EMBEDDINGS OF DI_{2} IN F_{4}

CARLES BROTO AND JESPER M. MØLLER

Abstract. We show that there is only one embedding of BDI_{2} in BF_{4} at the prime p = 3, up to self maps of BDI_{2}. We also describe the effect of the group of self-equivalences of BF_{4} at the prime p = 3 on this embedding and then show that the Friedlander’s exceptional isogeny composed with a suitable Adams map is an involution of BF_{4} whose homotopy fixed point set coincide with BDI_{2}

1. Introduction

A p-compact group is a p-complete analogue in homotopy theory of the concept of compact Lie group. The notion was introduced by Dwyer and Wilkerson in [12]. A p-compact group is a p-complete space BX whose loop space is F_{p}-finite, i.e. has finite mod p cohomology. The pcompletion of the classifying space of a compact connected Lie group is an example of a p-compact group but there are other, exotic, examples. Among other notions that p-compact groups share with compact Lie groups are that of a maximal torus, Weyl group, and maximal torus normalizer (see [12] for the definitions). There is also a well defined concept of homomorphism or monomorphism between p-compact groups. Since the introduction of this concept one main issue has been the description of the exotic examples. These are simply connected and irreducible p-compact groups that do not appear as the p-completion of a compact Lie group. Contrary to what happens with compact Lie groups these exotic examples can only be defined as p-complete spaces at a certain given prime p. We will focus our attention on one of these examples, namely, BDI_{2}. This is a 3-compact group with mod 3 cohomology ring

H^{∗}(BDI_{2}; F_{3}) ≅ P [x_{12}, x_{16}] , P^{1}x_{12} = x_{16} , (1.1)

the rank two Dickson algebra at the prime 3.

Theorem 1.2. Any non-trivial homomorphism from DI_{2} to F_{4} is a monomorphism and the group of automorphisms of DI_{2} acts simply transitively on the non-empty set of monomorphisms from DI_{2} to F_{4}.

There is then a monomorphism Bα: BDI_{2} → BF_{4} and it is unique up to composition with automorphisms of DI_{2}. We can rephrase it by saying that F_{4} contains a unique copy of DI_{2}.

If we compose the Friedlander’s exceptional isogeny of F_{4}, Bφ: BF_{4} → BF_{4} [14], with the Adams map ψ^{1/ζ}, where ζ^{2} = −2, then Bτ = Bψ^{1/ζ} ∘ Bϕ is a homotopy involution of BF_{4} that fixes Bα: BDI_{2} → BF_{4} up to homotopy.

Theorem 1.3. There is an action of Z/2 on BF_{4} induced up to homotopy by Bτ, for which Bα: BDI_{2} → BF_{4} induces a homotopy equivalence

BDI_{2} ≃ (BF_{4})^{hZ/2}

that identifies BDI_{2} with a homotopy fix point set of BF_{4}.

Notice that in particular the Bousfield-Kan spectral sequence for the homotopy groups of (BF_{4})^{hZ/2} degenerates to

π_{∗}(BDI_{2}) ≅ π_{∗}(BF_{4})^{Z/2}. (1.4)

As a further corollary we obtain Corollary 1.5. For any 3-complete space X, map(X, BDI_{2}) ≃ map(X, BF_{4})^{hZ/2}.

Date: February 24, 1999.

1991 Mathematics Subject Classification. 55R35, 55P15, 55P10.

Key words and phrases. Homotopy Lie groups, Dickson algebra.

C. Broto is partially supported by DGES grant 97-0203.

And applying this result to the homogeneous space F_{4}/DI_{2},

Corollary 1.6 (Harper [15]). At the prime 3 there is a splitting of H-spaces

F_{4} ≃ DI_{2} × F_{4}/DI_{2} .

Some historical considerations about the construction of this space will clarify our motivation. The way exotic examples are obtained is as realizations of graded polynomial algebras over F_{p} as mod p cohomology rings of p-completed spaces BX which will turn out to be p-compact groups. Steenrod [31] already pointed out the importance of realizing polynomial algebras as cohomology rings. The idea was reconducted by Clark-Ewing [8] and Wilkerson [34] who noticed the relevance of invariant theory to the realization problem.

Dickson algebras at odd primes are invariant algebras D(n) ≅ F_{p}[t_{1}, …, t_{n}]^{GL_{n}(F_{p})} by the action of the full general linear group of n by n matrices over F_{p} on a polynomial algebra on n independent generators of degree 2. They are shown to be again polynomial algebras of n independent generators of degrees 2(p^{n} − p^{i}) [10] [28, 33]. In [28] the authors already proved that Dickson algebras at odd primes can only appear as a cohomology ring if n = 1, in which case we obtain the cohomology of a p-local sphere of dimension 2(p − 1) or in case n = 2 and p = 3, when D(2) = F_{3}[t_{1}, t_{2}]^{GL_{2}(F_{3})} ≅ F_{3}[x_{12}, x_{16}] is the algebra of equation (1.1).

A space BDI_{2} realizing D(2) as mod 3 cohomology ring was first obtained by Zabrodsky in [36]. His method was technical and quite involved. It consisted in realizing geometrically the 3-local homotopy classes of BF_{4}, the classifying space of the exceptional Lie group F_{4}, that remain invariant under the action of the Friedlander’s exceptional isogeny. A more intelligible construction was provided by Aguadé [2]. The structure of BDI_{2} as a 3-compact group was exposed in [23]. However one important feature of Zabrodsky’s construction remained unexplained in these new approaches. Zabrodsky’s construction came together with a map BDI_{2} → BF_{4}, after 3-completion, turning H^{∗}(BDI_{2}; F_{3}) into a finitely generated H^{∗}(BF_{4}; F_{3})-module, thus representing a monomorphism of 3-compact groups. The original motivation of this note was an attempt to explain not the methods of Zabrodsky but the fact that BDI_{2} was obtained as a sort of invariant gadget in BF_{4}. This is achieved in Theorem 1.2 and equation (1.4).

The paper is organized as follows. Sections 2 and 3 are devoted to a careful investigation of p-compact group homomorphisms from DI_{2} to F_{4}. This leads to the proof of Theorem 1.2 In Section 4 we compute the cohomological structure of the homogeneous space F_{4}/DI_{2}. It turns out to coincide with that of the Harper’s molecule at the prime three, and this, in turn implies that both spaces have the same homotopy type. Details are included in an Appendix for convenience of the reader. Sections 5 and 6 are devoted to the Friedlander’s exceptional isogeny of (BF_{4})_{1/2}. It is shown that suitably composed with an Adams map becomes an involution that fixes DI_{2}. The effect of these maps in cohomological generators is determined and this leads to the proof of Theorem 1.3 and corollaries 1.5 and 1.6. Some straightforward calculations, mainly in Section two, were done with the aid of a computer using magma system. It should be easy to perform the same calculations with any other suitable program.

The authors are grateful to Bill Dwyer for a helpful discussion that led to the final form of Theorem 1.3, and also to Kasper Andersen for his help with the magma system. We kindly acknowledge the support of the Centre de Recerca Matemtica of the Institut d’Estudis Catalans and the Matematisk Institut of the Københavns Universitet that put the necessary means for the authors to meet together.

2. Admissible homomorphisms

Let X_{1} and X_{2} be p-compact groups [12] with maximal tori i_{1} : T_{1} = T (X_{1}) → X_{1}, i_{2} : T_{2} = T (X_{2}) → X_{2}, and corresponding Weyl groups W_{1} = W (X_{1}), W_{2} = W (X_{2}), respectively. Write Rep(X_{1}, X_{2}) = [BX_{1}, BX_{2}] for the set of conjugacy classes of morphisms X_{1} ⟶ X_{2}. The restriction map

Rep(X_{1}, X_{2}) ⟶^{\overline{i_{1}}} Rep(T_{1}, X_{2}) ⟵_{≅}^{\underline{i_{2}}} W_{2}\ Rep(T_{1}, T_{2})

Rep(DI_{2}, F_{4}) 3

where \overline{i_{1}} and the bijection \underline{i_{2}} [22, 3.4.(1)] are induced by i_{1} and i_{2}, takes f ∈ Rep(X_{1}, X_{2}) to f|T_{1} = W_{2}T (f) where T (f) is any lift

\xymatrix{T_{1} \ar[r]^{T (f)} \ar[d]_{i_{1}} & T_{2} \ar[d]^{i_{2}} \\ X_{1} \ar[r]_{f} & X_{2}} (2.1)

of f to a morphism between the maximal tori.

It is sometimes useful to know that certain morphisms T_{1} ⟶ T_{2} are not of the form T (f) for any f: X_{1} → X_{2}.

Lemma 2.2. (Cf. [18, 1.8]) Let f: X_{1} → X_{2} be a p-compact group morphism where p is odd and X_{1} is connected. Assume that

• π_{1}(T (f)) is injective, and

• p divides the order of the Weyl group W_{1}. Then p does not divide π_{1}(T (f)) in Hom(π_{1}(T_{1}), π_{1}(T_{2})).

Proof. By fixed point theory [13, 2.10, 2.14], f lifts to a morphism N_{p}(f): Syl_{p}(N_{1}) → Syl_{p}(N_{2}) of the p-normalizers. The assumption that π_{1}(T (f)) be injective implies, since W_{1} is faithfully represented in π_{1}(T_{1}) [12, 9.7], that π_{0}(N_{p}(f)) embeds the Sylow p-subgroup of W_{1} into W_{2}.

Choose a monomorphism μ: Z/p → Syl_{p}(N_{1}) such that also π_{0}(μ): Z/p → Syl_{p}(W_{1}) is injective. This is possible since the epimorphism Syl_{p}(N_{1}) ⟶ Syl_{p}(W_{1}) admits a section when p is odd [3]. Note that the composition N_{p}(f)μ is a monomorphism since it induces a monomorphism on component groups. Consider now the commutative diagram

\xymatrix{ & Syl_{p}(N_{1}) \ar[d]_{j_{p}} \ar[r]^{N_{p}(f)} & Syl_{p}(N_{2}) \ar[d] \\ Z/p \ar[ur]^{μ} \ar[r]^{j_{p}μ} \ar[dr]_{μ′} & X_{1} \ar[r]^{f} & X_{2} \\  & T_{1} \ar[u] \ar[r]_{T (f)} & T_{2} \ar[u]_{i_{2}}}

where μ′ is a lift of j_{p}μ [12, 4.7, 5.6]. Since N_{p}(f)μ is monomorphic, so is i_{2}T (f)μ′ by commutativity of the diagram. However, this map would be trivial were π_{1}(T (f)) divisible by p.

This lemma is also true for p = 2 provided the extra assumption

• there exists a monomorphism Z/2 ⟶ Syl_{2}(W_{1}) that factors through Syl_{2}(N_{1}) is added.

We now return to the general situation of (2.1). For any element w_{1} of the Weyl group W_{1} of X_{1},

\underline{i}_{2}(T (f) ∘ w_{1}) = i_{2} ∘ T (f) ∘ w_{1} = f ∘ i_{1} ∘ w_{1} = f ∘ i_{1} = i_{2} ∘ T (f) = \underline{i}_{2}(T (f))

so there must exist some Weyl group element w_{2} ∈ W_{2} such that T (f) ∘ w_{1} = w_{2} ∘ T (f). Thus the lift T (f) lies in the subset W_{2}\ Adm(T_{1}, T_{2}) where

Definition 2.3. Adm(T_{1}, T_{2}) = {a ∈ Rep(T_{1}, T_{2}) | a · W_{1} ⊆ W_{2} · a}

The fundamental group functor can be used to identify this set Adm(T_{1}, T_{2}) of admissible representations with the corresponding set Adm(π_{1}(T_{1}), π_{1}(T_{2})) ⊆ Hom(π_{1}(T_{1}), π_{1}(T_{2})) of admissible homomorphisms, i.e. homomorphisms a: π_{1}(T_{1}) → π_{1}(T_{2}) such that a · W_{1} ⊆ W_{2} · a.

We shall now determine the admissible homomorphisms in the particular case where p = 3, X_{1} = DI_{2} [23, 2], and X_{2} = F_{4} viewed as a 3-compact group.

The construction of BDI_{2} exhibits monomorphisms [23]

T_{1} ⟶ SU(3) ⟶ DI_{2} (2.4) of maximal rank where the morphism T_{1} = T (SU(3)) ⟶ SU(3) is a maximal torus for SU(3). Take the composite morphism (2.4) as the maximal torus for DI_{2}.

Let i_{2} : T_{2} = T (F_{4}) → F_{4} be a maximal torus for the simple compact Lie group F_{4} (defining a maximal torus for the corresponding 3-compact group).

There is a standard identification

π_{1}(T_{1}) ⟶^{≅} Σ_{0}(Z_{3}^{3}) = {(x_{1}, x_{2}, x_{3}) ∈ Z_{3}^{3} | x_{1} + x_{2} + x_{3} = 0}

and we shall use {(1, −1, 0), (0, 1, −1)} as a basis for the free Z_{3}-module Σ_{0}(Z_{3}^{3}) thus identifying Σ_{0}(Z_{3}^{3}) with Z_{3}^{2}. Under this identification we may assume [30, 7.3, Example 1] [23] that the Weyl group W_{1} = W (DI_{2}) of DI_{2} corresponds to the subgroup

W (DI_{2}) = ⟨α, σ, τ⟩ ⊆ Aut(Σ_{0}(Z_{3}^{3})) = GL(2, Z_{3})

whith generators

α = \frac{1}{1 + ζ} ( \begin{matrix}−1 + ζ & −ζ \\ ζ & 1 − ζ \end{matrix} ) , σ = ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} ) , τ = ( \begin{matrix}0 & −1 \\ −1 & 0 \end{matrix} )

where ζ^{2} + 2 = 0 and ζ ≡ 1 mod 3. The subgroup W (SU(3)) = ⟨σ, τ⟩ ≅ Σ_{3} is the Weyl group of SU(3).

Similarly, there is a standard identification [6] between the fundamental group π_{1}(T_{2}) of the maximal torus for F_{4} and the free abelian group

Σ_{2}(Z^{4}) = {(x_{1}, x_{2}, x_{3}, x_{4}) ∈ Z^{4} | x_{1} + x_{2} + x_{3} + x_{4} ∈ 2Z}

Under this identification, the Weyl group W_{2} = W (F_{4}) of F_{4} is carried to the group (of order 1152 = 384 · 3)

W (F_{4}) = W (B_{4})E ∪ W (B_{4})H_{1} ∪ W (B_{4})H_{2} (2.5)

where W (B_{4}) is the reflection group (of order 384 = 2^{4} · 4!) of all signed permutation matrices, and H_{1} and H_{2} are the matrices

H_{1} = \frac{1}{2} ( \begin{matrix}−1 & 1 & 1 & 1 \\ 1 & −1 & 1 & 1 \\ 1 & 1 & −1 & 1 \\ 1 & 1 & 1 & −1 \end{matrix} ) , H_{2} = \frac{1}{2} ( \begin{matrix}1 & 1 & 1 & 1 \\ 1 & 1 & −1 & −1 \\ 1 & −1 & 1 & −1 \\ 1 & −1 & −1 & 1 \end{matrix} )

Note that the group W (B_{4}) and the matrices H_{1} and H_{2} preserve the submodule Σ_{2}(Z^{4}) of Z^{4}. The operators H_{1} and H_{2} satisfy the relations H_{1}^{2} = E = H_{2}^{2}, H_{2}H_{1} = −H_{2}, H_{1}H_{2} = diag(−1, 1, 1, 1)H_{1}. For F_{4} viewed as a 3-compact group with maximal torus T_{2}, extensions of scalars provides an identification

π_{1}(T_{2}) ⟶^{≅} Σ_{2}(Z^{4}) ⊗ Z_{3} ⟶^{≅} Z^{4} ⊗ Z_{3} ≅ Z_{3}^{4}

taking the Weyl group W (F_{4}) onto the reflection group W (F_{4}) < GL(4, Z_{3}) as defined in (2.5). The linear map A(v): Σ_{0}(Z_{3}^{3}) → Z_{3}^{4} with matrix

A(v) = ( \begin{matrix}−ζ & 1 \\ ζ & 1 − ζ \\ 0 & 1 + ζ \\ −2 & 1 \end{matrix} ) , v ∈ Z_{3}, (2.6)

is admissible with respect to W (DI_{2}) and W (F_{4}) since A(v)w = χ(w)A(v) for all w ∈ W (DI_{2}) where χ: W (DI_{2}) → W (F_{4}) is the group homomorphism with the values

( \begin{matrix}0 & 0 & 0 & −1 \\ 0 & 0 & −1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ) , \frac{1}{2} ( \begin{matrix}−1 & −1 & 1 & −1 \\ 1 & −1 & −1 & −1 \\ −1 & 1 & −1 & −1 \\ 1 & 1 & 1 & −1 \end{matrix} ) , \frac{1}{2} ( \begin{matrix}−1 & −1 & 1 & 1 \\ −1 & 1 & −1 & 1 \\ 1 & −1 & −1 & 1 \\ 1 & 1 & 1 & 1 \end{matrix} )

on the generators α, σ, and τ, respectively.

Note that A(v) and −A(v) lie in the same orbit under the action of W (F_{4}) as −E ∈ W (F_{4}).

Lemma 2.7. Let v be a 3-adic integer.

1. The linear homomorphism A: Σ_{0}(Z_{3}^{3}) → Z_{3}^{4} is admissible with respect to W (DI_{2}) and W (F_{4}) if and only if A ∈ W (F_{4})A(v) for some 3-adic integer v ∈ Z_{3}.

2. The linear map A(v) is split injective if and only if v ∈ Z_{3}^{∗} is a 3-adic unit.

3. The map

{±1}\Z_{3}^{∗} ⟶ W (F_{4})\ Hom_{Z_{3}}(Σ_{0}(Z_{3}^{3}), Z_{3}^{4})
±v ⟶ W (F_{4})A(v)

is injective.

The proof, which we omit, is by direct (computer assisted) computation.

3. Constructing maps out of BDI_{2}

Let I = I(W (DI_{2}), W (SU(3))) be the category with two objects, 0 and 1, and morphism sets I(0, 0) = N_{W (DI_{2})}(W (SU(3))/W (SU(3)) ≅ Z(W (DI_{2})) ≅ Z/2, I(1, 1) = W (DI_{2}), I(0, 1) = W (DI_{2})/W (SU(3)), and I(1, 0) = ∅.

The space BDI_{2} can be constructed [23, 3.5] as the homotopy colimit of the I^{op}-space

[figure: B SU(3); (Z/2)^{op}; BT; W (DI_{2})^{op}; W (SU(3))^{op}\W (DI_{2})^{op}]

(3.1)

where Z/2 acts on B SU(3) as {ψ^{±1}}. We shall use diagram (3.1) in connection with Wojtkowiak obstruction theory [35] to prove existence and uniqueness of certain maps out of BDI_{2}.

The set of conjugacy classes of monomorphisms, Mono(DI_{2}, F_{4}) of monomorphisms DI_{2} ↣ F_{4} will turn out to be faithfully represented in the set Mono(SU(3), F_{4}) of conjugacy classes of monomorphisms SU(3) ↣ F_{4} which we now describe [25]:

There exists a bijection

{(u, v) ∈ (Z_{3})^{2} | u + v ≡ 1 mod 3} ⟶ Mono(SU(3), F_{4})
(u, v) ⟶ eψ^{(u,v)}

where ψ^{(u,v)} is the composite morphism

SU(3) ⟶^{Δ} SU(3) × SU(3) ⟶^{ψ^{u}×ψ^{v}} SU(3) × SU(3) ⟶ SU(3, 3)

and e: SU(3, 3) → F_{4} is the inclusion, described in [18, 3.3], of SU(3, 3) = SU(3) ×_{Z/3} SU(3) as a maximal rank subgroup of F_{4}. (The central Z/3 in SU(3) × SU(3) is generated by (zE, z^{−1}E) where z ≠ 1 is a third root of unity.) Furthermore, the injective restriction map

(3.2) Mono(SU(3), F_{4}) ⟶ Mono(T (SU(3)), F_{4}) = W (F_{4})\ Hom_{Z_{3}}(Σ_{0}(Z_{3}^{3}), Z_{3}^{4})

takes the monomorphism eψ^{(u,v)} to

W (F_{4}) ( \begin{matrix}−u & v \\ u & −u + v \\ 0 & u + v \\ −2v & v \end{matrix} ) (3.3)

Lemma 3.4. Let f ∈ Mono(DI_{2}, SU(3)) be a monomorphism. Then the restriction to SU(3),

f| SU(3) = eψ^{(ζv,v)}

for a uniquely determined 3-adic unit v ∈ Z_{3}^{∗}, v ≡ −1 mod 3.

Proof. According to (2.7), the restriction of f to the maximal torus,

f|T (DI_{2}) ∈ Mono(T (DI_{2}), F_{4}) = W (F_{4})\ Hom_{Z_{3}}(Σ_{0}(Z_{3}^{3}), Z_{3}^{4})

is of the form f|T (DI_{2}) = W (F_{4})A(v) for some 3-adic unit v ∈ Z_{3}^{∗}, uniquely determined up to sign. (The 3-adic number v must be a unit because A(v) is split injective as f|T (DI_{2}) is a monomorphism [26] [24].) But this means that the restriction f| SU(3) = eψ^{(ζv,v)} for a uniquely determined 3-adic unit v ∈ Z_{3}^{∗}, v ≡ −1 mod 3.

Let now v ∈ Z_{3}^{∗} be any 3-adic unit such that v ≡ −1 mod 3. Then the two homotopy classes eψ^{(ζv,v)} ∈ [B SU(3), BF_{4}] and W (F_{4})A(v) ∈ [BT (SU(3)), BF_{4}] form a homotopy coherent pair of maps out of the I^{op}-space (3.1) in the sense that • eψ^{(ζv,v)}|T (SU(3)) = W (F_{4})A(v)

• W (F_{4})A(v) is W (DI_{2})-invariant

• eψ^{(ζv,v)} is ⟨ψ^{−1}⟩-invariant where the last property follows from the computation

eψ^{(ζv,v)}ψ^{−1} = e(ψ^{−1} × ψ^{−1})ψ^{(ζv,v)} = eψ^{(ζv,v)}

which uses the identity e(ψ^{−1} × ψ^{−1}) = e from [18, 3.3]. Therefore, there is an induced I-space

(3.5)  [figure: BC_{F_{4}}(eψ^{(ζv,v)}); W (DI_{2})/W (SU(3)); Z/2; BC_{F_{4}}(W (F_{4})A(v)); W (DI_{2})]

of connected mapping spaces where

BC_{F_{4}}(eψ^{(ζv,v)}) = map(BSU(3), BF_{4})_{Beψ^{(ζv,v)}} ≃ BZ(SU(3))
BC_{F_{4}}(W (F_{4})A(v)) map(BT (SU(3)), BF_{4})_{B(W (F_{4})A(v))} ≃ BT (F_{4})

as eψ^{(ζv,v)} is centric and p-toric [25]. These two spaces are simple so we may apply the homotopy functor π_{t} to (3.5) to obtain the I-module \underline{π}_{t}(v).

Lemma 3.6. lim_{I}^{−s} \underline{π}_{t}(v) = 0 for all s ≤ 0, t ≥ 0, s + t ≥ −1 and for all 3-adic units v ∈ Z_{3}^{∗} with v ≡ −1 mod 3.

Proof. For the I-module \underline{π}_{1}(v),

[figure: Z/3; Z/2; W (DI_{2})/W (SU(3)); 0; W (DI_{2})]

we have lim_{I}^{−s} \underline{π}_{1}(v) = H^{−s}(Z/2; Z/3) which is trivial since Z/2 = ⟨ψ^{−1}⟩ acts non-trivially on the center Z/3 = Z(SU(3)). For the I-module \underline{π}_{2}(v),

[figure: 0; Z/2; W (DI_{2})/W (SU(3)); Z_{3}^{4}; W (DI_{2})]

we have lim_{I}^{0} \underline{π}_{2}(v) = 0, lim_{I}^{1} \underline{π}_{2}(v) = (Z_{3}^{4})^{Z(W (DI_{2})×W (SU(3))} = 0 as −E belongs to the center Z(W (DI_{2})), and lim_{I}^{−s} \underline{π}_{2}(v) = 0 for s ≤ −2 by [24, 3.8].

With this lemma in place we are ready for one of the main results of this paper.

Theorem 3.7. The following hold for the set Mono(DI_{2}, F_{4}) of conjugacy classes of monomorphisms DI_{2} ⟶ F_{4}:

1. The restriction map

Mono(DI_{2}, F_{4}) ⟶ Mono(SU(3), F_{4})

is an injection with {eψ^{(ζv,v)} | v ∈ Z_{3}^{∗}, v ≡ −1 mod 3} as its image.

2. The restriction map

Mono(DI_{2}, F_{4}) ⟶ Mono(T (DI_{2}), F_{4})

is an injection with {W (F_{4})A(v) | v ∈ Z_{3}^{∗}, v ≡ −1 mod 3} as its image.

Proof. 1. This follows from Wojtkowiak’s obstruction theory [35] as lim_{I}^{−s} \underline{π}_{t}(v) = 0 when s+t = 0 and s + t = −1.

2. The map is a composition

Mono(DI_{2}, F_{4}) ⟶ Mono(SU(3), F_{4}) ⟶ Mono(T (SU(3)), F_{4})

of two injections (3.2).

Let f(v): DI_{2} → F_{4}, v ∈ Z_{3}^{∗}, v ≡ −1 mod 3, denote the unique extension to DI_{2} of the monomorphism eψ^{(ζv,v)} : SU(3) → F_{4} on SU(3).

Corollary 3.8. 1. The group Out(DI_{2}) acts simply transitively on the set Mono(DI_{2}, F_{4}) of

monomorphisms.

2. The centralizer of any monomorphism DI_{2} ⟶ F_{4} is trivial.

3. The Weyl group [23] of any monomorphism DI_{2} ⟶ F_{4} is trivial.

Proof. 1. This is clear since [23] Out(DI_{2}) ≅ {v ∈ Z_{3}^{∗} | v ≡ 1 mod 3} and

f(v)ψ^{u} = eψ^{(ζv,v)}ψ^{u} = eψ^{(ζvu,vu)} = f(vu)

so that f(v)ψ^{u} = f(v) ⇔ u = 1.

2. The E_{2}-page of the Bousfield-Kan spectral sequence, E_{s}^{st} = lim_{I}^{−s} \underline{π}_{t}(v), converging to π_{s+t}(BC_{F_{4}}(f(v)DI_{2})) vanishes completely (3.6).

3. The Weyl group W_{F_{4}}(f(v)DI_{2}) is [23] the stabilizer subgroup for Out(DI_{2}) acting on f(v) ∈ Mono(DI_{2}, F_{4}). We have just seen that this stabilizer is trivial.

In a way, (3.8.1) says that F_{4} contains a unique copy of DI_{2}.

Finally, we determine the set Rep(DI_{2}, F_{4}) = [BDI_{2}, BF_{4}] of all maps up to homotopy.

Proposition 3.9. Any non-trivial morphism of DI_{2} to F_{4} is a monomorphism.

Proof. Let f: DI_{2} → F_{4} be any non-trivial morphism. Then f|T (DI_{2}) = W (F_{4})A(v) ∈ [BT (DI_{2}), BF_{4}] for some 3-adic integer v ∈ Z_{3}. This 3-adic integer, v, must be non-zero as f is non-trivial [22, 6.7] and even a 3-adic unit by (2.2). This shows that A(v) is split injective and hence [26, 3.6] [24, 5.2] that f is a monomorphism.

Proof of Theorem 1.2. We conclude from (3.7, 3.8, 3.9) that

Rep(DI_{2}, F_{4}) = {0} ∪ Mono(DI_{2}, F_{4})

is in bijection with the set {0} ∪ Z_{3}^{∗}/{±1}.

In the following, we let Bα: BDI_{2} → BF_{4} denote the monomorphism Bf(−1) corresponding (3.7) to the admissible homomorphism with matrix A(−1) (2.6).

4. The homogeneous space F_{4}/DI_{2}

In this section we compute the mod 3 cohomology of the exotic homogeneous space F_{4}/DI_{2}. The unstable mod 3 cohomology of F_{4} is known since Borel [5]:

H^{∗}(F_{4}; F_{3}) ≅ E[z_{3}, z_{7}, z_{11}, z_{15}] ⊗ P [z_{8}]/(z_{8}^{3})

with P^{1}z_{3} = z_{7}, βz_{7} = z_{8} and P^{1}z_{11} = z_{15}.

The 3-complete space BDI_{2} realizes the rank 2 mod 3 Dickson algebra meaning that

H^{∗}(BDI_{2}; F_{3}) ≅ P [x_{12}, x_{16}]

with P^{1}x_{12} = x_{16}.

The homogeneous space F_{4}/DI_{2} is defined as the homotopy fibre of the map BDI_{2} ⟶^{Bα} BF_{4}, and we have a sequence of fibrations

(4.1) DI_{2} ⟶^{ΩBα} F_{4} ⟶^{π} F_{4}/DI_{2} ⟶^{j} BDI_{2} ⟶^{Bα} BF_{4}

We now investigate the value of the functor H^{∗}(−; F_{3}) on this sequence.

Proposition 4.2. The effects on mod 3 cohomology of the maps π and j of (4.1) are as follows:

1. The unstable mod 3 cohomology algebra of F_{4}/DI_{2} is

H^{∗}(F_{4}/DI_{2}; F_{3}) ≅ E[z_{3}, z_{7}] ⊗ P [z_{8}]/(z_{8}^{3})

with P^{1}z_{3} = z_{7} and βz_{7} = z_{8}.

2. H^{∗}(π; F_{3}) is the obvious inclusion

H^{∗}(F_{4}/DI_{2}; F_{3}) ≅ E[z_{3}, z_{7}] ⊗ P [z_{8}]/(z_{8}^{3}) ⊆ E[z_{3}, z_{7}, z_{11}, z_{15}] ⊗ P [z_{8}]/(z_{8}^{3}) ≅ H^{∗}(F_{4}; F_{3})

of algebras.

3. H^{>0}(j; F_{3}) is the trivial homomorphism H^{>0}(BDI_{2}; F_{3}) ⟶ H^{>0}(F_{4}/DI_{2}; F_{3}).

Proof. It is a calculation with the Serre spectral sequence for the fibration F_{4} ⟶^{π} F_{4}/DI_{2} ⟶^{j} BDI_{2}. This starts at E_{2}^{p,q} ≅ H^{p}(BDI_{2}; F_{3}) ⊗ H^{q}(F_{4}; F_{3}).

By degree reasons the classes z_{3}, z_{7} and z_{8} in the vertical edge are pemanent cycles. Assume that z_{11} ∈ E_{2}^{0,∗} is a permanent cycle. Then z_{15} = P^{1}z_{11} is also a permanent cycle and therefore the spectral sequence collapses at E_{2}. But this is impossible because the fibre F_{4}/DI_{2} of the monomorphism Bα is F_{3}-finite [12].

Hence, the class z_{11} transgresses to ±x_{12} ∈ H^{∗}(BDI_{2}; F_{3}) ≅ E_{2}^{∗,0} and then z_{15} = P^{1}z_{11} trangresses to ±x_{16} = P^{1}(±x_{12}). After this last, the spectral sequence collapses to E_{∞}^{∗,∗} ≅ E_{∞}^{0,∗} ≅ E[z_{3}, z_{7}] ⊗ P [z_{8}]/(z_{8}^{3}) and the edge homomorphisms are an injection

H^{∗}(F_{4}/DI_{2}; F_{3}) ≅ E_{∞}^{0,∗} ≅ E[z_{3}, z_{7}] ⊗ P [z_{8}]/(z_{8}^{3})↣E_{2}^{0,∗} ≅ H^{∗}(F_{4}; F_{3})

and the trivial homomorphism

H^{∗}(BDI_{2}; F_{3}) ≅ E_{2}^{∗,0}↠E_{∞}^{0,∗} ≅ F_{3} ⊂ H^{∗}(F_{4}/DI_{2}; F_{3})

which proves the proposition.

Remark 4.3. The mod 3 cohomology of F_{4}/DI_{2} coincides with that of Harper’s molecule [16] at the prime 3 and this implies that they are actually homotopy equivalent up to 3-completion. Details will be worked out in Section 7.

5. Friedlander’s exceptional isogeny

In [14], E.M. Friedlander showed the existence of a self-homotopy equivalence Bφ of (BF_{4})_{1/2} that restricts to the maximal torus to the isogeny BT (φ): BT (F_{4}) → BT (F_{4}) determined by the matrix

π_{2}(BT (φ)) = (\begin{matrix}1 & −1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & −1 \\ 0 & 0 & 1 & 1 \end{matrix})

acting on π_{2}(BT (F_{4})) ≅ Z_{3}^{4}. The automorphism Bφ of the 3-compact group BF_{4} satisfies the relation Bφ ∘ Bφ ≃ Bψ^{2}. To see this, note that W (F_{4})T (φ)^{2} = W (F_{4})T (ψ^{2}) and recall [21] that BF_{4} has N-determined automorphisms.

Proposition 5.1. There is a homotopy commutative diagram

\xymatrix{BDI_{2} \ar[r]^{Bα} \ar[d]_{Bψ^{ζ}} & BF_{4} \ar[d]^{Bφ} \\ BDI_{2} \ar[r]_{Bα} & BF_{4}}

(5.2)

where ζ^{2} = −2 and Bψ^{ζ} ∈ [BDI_{2}, BDI_{2}] is the corresponding Adams map.

Proof. According to Theorem 3.7 it is enough to check the restriction to the maximal torus of BDI_{2}, and in fact,

π_{2}(BT (φ)) · π_{2}(BT (α)) = (\begin{matrix}1 & −1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & −1 \\ 0 & 0 & 1 & 1 \end{matrix}) · (\begin{matrix}ζ & −1 \\ −ζ & −1 + ζ \\ 0 & −1 − ζ \\ 2 & −1 \end{matrix})

= (\begin{matrix}2ζ & −ζ \\ 0 & −2 + ζ \\ −2 & −ζ \\ 2 & −2 − ζ \end{matrix}) = (\begin{matrix}−2 & 1 \\ 0 & −ζ − 1 \\ −ζ & 1 \\ −ζ & −ζ + 1 \end{matrix}) · ζ ∈ W (F_{4})(π_{2}(BT (α)) · ζ)

hence Bφ ∘ Bα = Bα ∘ Bψ^{ζ}.

Proposition 5.3. The isomorphism H^{∗}(Bφ; Q_{3}) of H^{∗}(BF_{4}; Q_{3}) ≅ P [x_{4}, x_{12}, x_{16}, x_{24}] is, up to decomposables, determined by

Bφ^{∗}(x_{4}) = 2x_{4} , Bφ^{∗}(x_{12}) = −8x_{12} , Bφ^{∗}(x_{16}) = 16x_{16} , Bφ^{∗}(x_{24}) = −64x_{24} .

Proof. This is a long but straightforward calculation, easier done with the aid of a computer. This result coincides also with calculation done by Adams and Mahmud [1, Table 2.14].

Define Bτ: BF_{4} → BF_{4} to be the automorphism Bτ = Bψ^{1/ζ} ∘ Bφ. Then (Bτ)^{2} is homotopic to the identity and Bτ ∘ Bα = Bα by (5.1).

Corollary 5.4. H^{4}(Bτ; Z_{3}) is multiplication by −1 on H^{4}(BF_{4}; Z_{3}).

Proof. The effect of H^{4}(Bτ; Q_{3}) on H^{4}(BF_{4}; Q_{3}) is given by

H^{4}(Bτ; Q_{3}) = H^{4}(Bφ; Q_{3}) ∘ H^{4}(Bψ^{1/ζ}; Q_{3}) = 2 · ζ^{−2} = −1

by (5.3) and because the unstable Adams operation ψ^{λ} induces multiplication by λ^{i} in degree 2i.

Let τ: F_{4}/DI_{2} → F_{4}/DI_{2} denote the self-homotopy equivalence of the exotic homogeneous space F_{4}/DI_{2} induced by Bτ on BF_{4} and the identity on BDI_{2}. This map makes the diagram

[figure: F_{4}/DI_{2}; τ; j; BDI_{2}; Bα; BF_{4}; Bτ; F_{4}/DI_{2}; j; BDI_{2}; Bα; BF_{4}]

commute up to homotopy.

Corollary 5.5. The involution H^{∗}(τ; F_{3}) of H^{∗}(F_{4}/DI_{2}; F_{3}) sends the generators z_{3}, z_{7}, and z_{8} to −z_{3}, −z_{7}, and −z_{8}, respectively.

Proof. That H^{∗}(τ; F_{3})(z_{3}) = −z_{3} follows from (5.4) since (4.2.2) the mod 3 cohomology of F_{4}/DI_{2} embeds in the mod 3 cohomology of F_{4}. As the two other generators are linked to z_{3} by Steenrod operations (4.2.1), H^{∗}(τ; F_{3})(z_{3}) must also act as multiplication by −1 on them.

We shall return to the homotopy involution Bτ in Section 6.

6. BDI_{2} as a homotopy fixed point space

The automorphism Bτ = Bψ^{1/ζ} ∘Bϕ of BF_{4} is a homotopy involution, (Bτ)^{2} ≃ 1, that homotopy fixes BDI_{2} in the sense that the diagram

[figure: BDI_{2}; Bα; BF_{4}; Bτ; BDI_{2}; Bα; BF_{4}]

(6.1)

commutes up to homotopy (5.1). This suggests that BDI_{2} should be fixed under Bτ in some sense. To make this precise, we need to rigidify the above two properties.

Proposition 6.2. We may assume that the maps Bα and Bτ satisfy the identities Bτ ∘ Bα = Bα and Bτ ∘ Bτ = 1.

Proof. Let D be the category

[figure: ·; a; ·; b]

with two objects and two non-identity morphisms, a and b, subject to the relations ba = a and bb = 1. The D-diagram

[figure: BDI_{2}; Bα; BF_{4}; Bτ] (6.3)

in the homotopy category of spaces is centric; in fact, all the relevant mapping spaces are weakly contractible (3.8.2) [23, 13]. Hence (6.3) has an essentially unique realization in the category of spaces [11, 1.1].

With such specific realizations of Bα and Bτ, let F_{4}/DI_{2} denote the homotopy fibre of Bα over any point fixed by Bτ, e.g. any point in the image of Bα. Then F_{4}/DI_{2} is Z/2-space and there is a homotopy fibre sequence

F_{4}/DI_{2} ⟶^{j} BDI_{2} ⟶^{Bα} BF_{4}

of Z/2-spaces and Z/2-maps which induces [7, XI.7.1] a homotopy fibre sequence

(6.4) (F_{4}/DI_{2})^{hZ/2} ⟶^{j^{hZ/2}} BDI_{2}^{hZ/2} ⟶^{(Bα)^{hZ/2}} BF_{4}^{hZ/2}

of homotopy fixed point spaces. Here,

BDI_{2}^{hZ/2} ≃ map(BZ/2, BDI_{2}) ≃ BDI_{2}

since Z/2 acts trivially on the 3-complete space BDI_{2}. As to the base space, the Bousfield-Kan spectral sequence for the homotopy fixed point space BF_{4}^{hZ/2} degenerates to the formula π_{∗}(BF_{4}^{hZ/2}) = π_{∗}(BF_{4})^{Z/2}. In particular, this space is non-empty and connected and its homotopy consists of the invariant part of the homotopy for BF_{4}, cf. [36]. The aim is to show that the fibre of (6.4) is contractible in order to obtain a homotopy equivalence between BDI_{2} and BF_{4}^{hZ/2}.

Proposition 6.5. The fibre, (F_{4}/DI_{2})^{hZ/2}, of fibration sequence (6.4) is contractible.

Proof. By the general result at the end of this section (6.8), it suffices to show that the the space E sitting in the homotopy pull back diagram

\xymatrix{E \ar[d] \ar[r] & F_{4}/DI_{2} \ar[d]^{(1,τ)} \\ F_{4}/DI_{2} \ar[r]_{Δ} & (F_{4}/DI_{2})^{2}}

(6.6)

is contractible. The Eilenberg-Moore spectral sequence E_{r}^{∗∗} ⇒ H^{∗}(E; F_{3}) associated to (6.6) is a second quadrant cohomological spectral sequence with

E_{2}^{−pq} = Tor_{p}^{R⊗R}(R, R)^{q}

where R = H^{∗}(F_{4}/DI_{2}; F_{3}) and where R is a right R ⊗ R-module through the cup product μ: R ⊗ R → R and a left R ⊗ R-module through μ ∘ (1 ⊗ τ^{∗}). We shall show that E_{2}^{∗∗} = F_{3}.

Let κ be the algebra isomorphism of R ⊗ R given by

κ(z_{i} ⊗ 1) = z_{i} ⊗ 1 − 1 ⊗ z_{i}, κ(1 ⊗ z_{i}) = 1 ⊗ z_{i}

Then κ satisfies the identity ε ⊗ 1 = μκ, where ε: R → F_{3} is the augmentation homomorphism, and hence (6.10)

E_{2}^{−p∗} = Tor_{p}^{R}(F_{3}, R)

where R is a left R-module via the algebra morphism

R ⟶^{1⊗η} R ⊗ R ⟶^{κ} R ⊗ R ⟶^{1⊗τ^{∗}} R ⊗ R ⟶^{μ} R

which takes z_{i} to −z_{i}, i = 1, 2, 3. In particular, R is a free R-module and the vanishing of the E_{2}-page for the Eilenberg-Moore spectral sequence follows.

Proof of Theorem 1.3. By (6.4, 6.5), the monomorphism Bα: BDI_{2} → BF_{4} induces a homotopy equivalence BDI_{2} ≃ (BF_{4})^{hZ/2}.

Proof of (1.4) and Corollary 1.5. It is a general rule that map(X, Y^{hG}) = map(X, Y)^{hG} and also that π_{i}(M^{hG}) = π_{i}(M)^{G} when M is p-complete and p ∤ |G|.

Proof of Corollary 1.6. As a consequence of (1.5) and exactness of the Barratt-Puppe sequence, the map j: F_{4}/DI_{2} → BDI_{2} is null-homotopic. The splitting is now constructed using a section of π: F_{4} → F_{4}/DI_{2} and the H-space structure of F_{4}.

In the homotopy splitting from Corollary 1.6 of F_{4}, F_{4}/DI_{2} ≃ K(3) is one of Harper’s H-spaces (7.5) and DI_{2} ≃ B_{5}(3) is one of the Mimura-Toda bundles of completed spheres over spheres [20].

We finish this section with the general results that we used above to compute the homotopy fixed point space (F_{4}/DI_{2})^{hZ/2}.

First, let G be a finite group and K a G-space. Define E to be the homotopy pullback

\xymatrix{E \ar[d] \ar[r] & K \ar[d]^{μ} \\ K \ar[r]_{Δ} & map(G, K)} (6.7)

where Δ takes x ∈ K to the constant map with value x and the action map μ takes x to the map μ(k)(h) = h^{−1}x, h ∈ G. These two maps are equivariant if we equip map(G, K) with the action (gu)(h) = u(g^{−1}h), g, h ∈ G, u ∈ map(G, K). Thus (6.7) is a diagram of G-spaces and G-maps.

Theorem 6.8. If K is BG-null, the G-map E ⟶ K induces a weak homotopy equivalence E^{hG} ⟶ K^{hG} of homotopy fixed point spaces.

Proof. Since the homotopy fixed point functor commutes with homotopy pull-backs [7, XI.4.3], there is an induced homotopy pull-back diagram

\xymatrix{E^{hG} \ar[d] \ar[r] & K^{hG} \ar[d]^{μ^{hG}} \\ K^{hG} \ar[r]_{Δ^{hG}} & map(G, K)^{hG}}

of homotopy fixed point spaces. The map Δ^{hG}: K^{hG} → map(G, K)^{hG} can be identified to the evaluation map map(BG, K) ⟶ K. Indeed, K^{hG} = map(BG, K), since the action is trivial in the lower left corner, and map(G, K)^{hG} = map(G_{hG}, K) = map(∗, K) = K. If K is BG-null, Δ^{hG} is a homotopy equivalence and so is then the top horizontal map E^{hG} ⟶ K^{hG}.

By an argument dual to that of [19, 5.1] [27, p. 282], we may identify E and the homotopy limit of the diagram K ⟶^{G} K consisting of the |G| maps K ⟶ K given by x ⟶ gx for g ∈ G.

The final result that we used above was an algebraic computation ensuring the vanishing of an Eilenberg-Moore spectral sequence.

Let R be a unital and augmented (graded) algebra over a field k. The structure maps

μ: R ⊗ R → R, η: k → R, ε: R → k

make R a right R ⊗ R-module and k a right R-module.

For any right R-module A, the tensor product A ⊗ R is a right R ⊗ R-module with right R ⊗ R-multiplication given by

(a ⊗ r) · (s ⊗ t) = as ⊗ rt

for all elements a ∈ A, r, s, t ∈ R (modified by the usual sign in the graded case). When A = k, in particular, right multiplication in k ⊗ R = R by s ⊗ t ∈ R ⊗ R is right multiplication in R by (ε ⊗ 1)(s ⊗ t) = ε(s)t.

Lemma 6.9. Let κ be an algebra isomorphism of R ⊗ R and ⊗_{R}(R ⊗ R) the left adjoint functor to the forgetful functor induced by the algebra homomorphism κ ∘ (1 ⊗ η): R → R ⊗ R. Then

1. A ⊗_{R} (R ⊗ R) = A ⊗ R with right R ⊗ R-multiplication (a ⊗ r)(s ⊗ t) = (a ⊗ r) · κ^{−1}(s ⊗ t)

2. Tor_{p}^{R}(A, B) = Tor_{p}^{R⊗R}(A ⊗_{R} (R ⊗ R), B) and Ext_{R}^{p}(A, B) = Ext_{R⊗R}^{p}(A ⊗_{R} (R ⊗ R), B) for all right R-modules A, all left R ⊗ R-modules B, and all p ≥ 0.

Proof. We have

U(κ(1 ⊗ η)) = U(1 ⊗ η)U(κ) and U(κ)L(κ(1 ⊗ η)) = L(1 ⊗ η)

where L(f) denotes the left adjoint to the forgetful functor U(f) induced by the algebra homomorphism f. Note that L(1⊗η)(A) = A⊗R. It follows that L(1⊗η) is exact; it also takes R-projectives to R⊗R-projectives because it is left adjoint to a (right) exact functor [32, 2.3.10]. The same is true of the functor L(κ(1 ⊗ η)) which differs from L(1 ⊗ η) by an isomorphism. Similarly, U(κ(1 ⊗ η)) is an exact functor that takes injectives to injectives. Therefore, the identities

A ⊗_{R} UB = LA ⊗_{R⊗R} B, Hom_{R}(A, UB) = Hom_{R⊗R}(LA, B)

where L = L(κ(1 ⊗ η)) and U = U(κ(1 ⊗ η)), prolong to the identities of (6.9.2).

Corollary 6.10. Suppose in addition to the assumptions of (6.9) that ε⊗1 = μκ. Then κ∘(1⊗η) induces an isomorphism

Tor_{p}^{R}(k, B) = Tor_{p}^{R⊗R}(R, B)

for all left R ⊗ R-modules B and all p ≥ 0.

Proof. Note that L(1 ⊗ η)(k) = U(ε ⊗ 1)(R) where R is considered a module over itself. Hence L(κ(1 ⊗ η))(k) = U(κ^{−1})L(1 ⊗ η)(k) = U(κ^{−1})U(ε ⊗ 1)(R) = U((ε ⊗ 1)κ^{−1})(R) = U(μ)(R) which is R as an R ⊗ R-module.

7. Appendix: Harper’s molecule

This section contains some comments on John Harper’s work on finite H-spaces. Our starting point will be the theorem below and our main result (7.3) says that the space K(p) is cohomologically unique in that it is determined up to homotopy equivalence by cohomological information. This will enable us (7.5) to identify K(3) and the fibre F_{4}/DI_{2} of the embedding Bα: BDI_{2} → BF_{4}.

Theorem 7.1. ([16, Theorem B]) For each odd prime p there exists a simply connected finite complex K(p) whose p-localization is an H-space and with

H^{∗}(K(p); F_{p}) ≅ E[x_{3}, x_{2p+1}] ⊗ P [x_{2p+2}]/(x_{2p+2})^{p} P^{1}x_{3} = x_{2p+1}, βx_{2p+1} = x_{2p+2}

It turns out that the p-completed homotopy type of the space K(p) is determined by its mod p cohomology as an algebra over the Steenrod algebra. This is proved by means of classical homotopy theory methods and we are sure that it is known to J. Harper and perhaps other people. We will provide a prove for the sake of completeness.

For this aim we sketch the construction of this p-completed homotopy type by G.E. Cooke and L. Smith [9]. They first introduce the stable two stage Postnikov system

\xymatrix{A(p) \ar[d] \ar[r] & ∗ \ar[d] \\ K(Z/p, 2p^{2} + 1) \ar[r]^{Ψ} & K(Z/p, 2p^{2} + 2p − 1) × K(Z, 2p^{2} + 2p + 1)}

where Ψ = (P^{1}ι, βP^{1}βι) (Notice that βP^{1}βι ∈ H^{2p^{2}+2p+1}(Z/p, 2p^{2} + 1; Z/p) is a mod p reduction of an integral class.) This is constructed in such a way that the composition

K(Z, 3) ⟶^{P^{p}P^{1}ι} K(Z/p, 2p^{2} + 1) ⟶^{Ψ} K(Z/p, 2p^{2} + 2p − 1) × K(Z, 2p^{2} + 2p + 1)

is null-homotopic and therefore the first map lifts (non-uniquely) to φ_{p}:

\xymatrix{ & A(p) \ar[d] \\ K(Z, 3) \ar[ur]^{φ_{p}} \ar[r]_{P^{p}P^{1}ι} & K(Z/p, 2p^{2} + 1)}

Now define Y (p) as the homotopy fibre

\xymatrix{Y (p) \ar[r] \ar[d]_{q_{p}} & ∗ \ar[d] \\ K(Z, 3) \ar[r]_{φ_{p}} & A(p)}

of the map φ_{p}.

The cohomology of the two stage Postnikov system A(p) is computed from results of [29]:

H^{∗}(A(p); F_{p}) ≅ E[ι] ⊗ P [βι, P^{1}βι] ⊗ H

where ι is restricted from the fundamental class of K(Z/p, 2p^{2}+1) and, in particular, deg ι = 2p^{2}+1 and H is a Hopf algebra that is (2p^{2} +4p−2)-connected. Now, φ_{p}^{∗}(ι) = P^{p}P^{1}ι_{3}, φ_{p}^{∗}(βι) = βP^{p}P^{1}ι_{3} and φ_{p}^{∗}(P^{1}βι) = P^{1}βP^{p}P^{1}ι_{3} = (βP^{1}ι_{3})^{p}, from which it is computed the cohomology of Y (p) in low dimensions:

Proposition 7.2. ([9, 1.2]) In dimensions less than or equal to 2p^{2} +2p+2 the mod p cohomology of Y (p) coincides with E[u, P^{1}u] ⊗ P [βP^{1}u]/(βP^{1}u)^{p} where u = q_{p}^{∗}ι_{3}.

It then follows that stage 2p^{2} + 2p + 2 in the homology decomposition tower [17, Chp. 8] [4] for Y (p) is a space K(p) = Y (p)^{2p^{2}+2p+2} with

H^{∗}(K(p); F_{p}) ≅ E[u, P^{1}u] ⊗ P [βP^{1}u]/(βP^{1}u)^{p} .

Dually, the Postnikov tower for K(p) has the form

\xymatrix{ & ⋮ \ar[d] \\  & K(p)_{2p^{2}+2p+3} \ar[d] \\ K(p) \ar[ur] \ar[r] \ar[dr] & K(p)_{2p^{2}+2p+2} \ar[d] \ar[r] & K(π_{2p^{2}+2p+3}(K(p)), 2p^{2} + 2p + 4) \\  & Y (p) \ar[r] & K(π_{2p^{2}+2p+2}(K(p)), 2p^{2} + 2p + 3)}

because K(p)_{2p^{2}+2p+1} = Y (p)_{2p^{2}+2p+1}^{2p^{2}+2p+2} = Y (p)_{2p^{2}+2p+1} = Y (p) appears at stage 2p^{2} + 2p + 1.

We now come to our main result of this section which is homotopy uniqueness at the prime p for K(p)_{p}^{∧}.

Theorem 7.3. Let X be a p-complete space with H^{∗}(X; F_{p}) ≅ H^{∗}(K(p); F_{p}) as algebras over the Steenrod algebra. Then X ≃ K(p)_{p}^{∧}.

Proof. Assume that X is a p-complete space with

H^{∗}(X; F_{p}) ≅ E[x_{3}, x_{2p+1}] ⊗ P [x_{2p+2}]/(x_{2p+2})^{p}

satisfying P^{1}x_{3} = x_{2p+1} and βx_{2p+1} = x_{2p+2}. Notice that X is 2-connected and since the mod p cohomology is finite dimensional, each π_{i}(X) and H^{i}(X; Z_{p}) is a finitely generated Z_{p}-module.

The Bockstein spectral sequence with p-adic coefficients applies. It collapses at

B_{∞} ≅ B_{2} ≅ E[x_{3}, x_{2p+1}(x_{2p+2})^{p−1}]

and shows that the top integral class is H_{2p^{2}+2p+2}(X; Z_{p}) ≅ H^{2p^{2}+2p+2}(X; Z_{p}) ≅ Z_{p}. (The cohomological dimension cd_{Z_{p}}(X) = 2p^{2} + 2p + 2.) In particular, H^{>2p^{2}+2p+2}(X; M) = 0 for any Z_{p}-module M and thus [X, K(p)_{p}^{∧}] = [X, Y (p)_{p}^{∧}] by obstruction theory.

We now use the p-completed version of the Cooke-Smith construction of K(p). Let

X ⟶^{x_{3}} K(Z_{p}, 3)

represent a generator of H^{3}(X; Z_{p}) ≅ Z_{p}.

Claim 7.4. [X, A(p)_{p}^{∧}] = {∗}.

Proof. This follows from the exact sequence of sets

0 = H^{2p^{2}+2p−2}(X; F_{p}) × H^{2p^{2}+2p}(X; F_{p}) ⟶ [X, A(p)_{p}^{∧}] ⟶ H^{2p^{2}+1}(X; F_{p}) = 0

obtained by mapping X into the fibration K(Z/p, 2p^{2} + 2p − 2) × K(Z_{p}, 2p^{2} + 2p) ⟶ A(p)_{p}^{∧} ⟶ K(Z/p, 2p^{2} + 1).

Since φ_{p}x_{3} is null-homotopic by Claim 7.4, x_{3} : X → K(Z_{p}, 3) lifts to a map

f: X → Y (p)_{p}^{∧}

that satisfies f^{∗}(u) = x_{3}. Hence f^{∗} is an isomorphisms in dimensions ≤ 2p^{2} + 2p + 2 and the corresponding map X ⟶ K(p)_{p}^{∧} an isomorphism on mod p cohomology.

In the special case where p = 3, we note that Harper’s molecule K(3)_{3}^{∧} and the exotic homogeneous space F_{4}/DI_{2} have isomorphic mod 3 cohomology algebras over the Steenrod algebra. Thus we may conclude that these two spaces are homotopy equivalent.

Proposition 7.5. F_{4}/DI_{2} and K(3)_{3}^{∧} are homotopy equivalent 3-complete spaces.

We do not know if K(p) for p > 3 is an exotic homogeneous space with respect to some monomorphism of p-compact groups.

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Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

   E-mail address: broto@manwe.mat.uab.es

Matematisk Institut, Universitetsparken 5, DK–2100 København Ø

   E-mail address: moller@math.ku.dk