Let α, β : P → SO(4) be two homomorphisms, such that Bα|_{BP_{T}} ≃ Bβ|_{P_{T}}, we see that there exists γ : P/P_{T} → Z(SO(4)) such that β is conjugate in SO(4) to the homomorphism:

P ⟶^{Δ} P × P ⟶^{Id×π} P × P/P_{T} ⟶^{α×γ} SO(4) × Z(SO(4)) ⟶^{μ} SO(4),

where μ is the multiplication in SO(4).

To prove the existence of γ is equivalent to proving that the induced map between the quotients by the center of SO(4) are conjugate in P U(2)^{2}, i.e. that the induced map

H^{2}(P/P_{T}; (LT_{SO(4)})_{2}^{∧}) → H^{2}(P/P_{T}; (LT_{P U(2)^{2}})_{2}^{∧})

is trivial.

We consider two different cases, when P/P_{T} has rank 1 or 2. The first one is the case of P_{5}. In that case, we are interested in extensions of the standard inclusion of the torus Bi : BT → (BP U(2)^{2})_{2}^{∧} to maps Bα : B\overline{P_{5}} → (BP U(2)^{2})_{2}^{∧}. The uniqueness of these extensions are classified by:

H^{∗}(\overline{P_{5}}/\overline{P_{5T}}; π_{∗} map(B\overline{P}_{5T}; (BP U(2)^{2})_{2}^{∧})_{Bi}) = H^{2}(ℤ/2; (LT_{P U(2)^{2}})_{2}^{∧}),

where the action is diagonal. To prove that this group is trivial, we apply lemma 5.1 and the exact sequence of ℤ/2 modules:

(LT_{P U(2)})_{2}^{∧} → (LT_{P U(2)^{2}})_{2}^{∧} → (LT_{P U(2)})_{2}^{∧}.

This finishes the proof of step 2 for P_{5}.

The second case is the case of P_{1}. Consider Q_{i} the 2-normalizer of the torus in SU(2), and \overline{Q_{i}} the quotient by the center. Then \overline{Q_{i}} is the 2-normalizer of the torus of P U(2). Now, the quotient of P_{1} by its center is \overline{P_{1}} = \overline{Q_{1}} × \overline{Q_{2}}, and therefore, any induced map \overline{α} : \overline{P_{1}} → (P U(2)^{2})^{∧} can be expressed as a matrix \overline{α} = (\overline{α}_{i,j}) where the morphism \overline{α}_{i,j} appears as the quotient by the center of the composition

Q_{i} ↪ P_{1} ⟶^{α} BSO(4)_{2}^{∧} ⟶^{π_{j}} (P U(2)_{j})_{2}^{∧}.

By construction, B\overline{α}_{i,i}|\overline{Q_{iT}} ≃ B\overline{β}_{i,i}|\overline{Q_{iT}}, so by lemma 5.2 B\overline{α}_{i,i} ≃ B\overline{β}_{i,i}.

Now, because the inclusion of Q_{1} ↪ P_{1} commutes with the inclusion Q_{2} ↪ P_{1}, it forces to \overline{α}_{i,j} ≡ \overline{β}_{i,j} and both are trivial. That implies B\overline{α} ≃ B\overline{β}, which finishes the proof of step 2.

Step 3. Different extensions are detected by cohomology.

We see that if H^{∗}Bα ≡ H^{∗}Bβ then the homomorphism γ : P/P_{T} → Z(SO(4)) defined in step 2 is trivial. By Lannes’ theory it is enough to prove that H^{∗}Bγ ≡ 0. Assume that it is non trivial, then H^{∗}Bγ : H^{∗}BZ(SO(4)) ↪ H^{∗}B(P/P_{T}). Note that also H^{∗}Bπ : H^{∗}B(P/P_{T}) ↪ H^{∗}BP by (2).

For w_{4} ∈ H^{∗}BSO(4) we then have that:

H^{∗}Bα(w_{4}) = H^{∗}Bβ(w_{4})
= H^{∗}Bα(w_{4}) + (H^{∗}Bπ) ∘ (H^{∗}Bγ)(ι^{4})
+H^{∗}Bα(w_{2})(H^{∗}Bπ) ∘ (H^{∗}Bγ)(ι^{2}).

An easy analysis of the cohomology of those centralizers shows that this equation can not hold because (H^{∗}Bπ) ∘ (H^{∗}Bγ)(ι) ≠ 0 as we noted.

So γ has to be trivial and therefore Bα ≃ Bβ.

6. The normalizer of the maximal torus.

In this section, we prove proposition 3.3, i.e. we extend the inclusion of maximal torus:

BT_{2}^{∧} ⟶^{f_{T}} BX

to an inclusion of its normalizer in G_{2} in BX:

BNT_{2}^{∧} ⟶^{f_{N}} BX

such that it fits in the following commutative diagram (also in cohomology):

[figure: BT_{2}^{∧}; Bi; f_{T}; BNT_{2}^{∧}^{∨}; f_{N}; (BG_{2})_{2}^{∧}  ϕ  ^{>} BX] (3)

We use similar arguments to that in [20].

Choosing a fixed-point of the W_{G_{2}}-action on BT_{2}^{∧}, the evaluation at this fixedpoint

map(BT_{2}^{∧}, BX)_{f_{T}} → BX

is W_{G_{2}}-equivariant with respect to the trivial W_{G_{2}}-action on BX, hence it produces a well defined map on the associated Borel construction Y := (map(BT_{2}^{∧}, BX)_{f_{T}})_{hW_{G_{2}}} and an extension:

\begin{matrix}BT_{2}^{∧} \\ [figure] & f^{T} & \\ Y & ^{f^{∨}} & \overline{g _{>}} & BX \\ BW_{G_{2}} & ∨ \end{matrix} (4)

We prove that Y is homotopy equivalent to (BNT)_{2}^{∘}, the fibrewise completion (see [2]) of BNT via the fibration:

BT → BNT → BW_{G_{2}}. (5)

The fibrations of the form BT_{2}^{∧} → Z → BW_{G_{2}} are determined (up to equivalence) by the W_{G_{2}}-action and a cohomological class in H^{3}(W_{G_{2}}; (LT_{G_{2}})_{2}^{∧}). Because fibrations (4) and (5) induce the same W_{G_{2}}-action on BT_{2}^{∧}, they are equivalent if and only if they are represented by the same cohomological class.

Fixed the W_{G_{2}}-action, the inclusion W_{SO(4)} ↪ W_{G_{2}} induces a natural map (via pull-back) between the fibrations with base space BW_{G_{2}} and the fibrations with base space BW_{SO(4)}, where the fibration (4) maps to the fibration:

(6) map(BT_{2}^{∧}, BSO(4)_{2}^{∧})_{Bi} → (map(BT_{2}^{∧}, BSO(4)_{2}^{∧})_{Bi})_{hW_{SO(4)}} → BW_{SO(4)},

(note that BT_{2}^{∧} ≃ map(BT_{2}^{∧}, BSO(4)_{2}^{∧})_{Bi}) and the fibration (5) maps to the fibration:

BT → BN_{SO(4)}T → BW_{SO(4)}. (7)

Moreover, the induced homomorphism in cohomology is injective ([20]):

H^{3}(W_{G_{2}}; (LT_{G_{2}})_{2}^{∧}) ↪ H^{3}(W_{SO(4)}; (LT_{SO(4)})_{2}^{∧}).

Now, the problem is reduced to check that the cohomological classes associated to the fibrations (6) and (7) are the same, that is, that both fibrations are equivalent. This becomes clear using the following lemma.

Lemma 6.1. Let G be a compact Lie group with maximal torus BT ⟶^{Bi} BG and Weyl group W_{G}. Let Z be a space which fits in a fibration BT_{p}^{∧} ⟶^{f} Z ⟶ BW_{G} where the action of W_{G} on BT_{p}^{∧} is the induced by the Weyl group one. Suppose that there exists a map Z ⟶^{g}BG_{p}^{∧} such that the following diagram commutes:

BT_{p}^{∧}
[figure: Bi]
Z^{f^{∨}}  \overline{g _{>}} BG_{p}^{∧}

Then Z ≃ BN_{G}T_{p}^{∘}, the fibrewise completion of the classifying space of N_{G}T, the normalizer of T in G.

Proof. By [20], 11.3, we know that

GrExt(W, T) ^{B >} Fib(BW, BT_{p}^{∧})
T → N → W ↦ BT_{p}^{∧} → BN_{p}^{∘} → BW

is surjective, hence Z ≃ BN_{p}^{∘}, the fibrewise completion of the classifying space of N, a group which fits in a short exact sequence T → N → W. Consider B̃T = EN/T ≃ BT. It is well known that the extensions of BT ⟶^{Bi} BG_{p}^{∧} to maps BN ⟶BG_{p}^{∧} are elements of map(B̃T , BG_{p}^{∧})_{Bi}^{hW_{G}}, but the standard inclusion of the normalizer of the torus induces an equivariant mod p equivalence between the 1-connected spaces map(B̃T , BN_{G}T)_{Bi} and map(B̃T , BG)_{Bi}, so the induced map between the homotopy fixed points is a mod p equivalence too. As we know that there exists an extension Z ≃ BN_{p}^{∘} ⟶^{f} BG_{p}^{∧} of BT_{p}^{∧} ⟶^{Bi} BG_{p}^{∧}, we have a map BN ⟶^{h} BN_{G}T such that the following diagram commutes:

BT = = = = = = = BT

BN_{G}T Bi_{∨}  \overline{h _{>}} BN^{f^{∨}}

BG_{p}^{∧}^{∨} = = = = = = BG_{p}^{∧}^{g^{∨}}

The map h induces a homomorphism between the fundamental groups of BN and BN_{G}T, that is, a self homomorphism of W_{G}, so we can close the commmutative diagram:

BT_{p}^{∧} = = = = = = BT_{p}^{∧}

BN_{G}T_{p}^{∘} Bi_{∨}  \overline{h _{>}} BN_{p}^{∘}^{f^{∨}}

BW_{G}^{∨}  \overline{>} BW_{G}.^{∨}

The lowest horizontal arrow induces an isomorphism on the fundamental groups because both actions on BT (that they are via the fundamental groups) are equivalent, and so BN_{G}T_{p}^{∘} ≃ BN_{p}^{∘} ≃ Z.

To finish, we have to check that the diagram 3 commutes in cohomology (it does in homotopy by construction). This follows from the diagram:

BSO(4)_{2}^{∧} = = = = = = = = = = = = = = = BSO(4)_{2}^{∧}
[figure]
BN_{SO(4)}T_{2}^{∧}

BNT_{2}^{∧}^{∨}
[figure]
(BG_{2})_{2}^{∧}^{∨}  ϕ  ^{>} BX^{∨}

and the fact that all the cohomological maps induced by the arrows are injective.

7. The map BG_{2} → BX.

In this section we prove proposition 3.4. Recall the situation:

In section 6, we contructed a commutative diagram:

BNT
[figure: Bi_{NT}; f_{N}]
BG_{2}  ϕ  ^{>} BX

that commutes also in cohomology. For any conjugacy class of 2-stubborn subgroups of G_{2}, we have a representative P included in the normalizer of the torus (see proposition 5.4) P ⟶^{i_{P}} NT ⟶G_{2}. Composition with the map f_{N}, produces a collection of maps:

f_{P} := f_{N} ∘ Bi_{P} : BP ⟶BNT ⟶BX,

that give us a diagram:

{BP }_{R_{2}(G_{2})} → BX

that commutes in cohomology by construction. We prove that it commutes up to homotopy.

Proof of proposition 3.4. Let c_{g} : G_{2}/P → G_{2}/P be a map of R_{2}(G_{2}) given by conjugation. We have to prove that the diagram:

BP \overline{Bc_{g} _{>}} BP

BNT^{Bi_{P}^{∨}}  BNT^{Bi_{P}^{∨}}

BX^{f_{N}^{∨}} = = = = = = = BX^{f_{N}^{∨}}

commutes up to homotopy. Without loss of generality, we can assume that the subgroup P is the representative of the conjugacy class that appears in proposition 5.4. Now define α := i_{P} ∘ c_{g}, we can reformulate the problem in the following way: given a homomorphism α such that H^{∗}(f_{N} ∘ Bα) ≡ H^{∗}(f_{N} ∘ Bi_{P}) and α is conjugate to i_{P} in G_{2}, then show that f_{N} ∘ Bα ≃ f_{N} ∘ Bi_{P}.

The group P_{T} = P ∩ T is a 2-toral subgroup. The restrictions α|_{P_{T}} and i_{P} |_{P_{T}} are cojugated in G_{2}, and hence by proposition 4.1 in [17], they are also conjugate in NT, i.e., f_{N} ∘ Bα|_{P_{T}} ≃ f_{N} ∘ Bi_{P}. The following proposition shows that the extensions of f_{N} ∘ Bi_{P} |_{P_{T}} to maps BP → BX are classified by cohomology, which finishes the proof.

Proposition 7.1. For any representative 2-stubborn subgroup P of G_{2}, the canonical map

π_{0}(map(BP, BX)_{Bα|BP_{T} ≃f_{N} Bi_{P}}) → hom(H^{∗}BX, H^{∗}BP)

is an injection.

Proof. As in the proof of proposition 5.4, the case of P_{6} is trivial by Lannes’ theory. Here we consider the case P ≠ P_{6}.

The quotient Q := P/P_{T} acts on { \atop BP_{T}} ≃ EP/P_{T} ≃ BP_{T} and therefore on map({ \atop BP_{T}}, ) ≃ map(BP_{T} , ), such that map(BP, ) ≃ map({ \atop BP_{T}}, )^{hQ}. Define i_{P_{T}} := i_{P} |_{P_{T}}, and consider the induced diagram

map({ \atop BP_{T}}, BNT)_{Bi_{P_{T}}}

[figure: f_{N}]
map({ \atop BP_{T}}, BG_{2})_{Bi_{N} Bi_{P_{T}}}  map({ \atop BP_{T}}, BX)_{f_{N} Bi_{P_{T}}} .

We show that the diagonal maps are Q-equivariant homotopy equivalences.

First consider the case of the left one. By proposition 5.4 (1) and [18], both mapping spaces are homotopy equivalent (up to 2-completion) to the classifying space of the centralizer in NT and G_{2}, of P_{T} via the indicated maps. From proposition 5.4 (4), we get that both mapping spaces are homotopy equivalent to BT_{2}^{∧}.

The case of the right one needs a similar argument. By proposition 5.4 (1) and by [7], map({ \atop BP_{T}}, BX)_{f_{N} Bi_{P_{T}}} is the classifying space of a 2-compact group. Moreover, by proposition 4.3 its Weyl group is Iso(f_{N}Bi_{P_{T}}). Now, note that by contruction f_{N} ∘ Bi_{P_{T}} = f_{T} |_{P_{T}} and Bi_{N} ∘ Bi_{P_{T}} = Bi_{T} |_{P_{T}}, and therefore

Iso(f_{N} ∘ Bi_{P_{T}}) = Iso(Bi_{N} ∘ Bi_{P_{T}}) = 0,

so map({ \atop BP_{T}}, BX)_{f_{N} Bi_{P_{T}}} is a connected 2-compact group of rank 2 with trivial

Weyl group, which implies that map({ \atop BP_{T}}, BX)_{f_{N} Bi_{P_{T}}} ≃ BT_{2}^{∧} and the diagonal map induces a homotopy equivalence.

The next step of the proof is to take homotopy fixed-points. We get the diagram:

map({ \atop BP_{T}}, BNT)_{Bi_{P_{T}}}^{hQ}

[figure: f_{N}]
map({ \atop BP_{T}}, BG_{2})_{Bi_{N} Bi_{P_{T}}}^{hQ}  map({ \atop BP_{T}}, BX)_{f_{N} Bi_{P_{T}}}^{hQ} ,

(8)

where both diagonal maps are again mod 2 equivalences because an equivariant mod 2 equivalence between 1-connected spaces induces a mod 2 equivalence between the homotopy fixed-point sets. Recall that:

1) The components of map({ \atop BP_{T}}, BG_{2})_{Bi_{N} Bi_{P_{T}}}^{hQ} are distinguished by mod 2 cohomology by proposition 5.4.

2) The diagram

BNT
[figure: Bi_{NT}; f_{NT}]
BG_{2}  ϕ  ^{>} BX

commutes in mod 2 cohomology.

3) Any map in map({ \atop BP_{T}}, BX)_{f_{N} Bi_{P_{T}}}^{hQ} has a lift to BNT.

4) The obstruction group which classifies the extensions up to homotopy is isomorphic to that associated to (BG_{2})_{2}^{∧}, for:

H^{2}(Q; π_{2} map({ \atop BP_{T}}, BX)_{f_{N} Bi_{P_{T}}}) ≅ H^{2}(Q; π_{2}BT_{2}^{∧})
≅ H^{2}(Q; π_{2} map({ \atop BP_{T}}, (BG_{2})_{2}^{∧})_{Bi_{N} Bi_{P_{T}}}).

All this together implies that the components of

map({ \atop BP_{T}}, BX)_{f_{N} Bi_{P_{T}}}^{hQ} ≃ map(BP, BX)_{Bα|BP_{T} ≃f_{N} Bi_{P}}

are also distinguished by mod 2 cohomology.

We finish the section by proving the proposition 3.5.

Proof of proposition 3.5. As before, the case P_{6} is trivial and so we consider the case P ≠ P_{6}. For any 2-stubborn subgroup P of G_{2}, the diagram (8) induces the homotopy equivalences

map(BP_{T} , BNT_{2}^{∧})
[figure: ≃; ≃]
map(BP, (BG_{2})_{2}^{∧})  map(BP, BX),

which depend on the chosen lift i_{P} : BP → BNT_{2}^{∧} of i_{P} : BP → (BG_{2})_{2}^{∧}, but two lifts differ by a conjugation c_{g}, and we proved that f_{N} ∘ Bi_{P} ≃ f_{N} ∘ Bc_{g} ∘ Bi_{P}. Therefore the above equivalences induce a well defined isomorphism

Π_{i}(G_{2})(G_{2}/P) ⟶^{≅} Π_{i}(X)(G/P)

which is compatible with the maps in R_{2}(G_{2}).

8. Proof of theorems 1.2 and 1.3.

In this section we deduce theorems 1.2 and 1.3 from theorem 1.5.

Proof of theorem 1.2. Let X be a 2-complete space such that H^{∗}X ≅ H^{∗}BG_{2} as A_{2}-algebras. Then H^{∗}X = 𝔽_{2}[y_{4}, y_{6}, y_{7}], so applying mod 2 EMss, we get that H^{∗}ΩX = Λ(x_{3}, x_{5}, x_{6}), hence X can be considered as the classifying space of the 2-compact group ΩX. Now by proposition 4.1, ΩX has the same Weyl group type as G_{2} and then applying theorem 1.5 we are finished.

Proof of theorem 1.3. Since H_{ℚ_{2}}^{∗}(BX) ≅ H_{ℚ_{2}}^{∗}(BT)^{W_{G_{2}}} ≅ ℚ_{2}[y_{4}, y_{12}] we get that H_{ℚ_{2}}^{∗}X = Λ_{ℚ_{2}}(x_{3}, x_{11}), so X is a mod 2 H-space of type (3, 11), that is, of rank two. Moreover, because the map π_{1}T → π_{1}X is surjective [16], and it factors through the covariants (π_{1}T)_{W_{X}} ≅ (π_{1}T)_{W_{G_{2}}} = 0, we have that X is simply connected.

The simply connected mod 2 H-spaces of rank two are classified in [15], so we get that H^{∗}X ≅_{A_{2}} H^{∗}G_{2} and the isomorphism respects the Hopf algebra structure. By Théorèmes 17.3 and 19.2 of [1], all generators can be taken to be universally transgressive and therefore H^{∗}BX ≅_{A_{2}} H^{∗}BG_{2}.

Now, applying proposition 4.1 we know that X has the same Weyl group type as G_{2} and by theorem 1.5, we get that BX ≃ (BG_{2})_{2}^{∧}.

Now, the integral homotopy type of BG_{2} is determined by the integral representation of its Weyl group.

Proof of corollary 1.4. By [19], we have to prove that the classifying space of L is in the same genus than BG_{2}. The rational case and for p ≠ 2 follows from the same article and the case p = 2 from theorem 1.3, which completes the proof.

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Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, AP. 59, 29080 Málaga, Spain