Contemporary Mathematics Volume 00, 0000

Maps between classifying spaces revisited

STEFAN JACKOWSKI, JAMES MCCLURE, AND BOB OLIVER

Abstract. In an earlier paper, we developed general techniques which can be used to study the set of homotopy classes of maps between the classifying spaces of two given compact Lie groups. Here, we describe more precisely the general strategy for doing this; and then, as a test of these methods, apply them to determine the existence and uniqueness of (potential) maps BG ⟶ BG′ studied earlier by Adams and Mahmud. We end with a complete description of the set of homotopy classes of maps from BG_{2} to BF_{4}.

In 1976, Adams & Mahmud [3] published the first systematic study of the problem of determining the homological properties of maps between classifying spaces of compact connected Lie groups. This was continued in later work by one or both authors: Adams [2] extended some of the results to the case of non-connected Lie groups by using complex K-theory; while Adams & Mahmud [4] identified further restrictions which could be made using real or symplectic K-theory.

Recently, in [21], the three of us developed new techniques for studying maps between classifying spaces: techniques based on new decompositions of BG for any compact Lie group G. The main application in [21] was to the problem of determining self maps of BG for any compact connected simple Lie group G. This problem had earlier been studied by several other people (cf. [26], [28], [16], [18], [19], and [23]). The main missing point was to show that the “unstable Adams operations” ψ^{k} : BG ⟶ BG are unique up to homotopy.

The main tools which now make possible a more precise study of maps between classifying spaces are a series of consequences of the proof of the Sullivan conjecture by Miller (cf. [13]), Carlsson [11], and Lannes [22]. The principal

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

0000-0000/00 $1.00 + $.25 per page

1991 Mathematics Subject Classification. Primary 55S37, 55R35; Secondary 55P60.

The first author was partly supported by a Polish scientific grant (RP I.10), the second author by a US NSF grant (DMS-8803279), and the third author by a Danish SNF grant (11-7792)

result which we use directly is the description, by Dwyer & Zabrodsky [14], Zabrodsky [33], and Notbohm [24], of map(BP, BG) when G is any compact Lie group and P is p-toral (an extension of a torus by a finite p-group). The idea of our approach was to combine these theorems of Dwyer-Zabrodsky and Notbohm with a new decomposition of BG: a decomposition which approximates BG at any prime p as a homotopy direct limit of classifying spaces of p-toral subgroups of G.

In this paper, we show how the same techniques can be used successfully in other situations, to get information about the existence and uniqueness of maps BG ⟶ BG′ when G and G′ are two distinct compact connected Lie groups. We first describe the general strategy for doing this, taking as our starting point the work of Adams & Mahmud in [3]. To illustrate how these tools work in practice, we then take those examples listed in [3] involving (potential) maps between classifying spaces of distinct rank; and use our methods to determine exactly which ones actually do exist.

We end with a complete description of the set of homotopy classes of maps from BG_{2} to BF_{4} (Example 3.4). In this case, there are four families of maps, of which two can be constructed (at least away from the prime 3) as composites of the inclusion G_{2} ↪ F_{4}, unstable Adams operations, and Friedlander’s “exceptional isogeny” on BG_{2}. The maps in the other two families come in pairs, where one map from each pair can be constructed in a similar fashion, but by using an inclusion which is defined only between the algebraic groups over \overline{F_{7}} instead of the inclusion of compact Lie groups. But the remaining map in each pair seems to be completely new, and cannot as far as we can tell be constructed as any composite of algebraically defined maps.

Many of the techniques used here carry over to the case where G is an arbitrary compact Lie group (in particular, a finite group). But since they are based on using Sullivan’s arithmetic pullback square for localizations and completions of BG′, we do always assume that G′ is connected.

Section 1

Throughout the paper, G and G′ will denote two compact Lie groups, where G′ is connected. We want to study the set [BG, BG′] of homotopy classes of maps from BG to BG′. We fix maximal tori T ⊆G and T′⊆G′, and let W = N(T)/T and W′ = N(T′)/T′ denote the Weyl groups. We also regard W and W′ as groups of automorphisms of T and T′, respectively (note, however, that the action of W need not be effective if G is not connected).

Our procedure for studying maps BG ⟶ BG′ can be broken up into three steps:

Step 1: Admissible maps

For the purposes of this paper, we define an admissible map from G to G′ to be a homomorphism ϕ : T ⟶ T′ such that for every w∈W there exists w′∈W′ such that w′ ∘ ϕ = ϕ∘w. The motivation for this definition comes from the following result, due in its original form to Adams & Mahmud [3].

Theorem 1.1. For any f : BG ⟶ BG′, there exists an admissible map ϕ : T ⟶ T′ such that the following square commutes up to homotopy:

BT ⟶^{Bϕ} BT′
↓^{incl}  ↓^{incl}
BG ⟶^{f} BG′.

(1)

Furthermore, for any other admissible map ϕ′ : T ⟶ T′ such that f|BT ≃Bϕ′, ϕ′ = w ∘ ϕ for some w∈W′.

Proof. By [3, Theorem 1.5], there is a homomorphism ϕ : T ⟶ T′ such that (1) commutes in ℚ-cohomology. Also, ϕ is admissible by [3, Corollary 1.8], and by [3, Theorem 1.7] is unique up to composition by an element in W′. And by a result of Notbohm [24, Proposition 4.1], two maps BT ⟶ BG′ are homotopic if they induce the same map in K-theory, and hence if they induce the same map in ℚ-cohomology. □

A partial converse to Theorem 1.1 is also proven in [3]: any admissible map ϕ : T ⟶ T′ is induced by some f : BG ⟶ BG′[\frac{1}{|W |}]; i.e., after inverting those primes dividing the order of the Weyl group. In fact, the definition of admissible maps given by Adams & Mahmud in [3] is more general than that given here, in order to allow for admissible maps corresponding to maps f : BG ⟶ BG′[\frac{1}{n}]; i.e., maps defined only after inverting some finite number of primes.

We want to determine which admissible maps extend to globally defined maps f : BG ⟶ BG′ (and if such maps exist, how many there are). When doing this, it will be useful to write [BG, BG′]_{ϕ} to denote the set of homotopy classes of maps which extend a given admissible map ϕ, and similarly for localizations and completions of BG′. The next proposition reduces the problem of describing [BG, BG′]_{ϕ} to the case of maps to the p-adic completions of BG′.

Proposition 1.2. For any set P of primes and any admissible map ϕ : T ⟶ T′, the map

[BG, BG′_{(P)}]_{ϕ} ⟶_{≅}^{ι} ∏_{p∈P}[BG, BG′_{p̂}]_{ϕ}

(induced by completion) is a bijection. Also, if p|∤||W |, then [BG, BG′_{p̂}]_{ϕ} has order 1. So in particular (if W ≠ 1)

[BG, BG′]_{ϕ} ≅ ∏_{p||W |}[BG, BG′_{p̂}]_{ϕ}.

Proof. The bijectivity of ι is shown in [21, Theorem 3.1], using Sullivan’s arithmetic pullback square for completions and localizations of the simplyconnected space BG′. The fact that [BG, BG′_{p̂}]_{ϕ} has order 1 whenever p|∤||W | is implicit in [30], although it is stated there only for G finite or connected. Another proof (for arbitrary G) is given in Theorem 1.12 below. □

Note that when G is finite, the last statement in Proposition 1.2 takes the form

[BG, BG′] ≅ ∏_{p||G|}[BG, BG′_{p̂}].

By [3, Theorem 2.21], for any admissible ϕ, there is a homomorphism ϕ̄ : W ⟶ W′ such that ϕ is ϕ̄-equivariant; i. e., ϕ̄(w) ∘ ϕ = ϕ ∘ w for each w∈W. In general, there can be more than one ϕ̄ : W ⟶ W′ for which ϕ is ϕ̄-equivariant. But in many cases ϕ̄ is uniquely determined, and the following lemma gives one condition for this to happen.

Lemma 1.3. Assume that ϕ : T ⟶ T′ is a \underline{regular} admissible map: an admissible map with either of the following equivalent properties:

(a) C_{G′}(ϕ(T)) = T′.

(b) Im[ϕ̃ : T̃ ⟶ T̃′] is not contained in the kernel of any of the roots of G′. (ϕ̃ denotes the universal covering map of ϕ.) Then there is a unique homomorphism ϕ̄: W ⟶ W′ for which ϕ is ϕ̄-equivariant.

Proof. See [3, Lemma 2.22]. Note, when showing that (b) implies (a), that the centralizer of any torus in the connected group G′ is connected (cf. [21, Proposition A.4]). □

We must now consider the problem: given an admissible map ϕ : T ⟶ T′, is it induced by a map f : BG ⟶ BG′ between the classifying spaces? And if so, is the map unique? By Proposition 1.2, ϕ can be extended to a map f if and only if it can be extended to f_{p} : BG ⟶ BG′_{p̂} for each prime p, and the extension is unique if it is unique for each completion.

Our procedure for answering these questions is based on p-local approximations of BG, by homotopy direct limits over certain orbit categories defined as follows:

Definition 1.4. A subgroup P ⊆G is \underline{p-toral} if it is an extension of a torus by a finite p-group. A subgroup P ⊆G is \underline{p-stubborn} if it is p-toral, and if N(P)/P is finite and has no nontrivial normal p-subgroups. We let O_{p}(G) denote the category of orbits G/P for p-toral P ⊆G (and Mor(G/P_{1}, G/P_{2}) is the set of all G-maps). And R_{p}(G) ⊆ O_{p}(Γ) denotes the full subcategory of orbits G/P for p-stubborn P ⊆G.

Our procedure for describing maps from BG to BG′_{p̂} is based on two results which are summarized in the following theorem. The first says that BG can be approximated as a limit of BP′s for p-stubborn subgroups P ⊆G, and the second describes the sets [BP, BG′] for p-toral P. For any pair of groups G_{1}, G_{2}, we let

Rep(G_{1}, G_{2}) := Hom(G_{1}, G_{2})/ Inn(G_{2})

denote the set of G_{2}-conjugacy classes of homomorphisms ρ : G_{1} ⟶ G_{2}.

Theorem 1.5. (i) (Jackowski, McClure, & Oliver [21]) For any compact Lie group G and any prime p, hocolim_{⟶}_{G/P ∈R_{p}(G)}(G/P) is 𝔽_{p}-acyclic. Hence the

projection map

hocolim_{⟶}(EG/P) ≅ EG ×_{G} ( hocolim_{⟶}_{G/P ∈R_{p}(G)}(G/P)) ⟶ BG

is an 𝔽_{p}-homology equivalence (and EG/P ≃BP).

(ii) (Dwyer & Zabrodsky [14], Notbohm [24]) For any p-toral group P, the map

B : Rep(P, G′) ⟶^{≅} [BP, BG′],  (1)

which sends ρ to Bρ, is a bijection. Also, for any prime p, the completion map [BP, BG′] ⟶ [BP, BG′_{p̂}] is injective, and is bijective if P is a finite p-group.

Proof. Point (i) is shown in [21, Theorem 1.4]. The bijectivity of (1) is shown in [14] (when P is a finite p-group) and [24] (when P is an arbitrary p-toral group). Note that this does not require the connectivity of G′.

If P is a finite p-group, then [BP, BG′_{p̂}] ≅ [BP, BG′] by obstruction theory (or the fact that BP is p-complete).

Now assume that P is an arbitrary p-toral group, and let f, f′ : BP ⟶ BG′ be such that (f)_{p̂} ≃ (f′)_{p̂}. Let T ⊆ P be the identity component. For each n ≥ 1, let T_{n} ⊆ T be the subgroup of elements of order dividing p^{n}. Choose subgroups P_{1} ⊆ P_{2} ⊆ ⋯ ⊆ P such that P_{n} ∩ T = T_{n} for each n, and such that the union of the P_{n} is dense in P (cf. [15, Corollary 1.2]). Then for each

n, (f|BP_{n})_{p̂} ≃ (f′|BP_{n})_{p̂}, and hence f|BP_{n} ≃ f′|BP_{n} (since P_{n} is a finite p-

group). Also, BP_{ p̂} ≃ hocolim_{⟶}((BP_{n})_{p̂}) (cf. [15, Proposition 2.3]); and so the obstructions to constructing a homotopy between f and f′ lie in

lim_{⟵}^{1}_{n}(π_{1}(map(BP_{n}, BG′)_{f|BP_{n}})).

By the theorem of Dwyer & Zabrodsky [14], if ρ_{n} : P_{n} ⟶ G′ is such that f|BP_{n} ≃ Bρ_{n}, then π_{1}(map(BP_{n}, BG′)_{f|BP_{n}}) ≅ π_{1}(BC_{G′}(Im(ρ_{n}))). In particular, these groups are all finite, so lim_{⟵}^{1} vanishes, and f ≃ f′. □

In Steps 2 and 3 below, we fix a prime p, and consider the problem of determining the set of homotopy classes of maps BG ⟶ BG′_{p̂} which extend a given admissible map ϕ : T ⟶ T′. Fix a maximal p-toral subgroup N_{p}(T)⊆G: i. e., a subgroup for which N_{p}(T)/T is a p-Sylow subgroup of N(T)/T. (See, e.g., [21, A.1 & A.2] for more details about maximal p-toral subgroups.)

Step 2: R_{p}-invariant representations.

An R_{p}-invariant representation on G is an element

ρ ∈ Rep(N_{p}(T), G′) = Hom(N_{p}(T), G′)/ Inn(G′)

with the property that the representations ρ|P combine to form an element in the inverse limit:

ρ̂ = (ρ|P)_{G/P ∈R_{p}(G)} ∈ lim_{⟵}_{G/P ∈R_{p}(G)} Rep(P, G′).

Equivalently, for any p-stubborn P ⊆G and any two homomorphisms i_{1}, i_{2} : P ↣ N_{p}(T) induced by inclusions and conjugation in G, ρ∘i_{1} is conjugate (in G′) to ρ∘i_{2}. Analogously, we say that ρ : N_{p}(T) ⟶ G′ is O_{p}-invariant if its restrictions define an element in above limit when taken over all G/P ∈ O_{p}(G) (see Definition 1.4). The next proposition says that these two conditions are equivalent.

Proposition 1.6. Any R_{p}-invariant representation ρ : N_{p}(T) ⟶ G′ is O_{p}- invariant. In particular, if ρ is R_{p}-invariant, and g, g′ ∈ N_{p}(T) have p-power order and are conjugate in G, then ρ(g) is conjugate to ρ(g′).

Proof. Fix a p-toral subgroup P_{0}⊆N_{p}(T), and an element x∈G such that xP_{0}x^{−1}⊆N_{p}(T). We must show that the homomorphisms

s_{1} = ρ|P_{0}, s_{2} = (ρ|xP_{0}x^{−1}) ∘ conj(x) : P_{0} ⟶ G′

are conjugate in G′. The last statement then follows from the special case P_{0} = ⟨g⟩.

Consider the G-complex

ER_{p}(G) = hocolim_{⟶}_{G/P ∈R_{p}(G)}(G/P).

Let Z_{0}⊆Z⊆ER_{p}(G) be its 0- and 1-skeletons. In other words, Z_{0} is the disjoint union of the orbits G/P in R_{p}(G), and Z consists of the mapping cylinders of all morphisms in R_{p}(G) attached to Z_{0}. Since ρ is R_{p}-invariant, the map

f_{0} = ∐_{G/P ∈R_{p}(G)} B(ρ|P) : EG ×_{G} Z_{0} ≃ ∐_{G/P ∈R_{p}(G)} BP ⟶ BG′

can be extended to a map

f : EG ×_{G} Z ⟶ BG′.

By [21, Theorems 2.14 & 1.4], (ER_{p}(G))^{P_{0}} is 𝔽_{p}-acyclic, and in particular connected. It is the homotopy colimit of the functor which sends G/P to (G/P)^{P_{0}}, and hence its 1-skeleton Z^{P_{0}} is also connected. It follows that any two G-maps G/P_{0} ⟶ Z are G-homotopic. And if we identify BP_{0} ≃ EG ×_{G} G/P_{0} in the usual way, then Bs_{i}≃f ∘ (EG ×_{G} α_{i}), where

α_{1}, α_{2} : G/P_{0} ⟶ G/N_{p}(T) ⊆ Z_{0} ⊆ Z

are defined by setting α_{1}(gP_{0}) = gN_{p}(T) and α_{2}(gP_{0}) = gx^{−1}N_{p}(T). Hence Bs_{1}≃Bs_{2}, and s_{1} is conjugate to s_{2} by Theorem 1.5(ii). □

By Theorem 1.5(ii), [BP, BG′] ≅ Rep(P, G′) for all p-toral P. So for any f : BG ⟶ BG′, f|BN_{p}(T) ≃ Bρ for some unique R_{p}-invariant representation ρ. Actually, the following slightly stronger version of this observation will be needed.

Proposition 1.7. For any f : BG ⟶ BG′_{p̂} extending ϕ : T ⟶ T′ (i.e., f ∈ [BG, BG′_{p̂}]_{ϕ}), f|BN_{p}(T) ≃ (Bρ)_{p̂} for some unique R_{p}-invariant representation ρ : N_{p}(T) ⟶ G′.

Proof. Choose finite p-subgroups P_{1} ⊆ P_{2} ⊆ ⋯ ⊆ N_{p}(T), as in the proof of Theorem 1.5(ii) above, such that the union of the P_{n} is dense in N_{p}(T). By Theorem 1.5(ii) (the Dwyer-Zabrodsky theorem), for each n, f|BP_{n} ≃ Bρ_{n} for some ρ_{n} : P_{n} ⟶ G′. We may choose the ρ_{n} such that ρ_{n}|T_{n} = ϕ|T_{n} for all n. Also, for each n, ρ_{n+1}|P_{n} and ρ_{n} are conjugate in G′, and hence by an element of C_{G′}(T_{n}). And since C_{G′}(T_{n}) = C_{G′}(T) for n sufficiently large, the ρ_{n} can be successively chosen such that ρ_{n+1}|P_{n} = ρ_{n} for all n.

Now let ρ : N_{p}(T) ⟶ G′ be the unique continuous extension of ∪_{n=1}^{∞}ρ_{n}. In particular, ρ|T = ϕ. And since BN_{p}(T)_{p̂} ≃ hocolim_{⟶}((BP_{n})_{p̂}) (cf. [15, Propo-

sition 2.3]), the same argument (involving lim_{⟵}^{1}) used in Theorem 1.5(ii) shows

that f|BN_{p}(T) ≃ (Bρ)_{p̂}.

The R_{p}-invariance of ρ now follows immediately from Theorem 1.5(ii): for any p-toral P ⊆ G and any σ_{1}, σ_{2} : P ⟶ G′, (Bσ_{1})_{p̂} ≃ (Bσ_{2})_{p̂} if and only if σ_{1} and σ_{2} are conjugate in G′. □

The question is now: given an admissible map ϕ : T ⟶ T′, can it be extended to an R_{p}-invariant representation, and if so to how many? As will be seen below, in many concrete cases, this can easily be determined using ad hoc methods. But we have no general techniques for doing this. In particular, this is the missing step if we want to construct the unstable Adams operations for the exceptional Lie groups using this procedure.

The following proposition gives one simple (but very useful) tool for showing that certain admissible maps do not extend to maps between classifying spaces.

Proposition 1.8. Let ϕ : T ⟶ T′ be an admissible map such that

(i) Ker(ϕ) is finite (i.e., ϕ̃: T̃ ⟶ T̃′ is injective), and

(ii) there exists an element t∈ Ker(ϕ) of p-power order, which is conjugate in G to some element in N_{p}(T)∖T. Then ϕ does not extend to any R_{p}-invariant representation from G to G′. In particular, Bϕ is not the restriction of any map BG ⟶ BG′_{p̂}.

Proof. Assume that ϕ extends to an R_{p}-invariant representation ρ : N_{p}(T) ⟶ G′. Since any p-toral subgroup of G′ is conjugate to a subgroup of N(T′), we may assume that Im(ρ) ⊆ N(T′) (this can be done without changing ρ|T, but that is not necessary here). By assumption, t is conjugate in G to some element t_{1}∈N_{p}(T)∖T. The conjugation action of t_{1} on T is nontrivial, and ρ sends T into T′ with finite kernel. Thus, the conjugation action of ρ(t_{1}) on T′ must be nontrivial. But since ρ is O_{p}-invariant by Proposition 1.6, ρ(t_{1}) is conjugate to ρ(t) = 1; and this is a contradiction.

The last statement follows from Proposition 1.7. □

Step 3: Computation of higher limits.

An R_{p}-invariant representation ρ determines a family of maps, compatible up to homotopy, from the BP ≃EG/P to BG′_{p̂}. The obstructions to extending these to a map from hocolim_{⟶}(EG/P) to BG′_{p̂} — and hence from BG to BG′_{p̂} — were described (in a much more general context) by Wojtkowiak in [29].

We use Wojtkowiak’s notation, and write Hgr to denote the category whose objects are groups, and where Mor_{Hgr}(G_{1}, G_{2}) = Rep(G_{1}, G_{2}). Also, Gr will denote the usual category of groups; and p-Hgr ⊆ Hgr and p-Gr ⊆ Gr the subcategories of finite p-groups. Then lim_{⟵}_{C}^{2}(F) is defined for any functor F :

C ⟶ Hgr, and lim_{⟵}_{C}^{1}(F) is defined for any F : C ⟶ Gr (cf. [29]).

Theorem 1.9. Fix an R_{p}-invariant representation ρ : N_{p}(T) ⟶ G′, and define functors

Π_{1}^{ρ} : R_{p}(G) ⟶ p-Hgr  and  Π_{n}^{ρ} : R_{p}(G) ⟶ ℤ_{(p)}-mod (n ≥ 2)

by setting

Π_{n}^{ρ}(G/P) = π_{n} (map(BP, BG′_{p̂})_{Bρ|P} ) ≅ \{ \begin{matrix}π_{1}(BC_{G′}(Im(ρ))) & if n = 1 \\ [π_{n}(BC_{G′}(Im(ρ)))]_{p̂} & if n ≥ 2. \end{matrix}

Then Bρ extends to a map f : BG ⟶ BG′_{p̂} if the higher limits lim_{⟵}^{n+1}(Π_{n}^{ρ}) vanish for all n ≥ 1. If there is such an extension f, then Π_{1}^{ρ} lifts to a functor R_{p}(G) ⟶ p-Gr, and f is unique up to homotopy if the limits lim_{⟵}^{n}(Π_{n}^{ρ}) vanish for all n ≥ 1.

Proof. The formula for the homotopy groups of map(BP, BG′_{p̂})_{Bρ|P} is implicit in Notbohm [24]; and is shown explicitly in [21, Theorem 3.2(iii)] using the results in [24]. Note in particular that for any p-toral P ⊆G′, π_{1}(BC_{G′}(P)) ≅ π_{0}(C_{G′}(P)) is a p-group (cf. [21, Proposition A.4]).

By Theorem 1.5(i),

[BG, BG′_{p̂}] ≅ [ hocolim_{⟶}_{G/P ∈R_{p}(G)}(EG/P), BG′_{p̂}].

So by Wojtkowiak [29], the obstructions to extending Bρ to a map f : BG ⟶ BG′_{p̂} lie in the higher limits lim_{⟵}^{n+1}(Π_{n}^{ρ}), and the obstructions to uniqueness in the lim_{⟵}^{n}(Π_{n}^{ρ}). Note that once we have found one map f which extends Bρ, then it can be used to consistently define base points for the map(BP, BG′_{p̂})_{f|BP}, and hence to lift Π_{1}^{ρ} to a functor to p-Gr (so that lim_{⟵}^{1}(Π_{1}^{ρ}) is defined). □

In fact, it is not hard to extend the arguments in [29] to construct a second quadrant spectral sequence which converges to the homotopy of map(BG, BG′_{p̂})_{ρ} — the space of maps which extend Bρ. See also [9] & [10]. Problems with nonabelian values for Π_{1}^{ρ} do not occur in any of the concrete examples we consider in Sections 2 and 3; but they do have to be considered in the proof of Corollary 1.11 below.

One might expect the computation of these higher limits to be quite hard in general. We did, however, succeed in [21] in developing some very powerful algorithms which are successful in making these computations in many cases. They are based on some functors Λ^{∗}(Γ; M), defined (for fixed p) for any finite group Γ and any ℤ_{(p)}[Γ]-module M.

More precisely, for any such Γ and M, let O_{p}(Γ) denote the category of orbits Γ/P for p-subgroups P ⊆ Γ. Let F_{M} : O_{p}(Γ) ⟶ ℤ_{(p)}-mod be the contravariant functor defined by setting F_{M}(Γ/1) = M, and F_{M}(Γ/P) = 0 for 1≠P ⊆ Γ. We then define

Λ^{∗}(Γ; M) := lim_{⟵}^{∗}_{O_{p}(Γ)}(F_{M}).

The following theorem explains the significance of these functors.

Theorem 1.10. Consider a contravariant functor F : R_{p}(G) ⟶ ℤ_{(p)}-mod.

(i) Assume, for some p-stubborn subgroup P ⊆G, that F vanishes except on the orbit type G/P. Then lim_{⟵}^{∗}(F) ≅ Λ^{∗}(N(P)/P; F (G/P)).

(ii) Assume, for some n ≥ 0, that Λ^{n}(N(P)/P; F (G/P)) = 0 for all pstubborn P ⊆G. Then lim_{⟵}^{n}(F) = 0.

(iii) If p|| Ker[Γ ⟶ Aut(M)]|, then Λ^{∗}(Γ; M) = 0. If p|∤||Γ|, then Λ^{0}(Γ; M) ≅ M^{Γ} and Λ^{n}(Γ; M) = 0 for all n ≥ 1. If p||Γ|, then Λ^{0}(Γ; M) = 0. If p^{2}|∤||Γ|, then Λ^{n}(Γ; M) = 0 for all n ≥ 2.

Proof. Point (i) is shown in [21, Lemma 5.4].

To see point (ii), let N_{p}(T) = P_{0}, P_{1}, …, P_{k} be conjugacy class representatives for all p-stubborn subgroups of G (there are finitely many by [21, Proposition 1.6]). Assume the P_{i} are arranged from smallest to largest; i. e., such that (P_{i}) ⊆ (P_{j}) implies i≤j. For each 0≤i≤k, define F_{i} by setting F_{i}(G/P) = F (G/P) if (P) = (P_{j}) for some j≤i, and F_{i}(G/P) = 0 otherwise. Then F_{0}⊆F_{1} ⊆ … ⊆F_{k} = F. By (i), for each i,

lim_{⟵}^{n}(F_{i}/F_{i−1}) ≅ Λ^{n}(NP_{i}/P_{i}; F (G/P_{i})) = 0.

The long exact sequence of higher limits for a short exact sequence of functors (cf, [21, Proposition 5.1]), can now be used to show that lim_{⟵}^{n}(F) = 0.

Finally, point (iii) is shown in [21, Propositions 5.5, 6.1, and 6.2]. □

Note that if we only are interested in the existence of maps extending ρ, then we need only compute higher limits lim_{⟵}^{n} over R_{p}(G) for n ≥ 2. The following corollary to Theorem 1.10 gives some simple sufficient conditions for these to vanish.

Corollary 1.11. Fix contravariant functors

F : R_{p}(G) ⟶ ℤ_{(p)}-mod, F_{1} : R_{p}(G) ⟶ p-Gr and F_{2} : R_{p}(G) ⟶ p-Hgr.

(i) Assume for each p-stubborn subgroup P ⊆G that either p|∤||N(P)/P |, or p|| Ker[N(P)/P ⟶ Aut(F (G/P))]|. Then lim_{⟵}^{1}(F_{1}) = 0, and lim_{⟵}^{n}(F) = 0 for all n ≥ 1.

(ii) Assume for each p-stubborn subgroup P ⊆G that either p^{2}|∤||N(P)/P |, or p|| Ker[N(P)/P ⟶ Aut(F (G/P))]|. Then lim_{⟵}^{2}(F_{2}) = 0, and lim_{⟵}^{n}(F) = 0 for all n ≥ 2.

Proof. For the functor F, which takes values in ℤ_{(p)}-mod, these claims follow immediately from Theorem 1.10.

Now let F_{i} be one of the functors F_{1} or F_{2}. Let G/P be minimal (i.e., P is minimal) among those objects upon which F_{i} is nonvanishing. Let F′_{i} ⊆ F_{i} be the subfunctor such that F′_{i}(G/P) = Z(F_{i}(G/P)), and such that F′_{i} vanishes on all other orbit types. Then F′_{i} is a functor to the category of finite abelian p-groups. So by Theorem 1.10, lim_{⟵}^{1}(F′_{i}) = 0 if p|∤||N(P)/P |, and lim_{⟵}^{2}(F′_{i}) =

0 if p^{2}|∤||N(P)/P |. We may assume inductively that lim_{⟵}^{i}(F_{i}/F′_{i}) = 0; and an

examination of the definitions of the nonabelian lim_{⟵}^{1} and lim_{⟵}^{2} in [29] now shows

that lim_{⟵}^{i}(F_{i}) = 0.

As a first illustration of the use of Theorems 1.5 and 1.10, we show how they apply in the case where p|∤||W |.

Theorem 1.12. (Adams & Mahmud, Wojtkowiak) If p|∤||W |, then any admissible map ϕ : T ⟶ T′ extends to a unique map f : BG ⟶ BG′_{p̂}.

Proof. The existence of an extension is shown in [3, Theorem 1.10], at least when G is connected; and existence and uniqueness when G is finite or connected is shown in [30]. We show here how the result follows from the theory just presented. When p|∤||W |, then the only p-stubborn subgroups of G are the maximal tori (any p-toral subgroup P ⊆G is contained in a maximal torus, and N(P)/P is finite). Thus, R_{p}(G) is equivalent to the category with one object G/T, with End(G/T)≅W. In particular, N_{p}(T) = T, and any admissible map ϕ is itself R_{p}-invariant. Finally, for each i, j ≥ 1,

lim_{⟵}^{i}(Π_{j}^{ϕ}) ≅ H^{i}(W; Π_{j}^{ϕ}(G/T)) = 0

since p|∤||W | and Π_{j}^{ϕ}(G/T) is p-local. □

We next consider the cases where p^{2}|∤||W |, or where G′ is a matrix group. These conditions hold for several of the examples from [3] which we study in Sections 2 and 3. In these cases, we again get some very strong results about the existence and uniqueness of maps BG ⟶ BG′_{p̂}.

Proposition 1.13. Fix a prime p. Assume that π_{0}(G) is a p-group, and that p^{2}|∤||W |. Then p^{2}|∤||N(P)/P | for all p-stubborn subgroups P ⊆G. Also, any R_{p}-invariant representation ρ : N_{p}(T) ⟶ G′ extends to a map BG ⟶ BG′_{p̂}.

Proof. Fix a p-stubborn subgroup P ⊆G. If P is contained in a maximal torus, then P is a maximal torus (N(P)/P is finite); and so p^{2}|∤||N(P)/P |. So we are left with the case where P is not contained in any maximal torus.

\underline{Step 1:} We start with a general observation. Let H ◁ P be any normal subgroup of finite index, let S ◁ P be the identity component, and let Aut(P, H) be the group of automorphisms of P leaving H invariant. Then

Ker [\frac{Aut(P, H)}{Inn(S)} ⟶^{(Res, Quot)} \frac{Aut(H)}{Inn(S)} × Aut(P/H)]

is a finite p-group. When P is finite, this is shown in [17, Corollary 5.3.3]. If H = S, then this holds since the kernel can be identified with H^{1}(P/S; S) (any automorphism α induces the 1-cocycle (g ↦ g^{−1}α(g))). And the general case now follows since the kernel of each of the maps

\frac{Aut(P, H)}{Inn(S)} ⟶ Aut(S)×Aut(P/S, H/S) ⟶ Aut(S)×Aut(H/S)×Aut(P/H)

is a finite p-group.

Assume in addition that H is normalized by N(P) (i.e., H ◁ N(P)). We claim that in this case,

Ker [N(P) ⟶ Aut(H) × Aut(P/H)] ⊆ P.  (1)

Let K denote this kernel. Since N(P)/P has no nontrivial normal p-subgroups (by definition of “p-stubborn”), it will suffice to show that the image of K in N(P)/P is a p-group. For any g ∈ K, conj(g) has p-power order in Aut(P, H)/ Inn(S), so conj(g^{p^{k}}) = conj(x) for some k and some x ∈ T. Hence g^{p^{k}}x^{−1} ∈ C_{G}(P) ⊆ P (see [21, Lemma 1.5]), and so g^{p^{k}} ∈ P.

\underline{Step 2:} Assume that p||N(P)/P | (otherwise we are done). Let P′/P be a p-Sylow subgroup of N(P)/P. Since P′ is p-toral, we may assume that P′⊆N_{p}(T). Since P ⊈ T (and p^{2}|∤||W |), we must have P′ ⊆ ⟨P, T ⟩. If P ∩T ◁N(P), then

P′ ∩ T ⊆ Ker[N(P) ⟶ Aut(P ∩ T) × Aut(P/(P ∩ T))];

and this contradicts (1).

Thus, P ∩T is not normal in N(P). Consider the Frobenius subgroup Φ(P): the subgroup generated by all commutators and p-th powers in P (cf. [17, §5.1]). Then C_{P} (Φ(P))◁N(P), and C_{P} (Φ(P))⊇P ∩T. Since P ∩T has index p in P and is not normal in N(P), this shows that C_{P} (Φ(P)) = P; i. e., that Φ(P)⊆Z(P).

In particular, the conjugation actions of P on Φ(P) and P/Φ(P) are trivial — the first since Φ(P) is central and the second since [P, P] ⊆ Φ(P). So there is a well defined map

(κ_{1}, κ_{2}) : N(P)/P ⟶ Aut(Φ(P)) × Aut(P/Φ(P)).

By (1) again, (κ_{1}, κ_{2}) is injective. Furthermore, P′/P ⊆ Ker(κ_{1}) (P′ ⊆ ⟨P, T ⟩ and Φ(P)⊆T), P′/P is a Sylow p-subgroup of N(P)/P; and so Im(κ_{1}) has order prime to p. And p|∤|| Ker(κ_{2})|, since (κ_{1}, κ_{2}) is injective.

\underline{Step 3:} Identify P/Φ(P) = V ≅ (𝔽_{p})^{n}, set Γ = Im(κ_{2}), and regard Γ as a subgroup of GL_{n}(𝔽_{p}). This subgroup has the following properties:

(a) any Sylow p-subgroup of Γ acts as the identity on some codimension one subspace of V. It suffices to check this on one Sylow p-subgroup of Γ = κ_{2}(N(P)/P); e.g., on κ_{2}(P′/P). And this acts via the identity on (P ∩T)/Φ(P), since P′ ⊆ ⟨P, T ⟩. Conversely, any two distinct Sylow p-subgroups of Γ fix distinct codimension one subspaces, since the stabilizer in Γ ⊆ GL_{n}(𝔽_{p}) of any codimension one subspace has a normal (and hence unique) Sylow p-subgroup.

(b) Γ contains no nontrivial normal p-subgroup. Since if 1 ≠ Q ◁ Γ is a nontrivial normal p-subgroup, then (κ_{1}, κ_{2})^{−1}(1 × Q) is a nontrivial normal psubgroup of N(P)/P (recall that p|∤||Im(κ_{1})|).

By (b), either p|∤||Γ| (and hence p|∤||N(P)/P |), or Γ contains at least 2 distinct Sylow p-subgroups A, B. In the second case, let H ⊆ Γ be the subgroup generated by A and B. Since V^{A} and V^{B} are distinct and have codimension 1 in V, V^{H} has codimension 2. Also, the conjugation map H → Aut(V/V^{H}) ≅ GL_{2}(𝔽_{p}) must be injective, since its kernel is a normal p-subgroup (Step 1 again); is therefore contained in A∩B, and hence fixes all of V. Since p^{2}|∤|| GL_{2}(𝔽_{p})|, p^{2}|∤||H|; and so p^{2}|∤||Γ| since H contains a Sylow subgroup of Γ. And finally, since p|∤|| Ker(κ_{2})|, this shows that p^{2}|∤||N(P)/P |.

\underline{Step 4:} It remains to prove the last statement. By Corollary 1.11(ii), lim_{⟵}^{n}(Π_{m}^{ρ}) = 0 for any R_{p}-invariant representation ρ : N_{p}(T) ⟶ G′, any m ≥ 1, and any n ≥ 2 (with n = 2 if m = 1). In particular, ρ lifts to a map BG ⟶ BG′. □

The following corollary lists some more consequences of Proposition 1.13, and also of character theory for representations.

Corollary 1.14. Fix a prime p. Assume that G is connected.

(i) Assume that G′≅ U(n), SU(n), SO(2n + 1), or Sp(n). Then a homomorphism ρ : N_{p}(T) ⟶ G′ is R_{p}-invariant if and only if for any pair g, g′∈N_{p}(T) of elements conjugate in G, ρ(g) and ρ(g′) are conjugate in G′. Also, an admissible map ϕ : T ⟶ T′ has (up to conjugacy) at most one extension to an R_{p}-invariant representation ρ : N_{p}(T) ⟶ G′.

(ii) Now assume that p^{2}|∤||W |; and that G′≅ SU(n) or U(n), or p is odd and G′≅ SO(2n + 1) or Sp(n). Then any given admissible map ϕ : T ⟶ T′ lifts to at most one map BG ⟶ BG′_{p̂}.

Proof. (i) When G′ is one of the groups U(n), SU(n), Sp(n), or SO(2n + 1), character theory applies to show that two homomorphisms ρ_{1}, ρ_{2} : P ⟶ G′ are conjugate if and only if ρ_{1}(g) and ρ_{2}(g) have the same trace for each g ∈ G, if and only if ρ_{1}(g) and ρ_{2}(g) are conjugate for each g ∈ G. When G′≅ SO(2n+1), this property holds because O(2n + 1)≅ SO(2n + 1) × ℤ/2. Note that it does not hold for G′≅ SO(2n).

By definition (and Proposition 1.6), a homomorphism ρ : N_{p}(T) ⟶ G′ is R_{p}-invariant if and only if for each p-toral P ⊆N_{p}(T), and each x∈G such that xP x^{−1}⊆N_{p}(T), the homomorphisms

ρ|P, (ρ|(xP x^{−1})) ∘ conj(x) : P ⟶ G′

are conjugate in G′. So by the remarks on character theory, ρ is R_{p}-invariant if and only if ρ(g) is conjugate to ρ(g′) for all pairs g, g′∈N_{p}(T) of elements conjugate in G such that either the closures of ⟨g⟩ and ⟨g′⟩ are p-toral, or both lie in T.

If ρ is R_{p}-invariant and g∈N_{p}(T) is arbitrary, then g = lim(g_{i}) for some sequence of elements g_{i}∈N_{p}(T) of p-power order, each g_{i} is conjugate to some g′_{i}∈T, and after restricting to a subsequence the g′_{i} converge to some g′∈T conjugate to g. Then ρ(g) = ρ(g′) (ρ is continuous); and hence ρ(g) = ρ(g′′) for any g′′∈N_{p}(T) conjugate to g.

By the same argument, if ρ, ρ′ : N_{p}(T) ⟶ G′ are both R_{p}-invariant and ρ|T = ρ′|T, then ρ(g) is conjugate to ρ(g′) for all g∈N_{p}(T) (since every element is conjugate to an element of T); and hence ρ is conjugate to ρ′.

(ii) If p^{2}|∤||W |, then lim_{⟵}^{n}(Π_{n}^{ρ}) = 0 for any R_{p}-invariant representation ρ and any n ≥ 2 (see Proposition 1.13).

If G′ = U(n), then the centralizer of any subgroup is a product of unitary groups, and hence connected. If G′ = Sp(n) or SO(2n+1) and p is odd, then the centralizer of any p-toral subgroup is a product of unitary groups and (possibly) one symplectic or special orthogonal group, and is again connected. And if G′ = SU(n), then the centralizer of any subgroup is generated by its connected component and Z(G′). Thus, in all of these cases, N(P)/P acts trivially on Π_{1}^{ρ}(G/P); and lim_{⟵}^{1}(Π_{1}^{ρ}) = 0 by Corollary 1.11(i).

This shows that any R_{p}-invariant representation lifts to a unique map BG ⟶ BG′_{p̂}. And an admissible map ϕ : T ⟶ T′ extends to at most one R_{p}-invariant representation (up to conjugacy in G′): since any two extensions must have the same character. □

Section 2

We now want to apply these procedures to the examples of admissible maps given by Adams & Mahmud in [3, Section 2]. For simplicity, we concentrate in this section on the question of which admissible maps can be realized as maps BG ⟶ BG′ (and at which primes p) — and pay less attention to the uniqueness question.

The first three examples in [3, §2] involve cases where all admissible maps are zero: and these maps can clearly be realized. Their next four examples involve unstable Adams operations ψ^{k} : BG ⟶ BG constructed by Wilkerson [28], and the “exceptional isogenies” of Friedlander [16].

An unstable Adams operation of degree k is a self-map ψ^{k} : BG ⟶ BG which extends the admissible map ϕ_{k} : T ⟶ T defined by ϕ_{k}(t) = t^{k}. The following theorem combines the results of several authors.

Theorem 2.1. For any compact connected group G and any integer k, there exists an unstable Adams operation of degree k on BG if and only if (k, |W |) = 1. Also, any two unstable Adams operations of the same degree k are homotopic.

Proof. For (k, |W |) = 1, the unstable Adams operations were first constructed by Sullivan [26] when G = U(n), and then by Wilkerson [28] for arbitrary connected G. The necessity of the condition (k, |W |) = 1 was shown by Ishiguro [19]. And the uniqueness of the maps was one of our main results in [21]. □

It is thus the remaining examples (2.8 to 2.11) which provide the main interest here. Those are the examples involving maps between simple groups of different rank. In all cases, we refer to [3] for more discussion.

In all of these examples, for any admissible map ϕ : T ⟶ T′, we let ϕ̃: T̃ ⟶ T̃′ denote its universal covering map. As will be seen, the only primes dividing the order of W (the Weyl group of G) are 2 and 3 in all cases. So any admissible map can be realized as a map BG ⟶ BG′_{p̂} for p ≥ 5 (see Theorem 1.12 above).

In the first three examples, G is one of the classical groups SU(3) or Sp(3). The p-stubborn subgroups of all of the classical groups U(n), SU(n), O(n), SO(n), and Sp(n) can be described explicitly (see [25]). For the results shown here, however, Proposition 1.13 can be used instead, to avoid having to list all pstubborn subgroups.

As one example, given here without proof, consider the group G = SU(3). Set ζ = exp(2πi/3), and consider the elements

A = ( \begin{matrix}1 & 0 & 0 \\ 0 & ζ & 0 \\ 0 & 0 & ζ^{2} \end{matrix} )  and  A_{1} = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} ) .

Then the only 3-stubborn subgroups of SU(3) are (up to conjugacy) the groups

N_{3}(T) = ⟨T, A_{1}⟩, and Γ_{3} = ⟨A, A_{1}⟩.

Note that Γ_{3} is a group of order 27 with center of order 3, and that N(Γ_{3})/Γ_{3}≅SL_{2}(𝔽_{3}).

In what follows, (x_{1}, …, x_{n}) denotes the usual coordinates in T̃ when G is one of the groups SU(n)⊆ U(n)⊆ Sp(n):

exp(x_{1}, …, x_{n}) = diag (exp(2πi · x_{1}), …, exp(2πi · x_{n})) .

Example 2.2. [3, Example 2.8] G = SU(3) and G′ = SU(6). For any k, m ∈ ℤ, define ϕ_{k,m} : T ⟶ T′ by setting

ϕ̃_{k,m}(x_{1}, x_{2}, x_{3}) = (kx_{1}, kx_{2}, kx_{3}, mx_{1}, mx_{2}, mx_{3}).

Then

(i) for p = 2, 3, ϕ_{k,m} extends to (f_{k,m})_{p̂} : B SU(3) ⟶ B SU(6)_{p̂} if and only if each of k, m is 0 or prime to p

(ii) ϕ_{k,m} extends to a map f_{k,m} : B SU(3) ⟶ B SU(6) if and only if each of k, m is 0 or prime to 6. In all cases, the extension is unique.

Proof. Point (ii) follows from point (i) and Proposition 1.2. And the uniqueness follows from Corollary 1.14(ii).

Fix p = 2 or 3. If each of k, m is 0 or prime to p, then there are unstable Adams operations ψ_{k}, ψ_{m} : B SU(3) ⟶ B SU(3)_{p̂} of degrees k and m, respectively (see [28]). By definition, ψ_{k}|BT ≃B(t ↦ t^{k}), and similarly for ψ_{m}. So if we let f_{k,m} be the composite

f_{k,m} : B SU(3) ⟶^{(ψ_{k},ψ_{m})} B SU(3)_{p̂} × B SU(3)_{p̂} ⟶^{⊕} B SU(6)_{p̂},

then f_{k,m}|BT ≃Bϕ_{k,m}.

We now prove the converse. If k ≡ m ≡ 0 (mod p), then all order p elements of T lie in Ker(ϕ_{k,m}). So by Proposition 1.8, ϕ_{k,m} extends to a map between the classifying spaces only if ϕ̃_{k,m} is noninjective, only if k = m = 0.

It remains to consider the case where p∤m, and p | k ≠ 0 (or vice versa). In this case, ϕ is regular (C_{G′}(ϕ(T)) = T′), and so by Lemma 1.3 there is a unique choice of homomorphism ϕ̄ : W ⟶ W′ for which ϕ is ϕ̄-equivariant (namely, ϕ̄(w) = w⊕w). Assume that Bϕ extends to a map f_{ p̂} : BG ⟶ BG′_{p̂}; then ϕ extends to an R_{p}-invariant representation ρ : N_{p}(T) ⟶ SU(6) by Proposition 1.7.

If p = 3, then set ζ = exp(2πi/3), and consider the matrices

A = ( \begin{matrix}1 & 0 & 0 \\ 0 & ζ & 0 \\ 0 & 0 & ζ^{2} \end{matrix} )  and  A_{1} = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} )

as before. Then ρ(A_{1})≡A_{1}⊕A_{1} (mod T′); and in particular Tr(ρ(A_{1})) = 0. On the other hand, since 3|k but 3∤m,

ρ(A) = ϕ(A) = diag(1, 1, 1, 1, ζ^{m}, ζ^{2m}),

and so Tr(ρ(A)) = 3. Thus, A and A_{1} are conjugate in SU(3) but ρ(A) and ρ(A_{1}) are not conjugate in SU(6). And since ρ is R_{p}-invariant, this contradicts Proposition 1.6.

If p = 2, then set

B = ( \begin{matrix}−1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ) , B_{1} = ( \begin{matrix}−1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) ,

and

S_{α}^{1} = \{( \begin{matrix}z^{−2} & 0 & 0 \\ 0 & z & 0 \\ 0 & 0 & z \end{matrix} ) : z ∈ S^{1}\} .

Then P = ⟨S_{α}^{1}, B, B_{1}⟩ is 2-stubborn in G, since

N(P) ⊆ C_{G}(S_{α}^{1}) ≅ U(2)

and

N_{G}(P)/P ≅ N_{U(2)}(P)/P = N_{SU(2)}(P ∩ SU(2))/(P ∩ SU(2))
≅ N_{SU(2)}(Q(8))/Q(8) ≅ Σ_{3}.

Since B is conjugate to B_{1} in U(2), there is an element in N(P) which sends B to B_{1} and centralizes S_{α}^{1}. Let x be such an element. Since ρ is R_{p}-invariant, there is an element y of SU(6) for which the composites ρ∘conj(x) and conj(y)∘ρ are equal. Then y centralizes ϕ(S_{α}^{1}); i.e., ρ(B_{1}) is conjugate to ρ(B) = ϕ(B) in

C_{SU(6)}(ϕ(S_{α}^{1})) = SU(6) ∩ ( U(1) × U(2) × U(1) × U(2)).

Also, ρ(B_{1})≡B_{1}⊕B_{1} (mod T′); and one now checks (by comparing traces) that this is impossible. □

Example 2.3. [3, Example 2.9] G = SU(3) and G′ = SU(6). For any k, m ∈ ℤ, define ϕ_{k,m} : T ⟶ T′ by setting

ϕ̃_{k,m}(x_{1},x_{2}, x_{3})

= (kx_{1}+mx_{2}, kx_{2}+mx_{3}, kx_{3}+mx_{1}, kx_{1}+mx_{3}, kx_{2}+mx_{1}, kx_{3}+mx_{2}).

Assume that k, m ≠ 0, and that k≠m (otherwise ϕ_{k,m} is one of the maps in [3, Example 2.8]). Then

(i) ϕ_{k,m} does not extend to any map (f_{k,m})_{ˆ2} : B SU(3) ⟶ B SU(6)_{ˆ2}.

(ii) ϕ_{k,m} extends to a map (f_{k,m})_{ˆ3} : B SU(3) ⟶ B SU(6)_{ˆ3} if and only if (k+m, 3) = 1; in which case the extension is unique.

Proof. Uniqueness follows from Corollary 1.14(ii): any admissible map extends to at most one homotopy class of maps between the classifying spaces.

Under the given assumptions on k and m, ϕ is always regular: C_{G′}(ϕ(T)) = T′. So by Lemma 1.3, ϕ̄: W ⟶ W′ ≅ Σ_{6} is uniquely defined. And one easily checks that the image of any element of W ∖1 is fixed-point free as a permutation on 6 objects. In other words, for any p, any extension of ϕ to ρ : N_{p}(T) ⟶ G′, and any g∈N_{p}(T)∖T, ρ(g) has no nonzero diagonal entries, and hence has trace zero.

p = 2: Consider the elements

B = ( \begin{matrix}−1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix} )  and  B_{1} = ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & −1 \end{matrix} )

in N_{p}(T). Then for any R_{p}-invariant representation ρ : N_{p}(T) ⟶ SU(6) extending ϕ = ϕ_{k,m},

ρ(B) = ϕ(B) = diag ((−1)^{k+m}, (−1)^{k}, (−1)^{m}, (−1)^{k}, (−1)^{k+m}, (−1)^{m}) .

Under the conditions on k and m, this cannot have trace zero. On the other hand, we noted above that T r(ρ(B_{1}))=0≠T r(ρ(B)). Since B_{1} and B are conjugate in G = SU(3), this contradicts the assumption that ρ is R_{2}-invariant (see Proposition 1.6).

p = 3: Set ζ = exp(2πi/3) again, and set

A = ( \begin{matrix}1 & 0 & 0 \\ 0 & ζ & 0 \\ 0 & 0 & ζ^{2} \end{matrix} )  and  A_{1} = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} )

in N_{p}(T). By definition,

ϕ(A) = diag(ζ^{m}, ζ^{k−m}, ζ^{−k}, ζ^{−m}, ζ^{k}, ζ^{m−k}).

Then T r(ϕ(A)) = 0 if and only if exactly one of the numbers k, m, k − m is a multiple of 3, if and only if 3∤(k+m).

If ϕ lifts to a map BG ⟶ BG′_{ˆ3}, then it extends to some R_{p}-invariant representation ρ : N_{3}(T) ⟶ G′. We saw earlier that Tr(ϕ(A_{1})) = 0, and so ϕ(A) must also have trace 0. And we just saw that this implies 3∤k + m.

Conversely, if 3∤k + m, then define ρ : N_{p}(T) = ⟨T, A_{1}⟩ ⟶ G′ by setting

ρ|T = ϕ, ρ(A_{1}) = A_{1} ⊕ A_{1}.

This is easily seen to be a well defined homomorphism, which is R_{p}-invariant by Corollary 1.14(i). So by Proposition 1.13, ρ extends to a map BG ⟶ BG′_{ˆ3}. □

As was shown by Adams & Mahmud [3, Proposition 2.16 and other remarks in Section 2], Examples 2.2 and 2.3 here give a complete list of all nonnullhomotopic maps BSU(3) ⟶ BSU(6) or BSU(3) ⟶ BSU(6)_{p̂}.

Example 2.4. [3, Example 2.10] G = Sp(3) and G′ = Sp(4). Define ϕ : T ⟶ T′ by setting

ϕ̃(x_{1}, x_{2}, x_{3}) = (x_{1} + x_{2} + x_{3}, −x_{1} + x_{2} + x_{3}, x_{1} − x_{2} + x_{3}, x_{1} + x_{2} − x_{3}).

Then ϕ extends to a unique map f_{ p̂} : B Sp(3) ⟶ B Sp(4)_{p̂} for all odd p, but does not extend to any map when p = 2. In particular, Bϕ is not the restriction of any map B Sp(3) ⟶ B Sp(4).

Proof. If p ≥ 5, then p∤|W |, and so ϕ extends to a (unique) map f_{ p̂} : B Sp(3) ⟶ B Sp(4)_{p̂} by Theorem 1.12.

If p = 2, then note first that diag(−1, −1, 1)∈ Ker(ϕ), and that this element is conjugate to elements in N(T)∖T. Hence, by Proposition 1.8, ϕ does not extend to any map B Sp(3) ⟶ B Sp(4)_{ˆ2}.

Now set p = 3. We can take N_{p}(T) = ⟨T, A_{1}⟩, where A_{1} = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} ).

Define ρ : N_{p}(T) ⟶ G′ = Sp(4) by setting ρ|T = ϕ, and

ρ(A_{1}) = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{matrix} ) = (1) ⊕ A_{1}.

Using Corollary 1.14(i), it is not hard to check that ρ is R_{p}-invariant. And by Proposition 1.13, ρ lifts to a map f_{ˆ3} : B Sp(3) ⟶ B Sp(4)_{ˆ3}; which is unique by Corollary 1.14(ii). □

Section 3

The last example (Example 2.11 in [3]) involves the groups G_{2} and F_{4}, and illustrates some of the techniques which can be used when working with exceptional Lie groups. This is the most interesting case of those we consider here. It is the only case where we find new maps BG ⟶ BG′ (without finite localization); i. e., maps which are not composites of maps constructed earlier. It is also the only case where we give a complete classification of the homotopy classes in [BG, BG′].

This result can be loosely described as follows. Following the notation of Adams & Mahmud, we say that a map f : BG_{2} ⟶ BF_{4} has type (k, m) if the following diagram commutes in rational cohomology:

B SU(3) ⟶^{(ψ_{k},ψ_{m})} B(SU(3)×_{C_{3}} SU(3))
incl↓  ^{incl}↓
BG_{2}  ⟶^{f}  BF_{4}.

Here, the vertical maps are induced by inclusions of subgroups (see Lemmas 3.2 and 3.3 below), and ψ_{k} and ψ_{m} are the unstable Adams operations on B SU(3). Also, type (k, m) is equivalent to type (−k, −m), but not to type (m, k) (the two SU(3) factors are embedded asymmetrically in F_{4}). Then every map from BG_{2} to BF_{4} has type (k, m) for some k and m. Aside from the null homotopic maps (k = m = 0), there are for each k > 0 prime to 6 unique homotopy classes of maps of type (k, 0) and (k, −k), and two homotopy classes each of maps of types (k, k) and (k, 2k).

The maps of type (1, 0) are induced by an inclusion ι : G_{2} ⟶ F_{4}, and the (k, 0) are the composites of Bι with unstable Adams operations on BG_{2}. Also, the maps of type (k, −k) can be obtained, though only away from the prime 3, as composites of maps of type (k, 0) with the “exceptional isogeny” on BG_{2} [16].

Of the remaining homotopy classes of maps, it is the two classes of type (1, 1) which are the most fundamental: the pairs of maps of type (k, k) and (k, 2k) are obtained by composing the maps of type (1, 1) with unstable Adams operations and (away from the prime 3) the exceptional isogeny on BG_{2}. In [32], Testerman constructs a homomorphism σ : G_{2}(\overline{𝔽_{7}}) ↪ F_{4}(\overline{𝔽_{7}}), which is remarkable because its image is a maximal connected subgroup of F_{4}(\overline{𝔽_{7}}). Via homotopy equivalences of Friedlander and Mislin [31], σ induces maps (BG_{2})_{p̂} ⟶ (BF_{4})_{p̂} of type (1, 1) for all primes p ≠ 7 (this is discussed in more detail at the end of the section). In other words, the homomorphism between algebraic groups provides a different construction of one of the two homotopy classes of maps of type (1, 1) (and a way of distinguishing the two classes).

The following lemma contains two general facts which are very useful when making computations in the exceptional Lie groups.

Lemma 3.1. Let G be any compact connected Lie group.

(i) (Borel) If G is simply connected, then the centralizer of any element in G — or the fixed point set of any automorphism — is connected.

(ii) Fix a maximal torus T ⊆G and an element g∈T. Let W = N_{G}(T)/T and W_{g} = N_{C(g)}(T)/T be the Weyl groups of G and of the centralizer C_{G}(g), respectively. Then the number of elements in T conjugate (in G) to g is just the Weyl group index [W : W_{g}].

Proof. (i) See [5, Theorem 3.4 and Corollary 3.5] (or [8, p.48, Théorème 1]). (ii) By [1, Lemma 4.33], two elements of T are conjugate in G if and only if

they are conjugate by an element of W. The result is then immediate. □

The groups G_{2} and F_{4} each comes in only one “version”: the simply connected groups are center-free (cf. Bourbaki [7, §VI.4], where the center of the simply connected group is denoted P (R)/Q(R)). From now on, we fix G = G_{2} and G′ = F_{4}; let T ⊆G and T′⊆G′ be maximal tori, and let W, W′ be the Weyl groups. As before, T̃ and T̃′ denote the universal covers of T and T′, respectively. We will also need to study the integral lattices of G and G′:

Λ = Ker[T̃ ⟶ T]  and  Λ′ = Ker[T̃′ ⟶ T′].

The next two lemmas collect for later use information about some of the subgroups of G_{2} and F_{4}. Note (cf. [8, p.40, Prop. 15]) that any isomorphism between the root systems and integral lattices of two compact connected Lie groups extends to an isomorphism between the groups themselves. This fact is very useful when working with the exceptional Lie groups, and in particular when identifying the centralizers of elements.

Lemma 3.2. We can identify T̃ = T̃(G_{2}) = {(x_{1}, x_{2}, x_{3}) ∈ ℝ^{3} : ∑x_{i} = 0} (with the usual inner product); such that the set of roots of G_{2} is

R = \{±(x_{i} − x_{j}), ±\frac{2x_{1}−x_{2}−x_{3}}{3} , ±\frac{2x_{2}−x_{1}−x_{3}}{3} , ±\frac{2x_{3}−x_{1}−x_{2}}{3} \} ⊆ T̃^{∗}

(where 1≤i<j≤3); and such that the integral lattice is

Λ = {(n_{1}, n_{2}, n_{3}) ∈ ℤ^{3} : ∑ n_{i} = 0} .

(i) All elements of order 2 in G_{2} are conjugate to each other, and the centralizer of any such element is isomorphic to SO(4) ≅ Sp(1)×_{C_{2}} Sp(1).

(ii) G_{2} contains a unique conjugacy class of subgroups isomorphic to (C_{2})^{3}. For any such A ≅ (C_{2})^{3} in G_{2}, C_{G_{2}}(A) = A and N_{G_{2}}(A)/A≅ GL_{3}(𝔽_{2}).

(iii) 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}

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

(iv) There is a unique conjugacy class of subgroup SU(3)⊆G_{2}. Also, there is a semidirect product SU(3)⋊C_{2}⊆G_{2}, where C_{2} acts on SU(3) by complex conjugation.

(v) G_{2} contains exactly two conjugacy classes of 3-stubborn subgroups, presented in the following list as subgroups of SU(3):

\begin{matrix}\underline{P} & \underline{N(P)/P} \\ ⟨T, A_{1}⟩ = N_{3}(T) & C_{2}×C_{2} \\ ⟨A, A_{1}⟩ & GL_{2}(𝔽_{3}). \end{matrix}

Here, A, A_{1} ∈ SU(3) again denote the matrices

A = ( \begin{matrix}1 & 0 & 0 \\ 0 & ζ & 0 \\ 0 & 0 & ζ^{2} \end{matrix} )  and  A_{1} = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} )

Proof. The description of the root system of G_{2} is given in Bourbaki [7, §VI.4.13] (the description there is slightly different, but is clearly equivalent). Note that the last three roots as given above are equal (as functions on T̃) to the coordinate functions ±x_{1}, ±x_{2}, and ±x_{3} — but the form given above makes clearer their lengths and angles relative to the other roots.

The description of Λ follows easily from the fact that the integral lattice is the group of elements in T̃ whose value on each root is integral. This property of the integral lattice holds in general for connected center-free groups (cf. [1, Proposition 5.3]).

(i) Set g̃ = (\frac{1}{2}, −\frac{1}{2}, 0) ∈ T̃, and let g = exp(g̃). Then g has order 2. By Lemma 3.1(i), C_{G}(g) is connected (and it clearly has maximal torus T). The roots of C_{G}(g) are precisely those roots of G which take integral values on g̃: namely, ±(x_{1} − x_{2}) and ±\frac{1}{3}(2x_{3} − x_{1} − x_{2}). Since these roots are orthogonal, the centralizer has type A_{1} × A_{1}. Also, the integral lattice is generated by the elements

\begin{matrix}e_{1} = (1, −1, 0) : & (x_{1} − x_{2})(e_{1}) = 2 & \frac{1}{3}(2x_{3} − x_{1} − x_{2})(e_{1}) = 0 \\ e_{2} = (−1, −1, 2) : & (x_{1} − x_{2})(e_{2}) = 0 & \frac{1}{3}(2x_{3} − x_{1} − x_{2})(e_{2}) = 2 \end{matrix}

and \frac{1}{2}(e_{1}+e_{2}). Upon comparing these with the integral lattices for Sp(1) ≅ SU(2) and SO(3), one sees that C_{G}(g) ≅ Sp(1)×_{C_{2}} Sp(1).

Finally, by Lemma 3.1(ii), the number of elements in T conjugate to g is |W |/4 = 3. Since this accounts for all elements of order 2 in T, we see that all elements of order 2 in G are conjugate to g.

(ii) Assume that A = ⟨g_{1}, g_{2}, g_{3}⟩ and A′ = ⟨g′_{1}, g′_{2}, g′_{3}⟩ are both isomorphic to (C_{2})^{3}. (Possibly A = A′ with different bases.) We want to show that there exists x∈G such that xg_{i}x^{−1} = g′_{i} for each i.

Since g_{1} is conjugate to g′_{1}, we may assume that g′_{1} = g_{1}. Then

A, A′ ⊆ C_{G_{2}}(g_{1}) = H_{1} ≅ Sp(1)×_{C_{2}} Sp(1).

Upon inspection, we see that all noncentral elements of order 2 in H_{1} are conjugate in H_{1} to (i, i). So we may assume (upon conjugating A and A′ by elements of H_{1}) that g′_{2} = g_{2} = (i, i).

We now have

A, A′ ⊆ C_{G_{2}}(g_{1}, g_{2}) = H_{2} = (H_{1})^{(i,i)} = ⟨S^{1}×_{C_{2}}S^{1}, (j, j)⟩.

Then g_{3} and g′_{3} are both conjugate (in H_{2}) to (j, j). This shows that A is conjugate to A′; and also that

C_{G_{2}}(A) = (H_{2})^{(j,j)} = A.

Also, N_{G_{2}}(A)/A≅ GL_{3}(𝔽_{2}), since we have just shown that any automorphism of A is an inner automorphism in G_{2}.

(iii) Fix a 2-stubborn subgroup P ⊆G_{2}, and let _{2}Z(P) be the 2-torsion subgroup of its center. If rk(_{2}Z(P)) = 1, say _{2}Z(P) = ⟨g⟩, then N_{G}(P) = N_{C(g)}(P), and so P is also 2-stubborn in C(g)≅ Sp(1)×_{C_{2}} Sp(1). By [21, Proposition 1.6], the 2-stubborn subgroups of Sp(1)×_{C_{2}} Sp(1) are precisely the groups of the form P′×_{C_{2}}P′′, where P′, P′′ are 2-stubborn in Sp(1). Also, the only 2-stubborn subgroups of Sp(1) are Q and N (just check the list of all 2-toral subgroups). Conversely, if P = P′×_{C_{2}}P′′ where P′ and P′′ are 2-stubborn in Sp(1), then N_{G}(P)/P ≅N(P′)/P′×N(P′′)/P′′, and so P is 2-stubborn in G = G_{2}. Note that the subgroups N×_{C_{2}}Q and Q×_{C_{2}}N are not conjugate in G_{2} — since if they were it would have to be via an element in the centralizer C_{G_{2}}(g).

Now assume that _{2}Z(P) = ⟨g_{1}, g_{2}⟩ has rank 2. Since C(g_{1}) is connected, ⟨g_{1}, g_{2}⟩ is contained in a maximal torus, and is hence the 2-torsion in some maximal torus T. One now checks that P ⊆C_{G_{2}}(g_{1}, g_{2}) = T ⋊C_{2}. This group is normal in N(P), and N(P)/P has no nontrivial normal 2-subgroups. Hence P = T ⋊C_{2}. Conversely, to check that this group is in fact 2-stubborn, note that N(P)/P ≅W/C_{2} ≅ Σ_{3}.

If rk(_{2}Z(P)) ≥ 3, then by part (ii), P ≅ (C_{2})^{3} and N(P)/P ≅ GL_{3}(𝔽_{2}). Now let α : G/P_{i} ⟶ G/P_{j} be any G-equivariant map. Choose x ∈ G such that

α(P_{i}) = xP_{j}; then x^{−1}P_{i}x ⊆ P_{j}. By inspection of the individual possibilities for (i, j), we see that P_{i} ⊆ P_{j}, and that x^{−1}P_{i}x is conjugate by an element of N(P_{j})/P_{j} to P_{i} (⊆ P_{j}). Hence, after replacing α by its composite with an automorphism of G/P_{j}, we may assume that x^{−1}P_{i}x = P_{i}. And in this case x ∈ N(P_{i}), and α is the composite of an automorphism of G/P_{i} with the map induced by the inclusion P_{i} ⊆ P_{j}.

(iv) Take g = exp(\frac{1}{3}, \frac{1}{3}, −\frac{2}{3}). Then C_{G_{2}}(g) has roots ±(x_{i} − x_{j}), and hence is isomorphic to SU(3). Also, since (x ↦ x^{−1})∈W (note that this is in the Weyl group of Sp(1)×_{C_{2}} Sp(1) ⊆ G_{2}), we see that complex conjugation on SU(3) is inner in G_{2}. Thus, there exists a∈N_{G_{2}}(⟨g⟩) such that conj(a) is complex conjugation on SU(3). Then a^{2}∈Z(SU(3)) = ⟨g⟩, and since |g| = 3 we may take a to have order 2. Thus, the normalizer of ⟨g⟩ is a semidirect product SU(3)⋊C_{2} ⊆ G_{2}.

Now let H be any other subgroup of G_{2} isomorphic to SU(3), and let g′ ∈ Z(H) ≅ C_{3} be a generator. We want to show that H is conjugate to the centralizer C_{G_{2}}(g) ≅ SU(3) just constructed, and it will suffice to show that g′ is conjugate to g. We may assume that g′ ∈ T. Since W ≅ Σ_{3} × C_{2} acts on T̃ by permuting coordinates and changing all signs, we see that g′ must be conjugate either to g = exp(\frac{1}{3}, \frac{1}{3}, \frac{−2}{3}), or to exp(\frac{1}{3}, \frac{−1}{3}, 0). And since the only roots in the centralizer of this last element are ±(2x_{3} − x_{1} − x_{2})/3, g′ must be conjugate to g.

(v) Fix a 3-stubborn subgroup P ∈ G_{2}. We may assume that it is contained in the maximal 3-toral subgroup N_{3}(T) = ⟨T, A_{1}⟩ ⊆ SU(3). Set z = diag(ζ, ζ, ζ); then z ∈ P since C_{G_{2}}(P) = Z(P) [21, Lemma 1.5]. Also, P ⊈ T (since N(P)/P is finite and T is not 3-stubborn); and so (up to conjugation) we may assume that A_{1} ∈ P. Furthermore, P ⫌ ⟨z, A_{1}⟩, since that subgroup is contained in a torus. By inspection, no cyclic subgroup of T strictly containing ⟨z⟩ is normalized by A_{1}, and so we must have ⟨z, A⟩ ⊆ P (the subgroup of 3-torsion in T). In particular, since C_{G_{2}}(z) = SU(3), C_{G_{2}}(P ∩ T) = C_{SU(3)}(P ∩ T) = T.

If P ∩ T ◁ N(P), then C_{G_{2}}(P ∩ T) = T and ⟨T, P ⟩ = N_{3}(T) are also normalized by N(P). In particular, N_{N_{3}(T)}(P)/P is normal in N(P)/P, is a 3-subgroup (cf. [21, Lemma A.3]), and must be trivial since P is 3-stubborn. And this is possible only if P = N_{3}(T) = ⟨T, A_{1}⟩.

Now assume that P ∩ T is not normal in N(P). Choose x ∈ N(P) such that x(P ∩ T)x^{−1} ≠ P ∩ T, and set Q = x(P ∩ T)x^{−1} for short. Choose any y ∈ Q ∖ T. Then y ∈ A_{1}^{±1}T, and [y, Q ∩ T] = 1 since Q is abelian. This implies that Q∩T ⊆ ⟨z⟩; and hence (since Q has index 3 in P) that P = ⟨A, A_{1}⟩. And in this case, P/⟨z⟩ ≅ (C_{3})^{2}, and one easily checks that the induced homomorphism

N_{G_{2}}(P)/P ⟶ Aut(P/⟨z⟩) ≅ GL_{2}(𝔽_{3})

is an isomorphism. □

One way frequently used to describe G_{2} directly is as the group of automorphisms of the Cayley numbers C: a division algebra on ℝ^{8} which contains the quaternions as a subalgebra ℍ ⊆ C. Under this description, the subgroup SU(3)⊆G_{2} can be regarded as the group of automorphisms α∈ Aut(C) such that α(i) = i (such an α acts complex linearly on the remaining coordinates). This shows that G_{2}/ SU(3)≅S^{6}: the set of unit vectors orthogonal to 1 ∈ C. Also, Sp(1)×_{C_{2}} Sp(1)≅ SO(4) can be regarded as the group of those α∈ Aut(C) for which α(ℍ) = ℍ. See [27, Appendix A.5] for more details.

Lemma 3.3. We can identify T̃′ = T̃(F_{4}) ≅ ℝ^{4} (with the usual inner product), such that the set of roots of F_{4} is

R′ = \{±x_{i} (1≤i≤4), ±x_{i}±x_{j} (1≤i<j≤4), \frac{1}{2} (±x_{1}±x_{2}±x_{3}±x_{4})\} ⊆ (T̃′)^{∗};

and such that the integral lattice is

Λ′ = {(n_{1}, n_{2}, n_{3}, n_{4}) ∈ ℤ^{4} : ∑ n_{i} even} .

Also, |W′| = 2^{7}·3^{2}, and W′ contains all signed permutations of the coordinates in ℝ^{4}. Furthermore:

(i) Every element in F_{4} is conjugate to its inverse.

(ii) If g̃ and g′ are two liftings to T̃′ of an element g∈T′, then

∥g̃∥^{2} ≡ ∥g′∥^{2} (mod ℤ) if |g| = 2,

and

∥g̃∥^{2} ≡ ∥g′∥^{2} (mod \frac{1}{3} ℤ) if |g| = 3.

Here, ∥g̃∥ denotes the norm of g̃ ∈ T̃′ ≅ ℝ^{4}, and |g| denotes the multiplicative order of g ∈ T′.

(iii) There are 2 conjugacy classes of elements of order 2 in F_{4}:

(I) ∥g̃_{I}∥^{2} ∈ ℤ, C_{F_{4}}(g_{I}) ≅ Spin(9)

(II) ∥g̃_{II}∥^{2} ∈ ℤ+\frac{1}{2} , C_{F_{4}}(g_{II}) ≅ Sp(1)×_{C_{2}} Sp(3) (iv) There is a unique conjugacy class of subgroups (C_{2})^{3}≅A⊆F_{4} with the

property that all elements of order 2 in A are conjugate to each other in F_{4}. For any such A, all elements of order 2 in A have type (II); C_{F_{4}}(A)≅A× SO(3), and N_{F_{4}}(A)/C_{F_{4}}(A) = Aut(A)≅ GL_{3}(𝔽_{2}).

(v) Set g = exp(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}, 1)∈T′. Then C_{F_{4}}(g) is the image of a map

θ : SU(3) ×_{C_{3}} SU(3) ↪ F_{4}

whose restriction to the maximal tori is given by

{ \atop (θ|T ×T)} ((x_{1}, x_{2}, x_{3}), (y_{1}, y_{2}, y_{3})) = (x_{1} + y_{3}, x_{2} + y_{3}, x_{3} + y_{3}, y_{1} − y_{2} = 2y_{1} + y_{3}).

Also, N_{F_{4}}(⟨g⟩) is a semidirect product (SU(3) ×_{C_{3}} SU(3))⋊C_{2}, where C_{2} acts on each SU(3) factor by complex conjugation.

Proof. The description of the root system of F_{4}, and the order of its Weyl group, are given in Bourbaki [7, §VI.4.9]. The Weyl group is generated by reflections in the kernels of the roots, and hence contains all signed permutations of the 4 coordinates (consider the roots x_{i} and x_{i} − x_{j}). Since F_{4} is centerfree, its integral lattice Λ′ is just the group of elements in T̃′ whose value on each root is an integer (cf. [1, Proposition 5.3]).

(i) As a special case of the above remarks, W′ contains the automorphism (t ↦ t^{−1}). Thus, g is conjugate to g^{−1} for all g∈T′; and hence for all g∈F_{4}.

(ii) This follows from the fact that if x ∈ \frac{1}{2}ℤ, y ∈ \frac{1}{3}ℤ, and x ≡ x′ and y ≡ y′ (mod ℤ), then

x^{2} ≡ (x′)^{2} (mod ℤ) and y^{2} ≡ (y′)^{2} (mod \frac{1}{3} ℤ)

(iii) See [21, Proposition 6.12]. By Borel’s theorem (Lemma 3.1), the centralizer of any element in F_{4} is connected. By Borel & Siebenthal [6] (and a study of the root system for F_{4}), F_{4} contains Spin(9) and Sp(1)×_{C_{2}} Sp(3) as maximal connected subgroups. Hence, these groups must be the centralizers of their central elements g_{I}, g_{II} of order 2. Alternatively, this can be shown by taking the following explicit elements, and computing the roots of their centralizers:

g_{I} = exp(0, 0, 0, 1): roots ±x_{i}, ±x_{i}±x_{j}

g_{II} = exp(\frac{1}{2}, \frac{1}{2}, 0, 0): roots ±x_{3}, ±x_{4}, ±x_{1}±x_{2}, ±x_{3}±x_{4}, \frac{±(x_{1}−x_{2})±x_{3}±x_{4}}{2}

A comparison of the orders of the Weyl groups (W′ = W (F_{4}) has order 2^{7} ·3^{2}) shows that each maximal torus contains three elements conjugate to g_{I} and 12 conjugate to g_{II}. So this accounts for all elements of order 2. The norms of g_{I} and g_{II} are immediate from the explicit formulas above.

(iv) Assume first that A = ⟨g_{1}, g_{2}, g_{3}⟩ ≅ (C_{2})^{3} is such that all elements of order 2 in A have type (I). Then A⊆C_{F_{4}}(g_{1})≅ Spin(9). Since Spin(9) is simply connected, C_{F_{4}}(g_{1}, g_{2}) is again connected by Lemma 3.1(i). Since any maximal torus of C_{F_{4}}(g_{1}, g_{2}) which contains g_{3} also contains g_{1} and g_{2}, A must be contained in a maximal torus of F_{4}. And this contradicts the fact that each maximal torus contains only 3 elements of type (I).

Now assume that A = ⟨g_{1}, g_{2}, g_{3}⟩ and A′ = ⟨g′_{1}, g′_{2}, g′_{3}⟩ are both isomorphic to (C_{2})^{3}, and that all elements of order 2 in A, A′ have type (II). (Possibly A = A′ with different bases.) Since g_{1} is conjugate to g′_{1}, we may conjugate A′ to arrange that g′_{1} = g_{1}. Then A, A′⊆C_{F_{4}}(g_{1}) = H_{1}≅ Sp(1)×_{C_{2}} Sp(3). Upon inspection, we see that all noncentral elements in H_{1} of type (II) are conjugate in H_{1} to

h_{1} = (i, diag(i, i, i))  or  h_{2} = (1, diag(−1, 1, 1)).

And g_{1}·h_{2} does not have type (II) (where g_{1} = (−1, I) = (1,-I)). Thus, g_{2}, g′_{2} are both conjugate to h_{1}, and we may assume (by conjugating by appropriate elements of H_{1}) that g′_{2} = g_{2} = h_{1}. We now have

A, A′ ⊆ C_{F_{4}}(g_{1}, g_{2}) = H_{2} ≅ C_{H_{1}}(h_{1}) ≅ ⟨S^{1}×_{C_{2}} U(3), (j, j · I)⟩.

Finally, we see that g_{3}, g′_{3} must both be conjugate (in H_{2}) to (j, jI). This shows that A is conjugate to A′; and that

C_{F_{4}}(A) ≅ (H_{2})^{(j,jI)} ≅ A × SO(3).

Also, N_{F_{4}}(A)/C_{F_{4}}(A)≅ GL_{3}(𝔽_{2}), since we have just seen that any automorphism of A is an inner automorphism in F_{4}.

(v) The centralizer C_{F_{4}}(g) is connected by Lemma 3.1(i), and it contains T′ as a maximal torus. The roots of C_{F_{4}}(g) are precisely those roots of F_{4} which take integer values on (\frac{1}{3}, \frac{1}{3}, \frac{1}{3}, 1): namely,

±(x_{i} − x_{j})(1 ≤ i < j ≤ 3), ±x_{4}, ±\frac{1}{2} (x_{1} + x_{2} + x_{3} ± x_{4}).  (1)

The homomorphism θ̃: T̃(SU(3) ×_{C_{3}} SU(3)) ⟶ T̃′, defined by setting

θ̃((x_{1}, x_{2}, x_{3}), (y_{1}, y_{2}, y_{3})) = (x_{1} + y_{3}, x_{2} + y_{3}, x_{3} + y_{3}, y_{1} − y_{2}),

is an isomorphism of vector spaces and integral lattices, and its dual sends the roots in (1) bijectively to the roots of SU(3) ×_{C_{3}} SU(3). Since an isomorphism between root systems and integral lattices induces an isomorphism between compact connected Lie groups (cf. [8, p.40, Prop. 15]), θ̃ extends to an isomorphism

θ : SU(3) ×_{C_{3}} SU(3) ⟶^{≅} C_{F_{4}}(g) ⊆ F_{4}.

By point (i), g is conjugate to g^{−1} in F_{4}. In particular, there exists a∈N_{F_{4}}(⟨g⟩) such that conj(a) is complex conjugation on both SU(3) factors in the centralizer. Then a^{2}∈Z(C_{F_{4}}(⟨g⟩)) = ⟨g⟩, and (since |g| = 3) we can take a of order 2. This shows that N_{F_{4}}(⟨g⟩) is a semidirect product (SU(3)×_{C_{3}} SU(3))⋊C_{2}; and finishes the proof of the lemma. □

Note, in part (iii) above, that θ(SU(3) × 1) is the factor which contains long roots of F_{4}: in keeping with the notation in [3, Example 2.11]. As noted in [3], there is a subgroup G_{2} of F_{4} such that θ(SU(3) × 1)⊆G_{2}⊆F_{4}. One way to see this is to consider the elements of order 2 in the maximal torus of θ(1× SU(3)): namely, g_{1} = exp(0, 0, 0, 1) and g_{2}, g_{1}g_{2} = exp(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, ±\frac{1}{2}). We have seen that C_{F_{4}}(g_{1})≅ Spin(9), and so θ(SU(3)×1)⊆C_{F_{4}}(g_{1}, g_{2})≅ Spin(8). If we regard G_{2} as the group of automorphisms of the Cayley numbers, then the induced inclusion G_{2}⊆ SO(8) lifts to Spin(8) (G_{2} being simply connected), and its restriction to SU(3)⊆G_{2} is the composite SU(3)⊆ U(3)⊆ U(4)⊆ SO(8). This is isomorphic to the above inclusion of θ(SU(3) × 1) into C(g_{1}, g_{2}); and so this inclusion factors through G_{2}.

The following result now completely classifies homotopy classes of maps from BG_{2} to BF_{4}.

Example 3.4. [3, Example 2.11] Set G = G_{2} and G′ = F_{4}. Use the inclusions SU(3)⊆G_{2} and SU(3) ×_{C_{3}} SU(3)⊆F_{4} defined above to identify

T̃ = T̃(G_{2}) = T̃(SU(3)) and T̃′ = T̃(F_{4}) = T̃(SU(3)) × T̃(SU(3)).

Under this identification, define ϕ = ϕ_{k,m} : T ⟶ T′ (any k, m ∈ ℤ) by setting

ϕ̃(x) = ϕ̃_{k,m}(x) = (kx, mx) (for any x ∈ T̃(G_{2}) = T̃(SU(3)))

Then

(i) ϕ extends to a map (f_{k,m})_{ˆ2} : BG_{2} ⟶ (BF_{4})_{ˆ2} if and only if either k = m = 0, or k is odd and m ∈ {0, ±k, 2k}. The extension is unique if m = 0 or m = −k; otherwise there are exactly two distinct homotopy classes of maps which extend ϕ.

(ii) ϕ extends to a map (f_{k,m})_{ˆ3} : BG_{2} ⟶ (BF_{4})_{ˆ3} if and only if each of k, m is 0 or prime to 3; and the extension is unique up to homotopy.

(iii) ϕ extends to a map f_{k,m} : BG_{2} ⟶ BF_{4} if and only if either k = m = 0, or (k, 6) = 1 and m ∈ {0, ±k, 2k}. The extension is unique if m = 0 or m = −k. If m = k or m = 2k, then there are exactly two distinct homotopy classes of maps which extend ϕ. Furthermore, for any f : BG_{2} ⟶ BF_{4}, f|BT ≃ Bϕ_{k,m} for some k and m.

Proof. We first check that the ϕ_{k,m} are admissible. By construction, they are equivariant with respect to the diagonal inclusion

W (SU(3)) ⟶^{Δ} W (SU(3)) × W (SU(3)) ⊆ W′.

Also, W = W (SU(3))×⟨w⟩, where w(t) = t^{−1} for all t ∈ T. Choose w′ ∈ W′ such that w′(t) = t^{−1} for all t ∈ T′ (Lemma 3.3(v)). Then the ϕ_{k,m} are equivariant with respect to the homomorphism

Δ × (w ↦ w′) : W ⟶ W′;

and hence are admissible.

Now let ϕ be any nontrivial admissible homomorphism from G_{2} to F_{4}. If ϕ is equivariant with respect to the diagonal inclusion Δ : Σ_{3} ⟶ Σ_{3}×Σ_{3} as above, then (by Schur’s lemma) ϕ must be equal to ϕ_{k,m} for some k, m. By [3, Theorem 2.21], there is some homomorphism ϕ̄ : W ⟶ W′ such that ϕ is ϕ̄-equivariant, and ϕ̄ is injective since ϕ̃ is. We will show that ϕ̄ can be chosen so ϕ̄|Σ_{3} is conjugate in W′ to Δ, and hence that ϕ is conjugate by an element of W′ to some ϕ_{k,m}.

To see this, fix elements a, b ∈ W (SU(3)) ⊆ W such that |a| = 3 and |b| = 2 (so bab^{−1} = a^{−1}). Write elements in W′ as matrices with respect to the standard basis for T̃′ ≅ ℝ^{4}; then

A_{1} = ( \begin{matrix}0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) and B_{1} = ( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )

generate the first Σ_{3} factor, and

A_{2} = ( \begin{matrix}\frac{1}{2} & −\frac{1}{2} & −\frac{1}{2} & −\frac{1}{2} \\ −\frac{1}{2} & \frac{1}{2} & −\frac{1}{2} & −\frac{1}{2} \\ −\frac{1}{2} & −\frac{1}{2} & \frac{1}{2} & −\frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & −\frac{1}{2} \end{matrix} ) and B_{2} = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )

generate the second. Note that A_{1} acts via rotation by 2π/3 on the plane P l_{1} = {(x, y, −x−y, 0)} and fixes P l_{2} = {(x, x, x, y)} = P l_{1}^{⊥}, while A_{2} rotates P l_{2} and fixes P l_{1}.

Since ⟨A_{1}, A_{2}⟩ is a Sylow 3-subgroup of W′, we may assume that ϕ̄(a) is one of the elements A_{1}, A_{2}, or A_{1}A_{2}. If ϕ̄(a) = A_{1}, then Im(ϕ̃) = P l_{1}, since this is the unique A_{1}-invariant subplane in T̃′ upon which A_{1} acts by rotation. The relation bab^{−1} = a^{−1} forces ϕ̄(b) to leave both P l_{1} and P l_{2} invariant, and to act on P l_{1} via a reflection. Define ϕ̄′ : W ⟶ W′ by setting ϕ̄′(a) = A_{1}A_{2}; ϕ̄′(w) = w′ (these are again the elements which act via (t ↦ t^{−1})); and ϕ̄′(b) = ϕ̄(b) or ϕ̄(b)·B_{2} according to which one acts as a reflection on P l_{2}. This is a well defined homomorphism — ⟨a, b⟩ ≅ Σ_{3} acts on each P l_{i} via the standard 2-dimensional representation — and ϕ is ϕ̄′-equivariant since Im(ϕ) = P l_{1}.

The same argument applies if ϕ̄(a) = A_{2}, and shows that ϕ̄ can always be chosen such that ϕ̄(a) = A_{1}A_{2}. Consider the homomorphism μ : W′ ⟶ GL_{4}(𝔽_{2}) induced by the action of W′ on Λ′/2Λ′, with respect to the basis

{(2, 0, 0, 0), (1, 1, 1, 1), (1, −1, 0, 0), (0, 1, −1, 0)}

of the integral lattice Λ′ ⊆ T̃′. It is not hard to see that Ker(μ) = {±I}, and that

Im(μ) = \{( \begin{matrix}X & Y \\ 0 & Z \end{matrix} ) : X, Z ∈ GL_{2}(𝔽_{2}), Y ∈ M_{2}(𝔽_{2})\} .

Also, μ(A_{1}) = ({I \atop 0} {0 \atop α}), μ(B_{1}) = ({I \atop 0} {0 \atop β} ), μ(A_{2}) = ({α \atop 0} {0 \atop I}), and μ(B_{2}) = ({β \atop 0} {0 \atop I}); where α = ({0 \atop 1} {1 \atop 1}) and β = ({1 \atop 0} {1 \atop 1}). Hence, since any element of order 2 in GL_{2}(𝔽_{2}) is conjugate to β (conjugate by an element of ⟨α⟩), we may assume that

μ(ϕ̄(a)) = ( \begin{matrix}α & 0 \\ 0 & α \end{matrix} ) and μ(ϕ̄(b)) = ( \begin{matrix}β & X \\ 0 & β \end{matrix} )

for some X. The relations b^{2} = 1 and bab = a^{−1} imply that X = 0 or β. Since ({I \atop 0} {α \atop I}) ({β \atop 0} {β \atop β} ) ({I \atop 0} {α \atop I})^{−1} = ({β \atop 0} {0 \atop β} ), we may assume that X = 0; i.e., that

ϕ̄(b) = ±B_{1}B_{2}. And if we set U = ({0 \atop 1} {−1 \atop 0}) ⊕ ({0 \atop −1} {1 \atop 0}) ∈ W′, then [U, A_{1}A_{2}] = I and UB_{1}B_{2}U^{−1} = −B_{1}B_{2}; and this finishes the proof that ϕ̄|Σ_{3} is conjugate to the diagonal inclusion.

It remains to determine which of the ϕ_{k,m} extend to maps from BG_{2} ⟶ (BF_{4})_{p̂} (for p = 2, 3), and to count the number of homotopy classes of such maps. As before, we identify

SU(3) = C_{G_{2}} (exp(\frac{1}{3} , \frac{1}{3} , \frac{−2}{3} )) ⊆ G_{2}.

Fix an embedding

θ : SU(3)×_{C_{3}} SU(3) ⟶ F_{4} :

where Im(θ) = C_{F_{4}} (exp(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}, 1)) and (by Lemma 3.3(v))

θ̃|T((x_{1}, x_{2}, x_{3}), (y_{1}, y_{2}, y_{3})) = (x_{1} + y_{3}, x_{2} + y_{3}, x_{3} + y_{3}, y_{1} − y_{2}).

Using this, we get the explicit formula

{ \atop ϕ_{k,m}}(x_{1}, x_{2}, x_{3}) = (kx_{1} + mx_{3}, kx_{2} + mx_{3}, kx_{3} + mx_{3}, mx_{1} − mx_{2}). (1)

Recall (Lemma 1.3) that ϕ is regular if and only if Im(ϕ̃) is not contained in the kernel of any root of F_{4}. Using this last criterion, one checks that ϕ is regular unless either k = 0, or m = 0, or k = −m, or k = 2m.

\underline{p = 3:} If ϕ = ϕ_{k,m} is trivial, then any extension f : BG_{2} ⟶ (BF_{4})_{ˆ3} of Bϕ_{k,m} is null homotopic (cf. [21, Theorem 3.11]).

Now fix (k, m) ≠ (0, 0), and assume that ϕ = ϕ_{k,m} lifts to a map f : BG_{2} ⟶ (BF_{4})_{ˆ3}. We first show that each of k, m must be 0 or prime to 3. If k and m are both multiples of 3, then all elements of order 3 in T lie in Ker(ϕ), and this contradicts Proposition 1.8. If one of k or m is prime to 3 and the other a nonzero multiple of 3, then ϕ is regular by the above remarks. So there is only one possible induced map ϕ̄: W ⟶ W′, and by the remarks at the beginning of the proof it must be Δ.

Again consider the matrices

A = ( \begin{matrix}1 & 0 & 0 \\ 0 & ζ & 0 \\ 0 & 0 & ζ^{2} \end{matrix} )  and  A_{1} = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} )

of order 3 in SU(3) ⊆ G_{2}. These are conjugate in SU(3); and, in fact, every element in ⟨T, A_{1}⟩ ∖ T is conjugate in SU(3) to A. From the previous paragraph, we see that ϕ̄(A_{1}T) = θ(A_{1}, A_{1})·T′. Hence, for any R_{p}-invariant representation ρ : N_{3}(T) = ⟨T, A_{1}⟩ ⟶ F_{4}, ρ(A_{1}) ≡ θ(A_{1}, A_{1}) (mod T′). Thus ρ(A_{1}) is conjugate to (A, A); but since ρ is R_{p}-invariant it is also conjugate to ρ(A) = (1, A^{±1}) or (A^{±1}, 1). And a norm computation, using Lemma 3.3(ii), shows that (A, A) = exp(−\frac{1}{3}, 0, −\frac{2}{3}, −\frac{1}{3}) cannot be conjugate in F_{4} to (A, 1) = exp(0, \frac{1}{3}, −\frac{1}{3}, 0) or (1, A) = exp(−\frac{1}{3}, −\frac{1}{3}, −\frac{1}{3}, −\frac{1}{3}).

\underline{Construction of R_{3}-invariant representations:} Now assume that k and m each is 0 or prime to 3. Extend ϕ = ϕ_{k,m} to

ρ : N_{3}(T) = ⟨T, A_{1}⟩ ⟶ SU(3)×_{C_{3}} SU(3) ⊆ F_{4}

by setting

ρ(A_{1}) = \{ \begin{matrix}θ(1, A_{1}) & if k = 0 \\ θ(A_{1}, 1) & if m = 0 \\ θ(A_{1}, A_{1}) & otherwise. \end{matrix}

This clearly gives a well defined homomorphism, and we claim that it is R_{3}- invariant.

We must show, for any pair P, P′ ⊆ N_{3}(T) of 3-stubborn subgroups of G_{2}, and any x ∈ G_{2} such that xP x^{−1} ⊆ P′, that the maps ρ|(xP x^{−1}) and conj(x)∘(ρ|P) are conjugate in F_{4}. Consider the 3-stubborn subgroups of G_{2} listed in Lemma 3.2(v). For any subgroup P′′ ⊆ ⟨T, A_{1}⟩ isomorphic to ⟨A, A_{1}⟩, P ∩ T = _{3}T (the 3-torsion subgroup); and hence P′′ is conjugate to ⟨A, A_{1}⟩ by an element of T. Hence, we need only consider the cases where x ∈ N(T) (where the maps are conjugate since ϕ is admissible); or where P = ⟨A, A_{1}⟩ and x ∈ N(P). And in this latter case, ρ|(xP x^{−1}) and conj(x)∘(ρ|P) are conjugate in Im(θ), since they have the same character in each SU(3)-factor.

\underline{Uniqueness of the R_{3}-invariant representations:} Assume that ρ′ : ⟨T, A_{1}⟩ ⟶ F_{4} is another R_{3}-invariant representation. If ϕ is regular, then

ρ′(A_{1}) ∈ ρ(A_{1})·C_{F_{4}}(T) = ρ(A_{1})·T′.

Also, ρ(A_{1}) = θ(A_{1}, A_{1}) in this case, and every element in ρ(A_{1})T′ is conjugate by an element of T′ to ρ(A_{1}). So ρ and ρ′ are conjugate.

Now assume that ϕ is not regular. Consider the elements g_{1} = exp(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}, 1) and g_{2} = exp(\frac{1}{3}, −\frac{1}{3}, 1, \frac{1}{3}), where in both cases C_{F_{4}}(g_{i}) ≅ SU(3)×_{C_{3}} SU(3). If m = 0 or k = 0, then ϕ(T) is the maximal torus in one of the factors of C_{F_{4}}(g_{1}), while if k = −m or k = 2m then ϕ(T) is the maximal torus in one of the factors of C_{F_{4}}(g_{2}). Fix η : SU(3)×_{C_{3}} SU(3) ⟶ F_{4} such that Im(ϕ) = η(T ×1). Note that (since g_{1} and g_{2} are conjugate by an element of W′), η is conjugate in F_{4} to θ, except possibly with the factors reversed. Now, C_{F_{4}}(ϕ(T)) = η(T ×_{C_{3}} SU(3)), and

Im(ρ), Im(ρ′) ⊆ η(⟨T, A_{1}⟩×_{C_{3}} SU(3)).

And after conjugating, we may assume that Im(ρ), Im(ρ′) ⊆ η(⟨T, A_{1}⟩×_{C_{3}}T).

Write ρ(A_{1}) = η(x, y) and ρ′(A_{1}) = η(x′, y′), where x, x′ ∈ A_{1}T, and y, y′ ∈ T. Since y^{3} = (y′)^{3} = 1, we may assume that y, y′ ∈ ⟨A⟩. Since A and A^{−1} are conjugate in SU(3), we may assume that y, y′ ∈ {1, A}. And since any element in A_{1}T is conjugate to A_{1}, we may assume x, x′ = A_{1}. Finally, ρ(A_{1}) and ρ′(A_{1}) must be conjugate in F_{4} — since both are conjugate to ϕ(A) — and hence η(A, y) must be conjugate to η(A, y′). And since η is conjugate in F_{4} to θ (except possibly with the factors switched), formula (1) can be used to show that y = y′, and hence that ρ = ρ′.

\underline{Existence and uniqueness of maps:} It remains to check that the appropriate higher limits vanish. By Lemma 3.2(v) again, G_{2} contains up to conjugacy two 3-stubborn subgroups: P_{1} = ⟨T, A_{1}⟩ and P_{2} = ⟨A, A_{1}⟩. Also, N(P_{1})/P_{1} has order prime to 3, and 3^{2}|∤||N(P_{2})/P_{2}|.

The main problem is to determine the centralizer of ρ(P_{2}). Set z = diag(ζ, ζ, ζ) ∈ SU(3) ⊆ G_{2}. If 3 ∤ k+m, then by (1),

ρ(z) = ρ(exp(\frac{1}{3} , \frac{1}{3} , −\frac{2}{3} )) = exp(±(\frac{1}{3} , \frac{1}{3} , \frac{1}{3} , 1));

and C_{F_{4}}(ρ(P_{2})) = C_{Im(θ)}(ρ(P_{2})). If 3|k+m, then ρ(z) = 1, and C_{F_{4}}(ρ(P_{2})) = C_{C_{F_{4}} (ρ(A))}(ρ(A_{1})), where

ρ(A) = ρ(exp(\frac{1}{3} , −\frac{1}{3} , 0)) = exp(\frac{k}{3} , \frac{−k}{3} , 0, \frac{2m}{3} )

is conjugate to exp(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}, 1). In either case, C_{F_{4}}(ρ(P_{2})) is the centralizer of a 3subgroup of SU(3)×_{C_{3}} SU(3), and hence is either connected or has 3 components. In particular, the action of N(P_{2})/P_{2} on

Π_{1}^{ρ}(G/P_{2}) ≅ π_{0}(C_{F_{4}}(ρ(P_{2})))

has order at most 2. By Corollary 1.11, lim_{⟵}^{i}(Π_{1}^{ρ}) = 0 for all i ≥ 1, and lim_{⟵}^{i}(Π_{n}^{ρ}) = 0 for all i, n ≥ 2. By Theorem 1.9, ρ (and hence ϕ) extends to a unique homotopy class of maps f : BG_{2} ⟶ (BF_{4})_{ˆ3}.

\underline{p = 2:} By [21, Theorem 3.11], the trivial homomorphism ϕ_{0,0} extends to a unique (null homotopic) map. So we restrict attention to the case ϕ = ϕ_{k,m} for (k, m) ≠ (0, 0).

Assume that ϕ extends to an R_{p}-invariant representation ρ. Note that N_{SU(3)}(T)∖T ⊆ N_{G_{2}}(T)∖T contains elements of order 2, which are conjugate to any given element of order 2 in T (Lemma 3.2(i)). So by Proposition 1.8, for each g∈T = T (G_{2}) of order 2, ϕ(g) ≠ 1. Also, ϕ(g) must have type (II) in F_{4} by Lemmas 3.2(i,ii) and 3.3(iv). This is the case if and only if k is odd: since by formula (1), ϕ(g) has type (I) if k is even and m is odd, and ϕ(g) = 1 if k and m are both even.

Assume now that k is odd.

\underline{Uniqueness of the R_{2}-invariant representation:} Fix some ϕ = ϕ_{k,m}, and assume that ρ, ρ′ : N_{2}(T) ⟶ F_{4} are two R_{2}-invariant representations which extend ϕ. We must show that they are conjugate.

Fix g ∈ T of order 2, set h = ϕ(g), and identify

C_{G_{2}}(g) = Sp(1)×_{C_{2}} Sp(1), N_{2}(T) = N×_{C_{2}}N, and C_{G_{2}}(h) = Sp(1)×_{C_{2}} Sp(3).

The images of ρ and ρ′ lie in C_{F_{4}}(h); and ρ and ρ′ lift to homomorphisms

ρ̃, ρ′ : N×N = ⟨S^{1}, j⟩ × ⟨S^{1}, j⟩ ⟶ Sp(1)× Sp(3).

The easiest way to check this last step is to take the pullbacks of Sp(1)× Sp(3) along ρ and ρ′, and check that they both must be isomorphic to N×N.

Since ρ is R_{2}-invariant, there is some x ∈ F_{4} such that xρ(z, j)x^{−1} = ρ(z, i) for all z ∈ S^{1}. In particular, x ∈ C_{F_{4}}(h), and hence lifts to x̃ ∈ Sp(1)× Sp(3) such that x̃ρ̃(z, j)x^{−1} = ±ρ̃(z, i) for all z. The sign (±) must be constant (by continuity); and since (1, i) is conjugate to (1, −i) in N×N we may assume that x̃ρ̃(z, j)x^{−1} = ρ̃(z, i) for all z. In particular, ρ(z, j) is conjugate to ρ′(z, j); and in fact ρ(z, z′j) is conjugate to ρ′(z, z′j) for all z, z′ ∈ S^{1} (since z′j is conjugate in N to j). The same argument applies to all other cosets of T in N_{2}(T). Thus, ρ̃ and ρ′ have the same characters in Sp(1) and Sp(3), and hence are conjugate. And hence ρ and ρ′ are conjugate.

\underline{Existence and uniqueness of maps:} Before going further, we must specify more precisely some of the identifications already used. Set g = exp(\frac{1}{2}, −\frac{1}{2}, 0) ∈ G_{2}. Then

ϕ_{k,m}(g) = exp(\frac{k}{2} , −\frac{k}{2} , 0, m) = \{ \begin{matrix}h_{1} = exp(\frac{1}{2}, −\frac{1}{2}, 0, 0) & if m is even \\ h_{2} = exp(\frac{1}{2}, −\frac{1}{2}, 0, 1) & if m is odd. \end{matrix}

By Lemmas 3.2(i) and 3.3(iii), we know that the centralizers of these elements are the images of embeddings

σ : Sp(1)×_{C_{2}} Sp(1) ⟶ G_{2} and τ_{1}, τ_{2} : Sp(1)×_{C_{2}} Sp(3) ⟶ F_{4}.

So if we choose N_{2}(T) ⊆ Im(σ), it then follows that Im(ρ) ⊆ Im(τ_{i}) (where i = 1, 2 depending on the parity of m).

Let S ⊆ Sp(1)×_{C_{2}} Sp(1) and S′ ⊆ Sp(1)×_{C_{2}} Sp(3) denote the standard maximal tori. Upon checking the roots of these centralizers, we see that σ and the τ_{i} can be chosen to satisfy the following formulas:

(σ̃|S)(x, y) = (x + y, −x + y, −2y)
({ \atop τ_{1}|S′})(x_{1}; x_{2}, x_{3}, x_{4}) = (x_{1} + x_{2}, −x_{1} + x_{2}, −x_{3} − x_{4}, −x_{3} + x_{4})  (2)
({ \atop τ_{2}|S′})(x_{1}; x_{2}, x_{3}, x_{4}) = (x_{1} + x_{2}, x_{1} − x_{2}, −x_{3} + x_{4}, −x_{3} − x_{4}).

For example, if m is even, then the roots of C_{F_{4}}(h_{1}) are ±x_{1}±x_{2}, ±x_{3}±x_{4}, ±x_{3}, ±x_{4}, ±\frac{1}{2}(x_{1} + x_{2}±x_{3}±x_{4}) (i.e., the roots of F_{4} whose value on (\frac{1}{2}, −\frac{k}{2} , 0, 0) is integral). And these are the roots of Im(τ_{1}).

The idea of the proof is now to push all computations into the centralizers C_{G_{2}}(g) and C_{F_{4}}(ϕ(g)). So we start by identifying the composite (τ_{1}|S′)^{−1} ∘ ϕ ∘ (σ|S). This splits into two cases, depending on whether m is even or odd.

\underline{Case 1. k odd and m even:} In this case, (1) and (2) yield the formula

((τ_{1}|S′)^{−1} ∘ ϕ ∘ (σ|S)) (x, y) = (kx; ky − 2my, ky + m(y − x), ky + m(x + y)) .

(3)

\underline{Case 1a. m ≠ 0, 2k:} Consider the subgroup P = σ(Q×_{C_{2}}N)⊆G_{2} (see Lemma 3.2(iii)). There is x ∈ N(P) such that x(i, 1)x^{−1} = (j, 1) and x centralizes σ(1×S^{1}). Hence, since ρ is R_{p}-invariant, ρσ(j, 1) must lie in and be conjugate to ρσ(i, 1) in

C_{F_{4}}(ϕσ(1 × S^{1})) = τ_{1} (C_{Sp(1)×_{C_{2}} Sp(3)}(exp(ℝ · (0; k − 2m, k + m, k + m))))
= τ_{1}(Sp(1)×_{C_{2}}(U(1)× U(2))).

(Recall that the centralizer of any subtorus is connected.) By (3),

τ_{1}^{−1}ρσ(i, 1) = τ_{1}^{−1}ϕσ(i, 1) = exp(\frac{k}{4} ; 0, −\frac{m}{4} , \frac{m}{4} ) = (±i, ( \begin{matrix}1 & 0 & 0 \\ 0 & ±1 & 0 \\ 0 & 0 & ±1 \end{matrix} ))

and so τ_{1}^{−1}ρσ(j, 1) must have the form

τ_{1}^{−1}ρσ(j, 1) = (z, ( \begin{matrix}1 & 0 & 0 \\ 0 & ±1 & 0 \\ 0 & 0 & ±1 \end{matrix} ))

for some z ∈ Sp(1) conjugate to i. In addition (since ρ is a homomorphism) this element must act on

τ_{1}^{−1}ρσ(S^{1} × 1) = exp(ℝ·(k; 0, −m, m))

via (z ↦ z^{−1}); and this is impossible.

\underline{Case 1b. m = 2k:} In this case, (3) takes the form

((τ_{1}|S′)^{−1} ∘ ϕ ∘ (σ|S)) (x, y) = (kx; −3ky, 3ky − 2kx, 3ky + 2kx) .  (3a)

We extend this to a homomorphism ρ̄: N×_{C_{2}}N ⟶ Sp(1)×_{C_{2}} Sp(3) by setting

ρ̄(z, 1) = (z^{k}, ( \begin{matrix}1 & 0 & 0 \\ 0 & z^{−2k} & 0 \\ 0 & 0 & z^{2k} \end{matrix} ))  ρ̄(1, z) = (1, ( \begin{matrix}z^{−3k} & 0 & 0 \\ 0 & z^{3k} & 0 \\ 0 & 0 & z^{3k} \end{matrix} ))

ρ̄(j, 1) = (j, ( \begin{matrix}−1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ))  ρ̄(1, j) = (1, ( \begin{matrix}−j & 0 & 0 \\ 0 & 0 & j \\ 0 & j & 0 \end{matrix} )) .

and set ρ = τ_{1} ∘ ρ̄: N×_{C_{2}}N ⟶ F_{4}. Note in particular that

ρ̄(i, 1) = (±i, ( \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & −1 \end{matrix} ))  and  ρ̄(1, i) = (1, ( \begin{matrix}±i & 0 & 0 \\ 0 & ∓i & 0 \\ 0 & 0 & ∓i \end{matrix} )) .

We claim that the composite

ρ = τ_{1} ∘ ρ̄: N_{2}(T) = N×_{C_{2}}N ⟶ F_{4}

is R_{2}-invariant. To show this, it suffices using Lemma 3.2(iii) to check that for any of the subgroups P_{i} ⊆ N×_{C_{2}}N listed there (i = 1, …, 6), and any x ∈ N(P_{i}), the homomorphisms τ_{1} ∘ ρ|P_{i} and τ_{1} ∘ (ρ|P_{i}) ∘ conj(x) are conjugate in F_{4}.

The case P_{6} ≅ (C_{2})^{3} follows from Lemma 3.3(iv), once one has checked that all elements of order 2 are sent to elements of type (II) in F_{4}. To see this, note that for any 1 ≠ x ∈ P_{6}, ρ(x) is conjugate in Im(τ_{1}) to ϕσ(i, i) or ϕσ(1, −1), both of which have type (II).

For i = 1, …, 4, this is straightforward, and the homomorphisms are in fact always conjugate in Sp(1)×_{C_{2}} Sp(3) (i.e., before composing with τ_{1}). In the case P_{1} = N×_{C_{2}}N, there is nothing to prove (N(P_{1})/P_{1} = 1). In the next two cases, P_{2} = N×_{C_{2}}Q and P_{3} = Q×_{C_{2}}N, the arguments can be greatly simplified by conjugating elements of Sp(3) with the matrices

( \begin{matrix}1 & 0 & 0 \\ 0 & j & 0 \\ 0 & 0 & 1 \end{matrix} )  and  ( \begin{matrix}j & 0 & 0 \\ 0 & i/\sqrt{2} & i/\sqrt{2} \\ 0 & −1/\sqrt{2} & 1/\sqrt{2} \end{matrix} )

respectively. Note also that the centralizer in U(2) of the matrix ({0 \atop −j} {j \atop 0}) is SU(2). The last case, P_{4} = Q×_{C_{2}}Q, follows from the two previous ones.

It remains to consider the case P_{5} = ⟨T, γ⟩, where we write γ = σ(j, j) for short. Fix any x ∈ N(P_{5}) = N(T). Since ϕ is admissible, there is y ∈ N(T′) ⊆ F_{4} such that ρ(xax^{−1}) = yρ(a)y^{−1} for all a ∈ T. Also, ρ(γ), yρ(γ)y^{−1}, and ρ(xγx^{−1}) all have the same conjugation action on ρ(T), and hence lie in the same coset of C_{F_{4}}(ρ(T)) = T′. From the formula ρ(γ) = τ_{1}(j, j·I), we see that any two elements in ρ(γ)T′ are conjugate by an element of T′. And this shows that conj(y′) ∘ ρ = ρ ∘ conj(x) (on P_{5} = ⟨T, γ⟩) for some y′ ∈ T′y.

The higher limits lim_{⟵}^{i}(Π_{n}^{ρ}) can now be computed using Theorem 1.10 (and the formula for Π_{n}^{ρ} in 1.9); together with the computation Λ^{∗}(GL_{3}(𝔽_{2}); (𝔽_{2})^{3}) = 0 of [21, Proposition 6.3]. The results are summarized in the following table:

\underline{P}  \underline{N(P)/P} \underline{C_{F_{4}}(ρ(P))} \underline{Λ^{0}(NP/P; Π_{1}^{ρ}(−))} \underline{Λ^{1}(NP/P; Π_{1}^{ρ}(−))}
N×_{C_{2}}N  1  (C_{2})^{2}  (ℤ/2)^{2}  0
N×_{C_{2}}Q 1 × Σ_{3}  (C_{2})^{2}  0  0
Q×_{C_{2}}N Σ_{3} × 1  (C_{2})^{3}  0  ℤ/2
Q×_{C_{2}}Q Σ_{3} × Σ_{3}  (C_{2})^{3}  0  0
T ⋊ C_{2}  Σ_{3}  _{2}(T′) ≅ (C_{2})^{4}  0  (ℤ/2)^{2}
(C_{2})^{3} GL_{3}(𝔽_{2}) (C_{2})^{3}× SO(3)  0  0.

Note for example that whenever there is P′ ⫌ P such that C(ρ(P′)) = C(ρ(P)), then (N(P) ∩ P′)/P acts trivially on all homotopy groups of BC(ρ(P)), and hence Λ^{∗}(NP/P; Π_{n}^{ρ}(G/P)) = 0 for all n by Theorem 1.10(iii).

In all other cases (n > 1 or i > 1), Λ^{i}(N(P)/P; Π_{n}^{ρ}(G/P)) = 0. Since the Λ^{i} are the higher limits of the quotient functors of a certain filtration of Π_{n}^{ρ} (see Theorem 1.10(i)), these computations show that lim_{⟵}^{1}(Π_{1}^{ρ}) = ℤ/2, and that

lim_{⟵}^{i}(Π_{n}^{ρ}) = 0 for all (i, n) ≠ (1, 1). It follows that ρ (and hence ϕ_{k,2k}) extends to exactly two homotopy classes of maps (f_{k,2k})_{ˆ2}, (f′_{k,2k})_{ˆ2} : BG_{2} ⟶ (BF_{4})_{ˆ2}. This last step, the existence in this situation of exactly two extensions, follows from the arguments in Wojtkowiak [29], even though it is not stated explicitly there.

\underline{Case 1c. m = 0:} In this case, formula (3) simplifies to give

((τ_{1}|S′)^{−1} ∘ ϕ ∘ (σ|T)) (x, y) = (kx; ky, ky, ky).  (3b)

In other words, τ_{1}^{−1}ϕσ(z, w) = (z^{k}, w^{k}·I) for any (z, w)∈S^{1}×_{C_{2}}S^{1}. This can now be extended to a homomorphism ρ : σ(N×_{C_{2}}N) ⟶ F_{4} (N = ⟨S^{1}, j⟩) by setting ρσ(j, 1) = τ_{1}(j, I) and ρσ(1, j) = τ_{1}(1, j·I).

We check that ρ is R_{2}-invariant, by again referring to the list of 2-stubborn subgroups of G_{2} in Lemma 3.2(iii). Invariance with respect to conjugation and inclusion of the subgroups N×_{C_{2}}Q, Q×_{C_{2}}N, and Q×_{C_{2}}Q is easily checked (and holds within Im(τ_{1})≅ Sp(1)×_{C_{2}} Sp(3)). Invariance for the subgroup T ⋊C_{2} follows automatically from the fact that ϕ is admissible. And invariance for the subgroup (C_{2})^{3}⊆G_{2} follows from Lemma 3.3(iv) (uniqueness of (C_{2})^{3}⊆F_{4}) — after checking that all elements of order 2 in N×_{C_{2}}N are sent to elements of type (II) in F_{4}.

The higher limits lim_{⟵}^{i}(Π_{n}^{ρ}) can again be computed using Theorem 1.10 (and 1.9); together with the computation Λ^{∗}(GL_{3}(𝔽_{2}); (𝔽_{2})^{3}) = 0 of [21, Proposition 6.3]. This time, we get the following table:

\underline{P}  \underline{N(P)/P} \underline{C_{F_{4}}(ρ(P))} \underline{Λ^{0}(NP/P; Π_{1}^{ρ}(−))} \underline{Λ^{1}(NP/P; Π_{1}^{ρ}(−))}
N×_{C_{2}}N  1  O(3)  ℤ/2  0
N×_{C_{2}}Q 1 × Σ_{3}  O(3)  0  0
Q×_{C_{2}}N Σ_{3} × 1  O(3)  0  0
Q×_{C_{2}}Q Σ_{3} × Σ_{3}  O(3)  0  0
T ⋊ C_{2}  Σ_{3}  (C_{2})^{2}× SO(3)  0  ℤ/2
(C_{2})^{3} GL_{3}(𝔽_{2}) (C_{2})^{3}× SO(3)  0  0.

Also, Λ^{i}(N(P)/P; Π_{n}^{ρ}(G/P)) = 0 whenever i > 1, or n > 1 and i > 0. So in this case, lim_{⟵}^{0}(Π_{n}^{ρ}) ≅ π_{n}(B SO(3)) for all n, and lim_{⟵}^{i}(Π_{n}^{ρ}) = 0 whenever i > 0. Thus, by Theorem 1.9, ρ, and hence ϕ_{k,2k}, extend to a unique homotopy classe

of map (f_{k,0})_{ˆ2} : BG_{2} ⟶ (BF_{4})_{ˆ2}.

\underline{Case 2. k odd and m odd:} These cases can be handled by the same straightforward procedure as that carried out in Case 1. But it is much simpler to use Case 1 directly, together with the relation

ϕ_{k,m} ∘ ϵ = ϕ_{−k+2m,k+m}

given in [3, Example 2.11]. Here, ϵ denotes the restriction to the maximal torus T ⊆ G_{2} of the “exceptional isogeny” Φ : (BG_{2})_{ˆ2} ⟶ (BG_{2})_{ˆ2} constructed by Friedlander [16]. Since Φ is a homotopy equivalence, we can define (for any odd k):

(f_{k,k})_{ˆ2} = Φ^{−1} ∘ (f_{k,2k})_{ˆ2} and (f_{k,−k})_{ˆ2} = Φ ∘ (f_{−k,0})_{ˆ2}.

And the same argument shows that (f_{k,−k})_{ˆ2} is unique, and that ϕ_{k,k} extends to exactly two maps.

Conversely, if k and m are odd and (f_{k,m})_{ˆ2} is defined, then (f_{−k+2m,k+m})_{ˆ2} = Φ ∘ (f_{k,m})_{ˆ2} is also defined. So either k + m = 0 or k + m = 2(−k + 2m) by Case 1. And these relations imply that m = ±k. □

Among the maps f_{k,m} : BG_{2} ⟶ BF_{4} constructed above, f_{1,0} is (aside from f_{0,0}) the only one which is induced by a homomorphism between the (real) compact Lie groups. See the remarks before Example 3.4 on embedding of G_{2} as a subgroup of F_{4}. It is not hard, using Lemmas 3.2(ii) and 3.3(iv), to show that any two such subgroups are conjugate.

There are irreducible complex representations V^{27} of G_{2} and W^{26} of F_{4} (with dimensions as given by the superscripts). It is not hard to check that V |T ≅ (W |ϕ_{1,1}(T)) ⊕ ϵ, where ϵ denotes the trivial 1-dimensional representation (and that this relation holds for no other ϕ_{k,m}). This shows in particular that there is no homomorphism of type (1, 1): since W |ϕ_{1,1}(T) is not the weight system (restriction to T) of any G_{2}-representation. In characteristic 7, the corresponding representation V (\overline{𝔽_{7}}) of G_{2}(\overline{𝔽_{7}}) contains a 1-dimensional fixed subspace, and Testerman’s embedding σ : G_{2}(\overline{𝔽_{7}}) ⟶ F_{4}(\overline{𝔽_{7}}) in [32] is characterized by the property that W (\overline{𝔽_{7}})|σ(G_{2}(\overline{𝔽_{7}})) ≅ V (\overline{𝔽_{7}})/\overline{𝔽_{7}}. Hence the map

σ_{∗} : (BG_{2})_{p̂} ≃ BG_{2}(\overline{𝔽_{7}})_{p̂} ⟶ BF_{4}(\overline{𝔽_{7}})_{p̂} ≃ (BF_{4})_{p̂},

defined for any p ≠ 7 using the equivalences of Friedlander and Mislin [31, Theorem 1.4], has type (1, 1).

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Instytut Matematyki, Uniwersytet Warszawski, ul. Banacha 2, 02-097 Warszawa, Poland

Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 USA

Département de Mathématiques, Université de Paris XIII, 93430 Villetaneuse, France