MAPS BETWEEN CLASSIFYING SPACES AND APPLICATIONS

by Dietrich Notbohm

Abstract. For connected Lie groups G and H the calculation of the mapping space map(BG, BH) can be reduced to the case of simply connected Lie groups. This reduction method allows some applications. For example a homotopy classification of self maps BG⟶BG which induce rational homotopy equivalences and a classification of fake Lie groups up to homotopy.

1. Introduction.

Let G and H be compact connected Lie groups with classifying spaces BG and BH. Recently the mapping space map(BG, BH) has been studied intensively by many people. In this paper we will contribute the following theorem, which is more a ‘philosophical’ statement than an explicit result.

Theorem. The calculation of the mapping space map(BG, BH), where G and H are compact connected Lie groups, can be reduced to the case of simply connected Lie groups.

Of course this statement needs some illustrations. We have to explain how this reduction can be carried out.

It is well known that for every compact connected Lie group G there exists a finite cover G̃ sitting in an exact sequence

1⟶K_{G}⟶G̃ ⟶^{q_{G}} G⟶1 ,

such that G̃ = G_{s} × T is a product of a simply connected Lie group and a torus. K_{G} is finite and central in G̃. We call such coverings finite universal. The group G̃ is unique up to isomorphism, but the map q_{G} : G̃⟶G is not unique. One can compose q_{G} with a finite self cover of T.

Now let

K_{G} ⟶ G̃ ⟶^{q_{G}} G

and

K_{H} ⟶ H̃ ⟶^{q_{H}} H

be finite universal coverings. We get induced maps

Bq_{G}^{∗} : map(BG, BH) ⟶ map(BG̃, BH)

and

Bq_{H}_{∗} : map(BG, BH̃) ⟶ map(BG, BH) .

The analysis of these two maps allows the reduction to the case of universal Lie groups which are products of a simply connected Lie groups and a torus.

Typeset by AMS-TEX

1.1 Proposition. Let K ⟶ G̃ ⟶^{q} G be a finite covering (universal is not necessary).

(1) The map

Bq^{∗} : map(BG, BH) ⟶ map(BG̃, BH)

is a pseudo equivalence.

(2) f̃: BG̃⟶BH is in the image of π_{0}(Bq^{∗}) iff f̃ | BK is homotopically trivial.

Remark. A map f : X⟶Y is called a pseudo equivalence if it induces an injection between the components and if π_{n}(X, x)⟶π_{n}(Y, f(x)) is an isomorphism for all x ∈ X and all n ≥ 1.

1.2 Proposition. Let K ⟶ H̃ ⟶^{q} H be a finite covering (universal is not necessary).

(1) The map

Bq_{∗} : map(BG, BH̃) ⟶ map(BG, BH)

induces a π_{0}-injection.

(2) For every f̃: BG ⟶ BH̃ there exists a homotopy fibration

BK⟶map(BG, BH̃)f̃⟶map(BG, BH)_{Bq∘f̃} .

Dwyer and Zabrodsky calculated map(BP, BG) for a finite p-group P [D-Z]. In [N] this result was generalized to the case of p-toral groups, which are extensions of a torus by a finite p-group. To describe these mapping spaces, the following construction was introduced in [D-Z].

Let ρ : P ⟶H be a homorphism between groups. Denote by C_{H}(ρ) the centralizer of ρ and by Rep(P, H) the set of representations of P in H, i.e. the set of conjugation classes of homorphisms. The obvious homomorphism C_{H}(ρ) × P ⟶H passes to a map between the classifying spaces, which has as adjoint the map

ad : BC_{H}(ρ)⟶map(BP, BH)_{Bρ} .

1.3 Theorem. ( [D-Z], [N]) For a p-toral group P and a compact Lie group H the map

ad : ∐_{ρ∈Rep(P,H)} BC_{H}(ρ) ⟶ map(BP, BH)

is a mod-p equivalence.

Remark. A map f : X⟶Y is called a mod-p equivalence, if it induces an isomorphism in H^{∗}( ; ℤ/p).

This theorem allows us to carry out the next step in the program. We will pass from universal Lie groups to simply connected Lie groups.

1.4 Proposition. Let T be a torus and let G and H be compact connected Lie groups. Then the map

∐_{ρ∈Rep(T,H)} map(BG, BC_{H}(ρ)) ⟶^{ad_{∗}} map(BG, map(BT, BH)) = map(BG × BT, BH)

is a mod-p equivalence for every prime p.

For any homomorphism ρ : T ⟶H the centralizer C_{H}(ρ) is generated by all maximal tori which contains the image of ρ [B–tD]. Thus C_{H}(ρ) is connected. If C_{H}(ρ) is not simply connected, we can use proposition 1.2 to pass to the finite universal cover C̃_{H}(ρ) which may contain a torus T as a factor. Splitting off the factor map(BG, BT) ≃ ∐_{H^{2}(BG;π_{2}(BT))} BT of map(BG, BC̃_{H}(ρ)) leads to the situation, where the groups in the target and in the domain are simply connected. The equivalence follows because BT is an Eilenberg-MacLane space of degree 2.

Now we apply these results to self maps of BG. First we investigate the canonical map

K : [BG, BG] ⟶ Hom_{λ}(K(BG), K(BG)) ,

where [ , ] denotes the set of homotopy classes of maps and Hom_{λ}( , ) denotes λ-ring homomorphisms. A λ-ring homomorphism θ : K(BG)⟶K(BG) is called real if θ maps the image of KO(BG) ⟶ K(BG) into itself.

1.5 Theorem. Any real λ-ring isomorphism

θ : K(BG) ⟶ K(BG)

is contained in the image of K, and hence is realizable by a self map of BG.

This theorem has a localised version as well as a completed version (see theorem 5.3). Similiar results as the completed version are also proved by Wojtkowiak [Wo].

As a second application we will show that the map

K : [BG, BG]⟶Hom(K(BG), K(BG))

is injective on some part of [BG, BG]. This is a generalisation of a result of S. Jackowski, J.E. McClure and R. Oliver [J–M–O 1], which we recall now.

For a simple Lie group G, Hubbuck proved that every self map f : BG⟶BG is the trivial map on rational cohomology or looks in rational cohomology like a composition ψ^{k} ∘ Bα^{∗} of an Adams operation of degree k, k ≥ 1, and a map induced by an outer automorphism α [ H ]. K. Ishiguro showed that only those k can occur, which are relative prime to the order of the Weyl group W_{G} of G [I]. This establishes a map

[BG, BG] ⟶ {0} ∐ Out(G) × {k > 0 : (k, |W G|) = 1} .

The Dwyer-Zabrodsky construction induces a map BC(G)⟶map(BG, BG)_{id}, where C(G) is the center of G. Denote by Out(G) the set of outer homomorhism and map(BG, BG)_{α,k} the component of ψ^{k} ∘ Bα.

1.6 Theorem. ( [J–M–O 1]) Let G be a simple Lie group.

(1) The map

[BG, BG] ⟶ {0} ∐ Out(G) × {k > 0 : (k, |W G|) = 1}

is a bijection.

(2) The composition

BC(G) ⟶^{ad} map(BG, BG)_{id} ⟶^{ψ^{k}Bα} map(BG, BG)_{α,k}

is a mod-p equivalence for all primes p.

For any compact connected Lie group we denote by HE_{ℚ}(BG) the set of homotopy classes of self maps BG⟶BG which are rational equivalences. If G is simple then HE_{ℚ}(BG) is the set of all nontrivial self maps.

1.7 Theorem. Let G be a compact connected Lie group

(1) The map

K : HE_{ℚ}(BG) ⟶ Hom_{λ}(K(BG), K(BG))

is an injection.

(2) For every f ∈ HE_{ℚ}(BG) the composition

BC(G)⟶map(BG, BG)_{f}

is a mod-p equivalence for every prime p.

Remark. This result is also proved independently by Jackowski, McClure and Oliver [J– M–O 2]

As a last application we will classify up to homotopy the spaces in the adic genus genus_{ℚ}^{∧}(BG) of BG, where G is a compact connected Lie group. A nilpotent CW –complex of finite type Y is in the adic genus of BG if the p–completed spaces Y_{p}^{∧} and BG_{p}^{∧} for all primes p are homotopy equivalent, as well as Y_{ℚ} and BG_{ℚ}.

1.8 Theorem. Let Y_{1} and Y_{2} be in the adic genus of BG. Y_{1} ≃ Y_{2} if and only if there exist a real λ-ring isomorphism K(Y_{1}) ≅ K(Y_{2}).

A finite loop space X = ΩBX is called an adic fake Lie group of type G, G a compact connected Lie group, if BX ∈ genus_{ℚ}^{∧}(BG).

Two finite loop spaces X_{1} = ΩBX_{1} and X_{2} = ΩBX_{2} are called homotopy equivalent if BX_{1} and BX_{2} are homotopy equivalent. Thus theorem 1.8 answers an open question about fake Lie groups of [ N-S 1,2,3], namely to classify fake Lie groups up to homotopy.

The paper is organized as follows: Propositions 1.1, 1.2, and 1.4 are proved in the next section. The applications are discussed in same the order as they are stated, each in one section.

Whenever we deal with localizations X_{(p)} or completions X_{p}^{∧} we do this in the sense of [ B-K ], and define X^{∧} := ∏_{p} X_{p}^{∧}.

2. The reduction to simply connected Lie groups.

Let K⟶G⟶\overline{G} be an exact sequence of compact Lie groups, with G and \overline{G} connected and K finite, and let BK⟶BG⟶B\overline{G} be the associated fibration between the classifying spaces. We think of the classifying spaces as given by the Milnor models. \overline{G} acts on B̃K := EG/K from the right. B̃K ≃ BK and B̃K/\overline{G} ≃ BG. The \overline{G}-action on B̃K is free. Hence we get, for any connected space X with trivial \overline{G}-action, a sequence of homotopy equivalences

map(BG, X) ≃ map\overline{G}(B̃K, X)
≃ map\overline{G}(E\overline{G} × B̃K, X)
≃ map\overline{G}(E\overline{G}, map(B̃K, X))
= map(B̃K, X)^{h\overline{G}} .

Here, map\overline{G}( , ) is the space of equivariant maps, \overline{G} acts on map(B̃K, X) via the action on the source, and we may take the last equation as the definition of homotopy fixed-point sets.

The component map(B̃K, X)_{c} of the constant map is fixed under the action. The canonical inclusion X⟶map(B̃K, X)_{c} is equivariant and induces a map

map(B\overline{G}, X) = X^{h\overline{G}} ⟶ map(B̃K, X)_{c}^{h\overline{G}} ⟶ map(BG, X)

between homotopy fixed-point sets. The last map is a pseudo equivalence; it is the inclusion of some component into whole the mapping space. This proves the following lemma

2.1 Lemma. If X ⟶ map(BK, X)_{c} is a homotopy equivalence, then

map(B\overline{G}, X) ⟶ map(BG, X)

is a pseudo equivalence.

Sufficient conditions on X for satisfying the assumptions may be found in [D–W].

Now we are prepared for

Proof of proposition 1.1. Let K⟶G⟶\overline{G} be an exact sequence of compact Lie groups, where K finite. By lemma 2.1 we need only show that for a compact connected Lie group H,

BH⟶map(BK, BH)_{c}

is a homotopy equivalence. This follows from the Sullivan conjecture [M], which implies that the loop space of the pointed mapping space map_{∗}(BK, BH)_{c} is contractible. □

There is also a p-completed version of proposition 1.1, even for K beeing a compact Lie group. In [F–M] is shown that map_{∗}(BK, H_{p}^{∧}) is contractible, and hence, that BK⟶map(BK, BH_{p}^{∧})_{c} is homotopy equivalence. Now, the same proof works to show

2.2 Proposition. Let K ⟶ G ⟶^{q} \overline{G} be an exact sequence of compact Lie groups and H a compact connected Lie group. Then,

map(B\overline{G}, BH_{p}^{∧}) ⟶ map(BG, BH_{p}^{∧})

is a pseudo equivalence.

Proof of proposition 1.2. Let K⟶H̃ ⟶^{q} H be a finite cover of the compact connected Lie group H. Then K is central in H̃ and finite. The fibration

BK⟶BH̃⟶BH

is a principal fibration. By Thom–theory [T], as revisited in [N–S 4], every map f̃ : BG⟶BH̃, G connected, establishes a principal fibration

Bq_{∗} : map(BG, BH̃)f̃⟶map(BG, BH)_{f}

with structure group map(BG, BK). f is defined by f := Bq ∘ f̃. Since BG is 1– connected, map(BG, BK) consists only of the component of the constant map. The evaluation map e : map(BG, BK)⟶BK is a homotopy equivalence and has as inverse the canonical section s : BK⟶map(BG, BK). This proves part (ii) of proposition 1.2. To show that Bq_{∗} : map(BG, BH̃)⟶map(BG, BH) is a π_{0}-injection we use obstruction theory. The obstruction groups H^{∗}(BG; π_{∗}(BK)) for lifting a homotopy vanish because BG is 1-connected and BK is an Eilenberg-McLane space of degree 1. This completes the proof. □

2.3 Corollary. Let G and H be compact connected Lie groups and let K⟶H̃⟶H be a finite cover of H. Then there exists a finite universal cover G̃⟶G of G, such that every map f : BG⟶BH can be lifted to a map f̃ : BG̃⟶BH̃. Moreover, two maps f, g : BG⟶BH are homotopic if and only if the lifts f̃ and g̃ are homotopic.

Proof. We choose a finite universal covering q : G̃⟶G, where G̃ = G_{s} × T is a product of a simply connected Lie group and a torus. We can compose Bq with the map id × ψ^{k} : BG_{s} × BT ⟶BG_{s} × BT, where k is the order of K and ψ^{k} the Adams map inducing multiplikation with k in H^{2}(BT). Then, for every map f : BG⟶BH there is only one obstruction group H^{2}(BG̃; K) ≅ H^{2}(BT; K) for a lift of f ∘ Bq. If we try to lift the map f ∘ Bq ∘ (id × ψ^{k}), every obstruction must vanish. This proves the existence of lifts. The uniqueness of the lifts follows from the fact that in the two maps

map(BG̃, BH̃) ⟶ map(BG̃, BH) ⟵ map(BG, BH)

are π_{0}-injections (proposition 1.1 and 1.2). □

Finally, we proof proposition 1.4.

Proof of proposition 1.4. We only have to consider the case of a fixed homomorphism ρ : T ⟶H. First, we show that ad_{∗} is a π_{0}–injection. The centralizer C_{H}(ρ) is connected, because it is generated by all maximal tori of H which contain the image of ρ [B–tD; IV,2.3]. Using completions we can build the commutative diagram

[BG, BC_{H}(ρ)]  ⟶  [BG, BC_{H}(ρ)^{∧}]
↓  ↓
[BG, map(BT, BH)_{Bρ}] ⟶ [BG, map(BT, BH)_{Bρ)}^{∧}] .

The homotopy groups of the homotopy fiber F of BC_{H}(ρ)⟶BC_{H}(ρ)^{∧} are all rational vector spaces, which are concentrated only in odd dimensions [F–M]. The rational cohomology of BG is concentrated in even degrees. Thus, all the obstruction groups H^{∗}(BG; π_{∗}(F)) for lifting homotopies vanish. That is that the top row is an injection. The right column is a bijection by theorem 1.3, and therefore ad_{∗} is a π_{0}–injection.

Next, we show that ad_{∗} is also a surjection. This is based on the following lemma.

2.4 Lemma. For two tori T_{1} and T_{2} the map

ad_{∗} : ∐_{ρ∈Rep(T_{2},H)} map(BT_{1}, BC(ρ)) ⟶ map(BT_{1}, map(BT_{2}, BH))

is a π_{0}-bijection.

Proof. Again, we only have to consider the case of a fixed homomorphism ρ : T_{2}⟶H. For every map f : BT_{1} × BT_{2}⟶BH there exists a homomorphism α : T_{1} × T_{2}⟶H, such that f ≃ Bα (theorem 1.3). Let τ := α |_{T_{1}}. Then we know that both, the maximal torus of H and the image im(τ) of τ, are contained in C_{H}(ρ). That is that Bτ lifts to a map Bτ′ : BT_{1}⟶BC_{H}(ρ). Obviously, ad_{∗}(Bτ′) = Bα ≃ f, i.e ad_{∗} is surjective. The injectivity was already proved above. □

Let f : BG × BT ⟶BH be a map such that f|_{BT} ≃ Bρ. By theorem 1.3, we can choose a homomorphism α : T_{G} × T ⟶H with the properties f|_{BT_{G}} × BT ≃ Bα and α|_{T} = ρ. For the diagram

BT_{G} × BT ⟶^{Bα} BT_{H}
↓  ↓
BG × BT ⟶^{f} BH

there is an associated homomorphism β : W_{G}⟶W_{H} between the Weyl groups such that Bα is equivariant [A–M; 2.21]. C(ρ) is a connected subgroup of H of maximal rank. This implies that T_{C_{H}(ρ)} = T_{H} and W_{C_{H}(ρ)} is a subgroup of W_{H}. Because W_{G} acts trivially on BT, β splits over W_{C_{H}(ρ)}.

Let Bτ be the restriction of Bα to BT_{G}. Then, Bτ splits over BC_{H}(ρ), is equivariant with respect to β, and ad_{∗}(Bτ) = Bα. Passing to the Weyl group invariants in rational cohomology yields a map

θ : H^{∗}(BC(ρ); ℚ) ⟶ H^{∗}(BG; ℚ) .

θ has a realization

f′_{ℚ} : BG_{ℚ}⟶BC(ρ)_{ℚ} ,

which fits into

(BT_{G})_{ℚ} ⟶^{Bτ} (BT_{C_{H}(ρ)})_{ℚ}
↓  ↓
BG_{ℚ} ⟶^{f′_{ℚ}} BC_{H}(ρ)_{ℚ} .

Therefore, ad_{∗}(f′_{ℚ}) and f_{ℚ} induce the same map in rational cohomology and are homotopic.

From the homotopy equivalence BC(ρ)^{∧} ≃ (map(BT, BH)_{Bρ})^{∧} we get a map f′^{∧} : BG^{∧}⟶BC(ρ)^{∧} with ad_{∗}(f′^{∧}) ≃ f ∧.

H^{∗}(BG; ℚ)⟶H^{∗}(BT_{G}; ℚ) is injective. Restricting all the maps to BT_{G} lemma 2.4 implies that all maps agree over the adeles. Using the arithmetic square we glue f′_{ℚ} and f′^{∧} together and get a map f′ : BG⟶BC(ρ) which has the property ad_{∗}(f′) ≃ f as desired. This shows that ad_{∗} is a surjection.

To finish the proof of proposition 1.4, we can now fix a map f′ : BG⟶BC_{H}(ρ) with adjoint f : BG × BT ⟶BH respectively f : BG⟶map(BT, BH). In [B–N] it is shown that, for any map g : BG⟶BH, G a compact Lie group and H compact connected Lie group, map(BG, BH_{p}^{∧})_{g} is p–complete and that map(BG, BH)_{g} ⟶ map(BG, BH_{p}^{∧})_{g} is a homotopy equivalent after completion. Hence, after completion, the Dwyer–Zabrodsky map induces an equivalence

BC_{H}(ρ)_{p}^{∧} ⟶^{≃} map(BT, BH_{p}^{∧})_{B}ρ .

Now, in the commutative diagram

map(BG, BC_{H}(ρ))_{f} ⟶^{ad_{∗}} map(BG, map(BT, BH)_{Bρ})_{f′} ≃ map(BG × BT, BH)_{f′}
↓  ↓  ↓
map(BG, BC_{H}(ρ)_{p}^{∧})_{f} ⟶^{ad_{∗}} map(BG, (map(BT, BH)_{Bρ})_{p}^{∧})_{f′}≃ map(BG × BT, BH_{p}^{∧})_{f′}

the lower horizontal map is an equivalences. The second equivalence in the lower row also follows from [B–N]. The vertical arrows become equivalences after completion. This finishes the proof of proposition 1.4. □

3. λ–ring isomorphisms of K(BG).

Let G be a compact connected Lie group. According to [W 2], any λ-ring homomorhism

θ : K(BG) ⊗ ℤ_{p}^{∧} ⟶ K(BG) ⊗ ℤ_{p}^{∧}

induces a commutative diagram

(∗)  \begin{matrix}K(BT_{G}) ⊗ ℤ_{p}^{∧} & ⟶^{θ_{T}} & K(BT_{G}) ⊗ ℤ_{p}^{∧} \\ ↑ & & ↑ \\ K(BG) ⊗ ℤ_{p}^{∧} & ⟶^{θ} & K(BG) ⊗ ℤ_{p}^{∧} & , \end{matrix}

where T_{G} ↪ G is the maximal torus. In the case of product of a simply connected Lie group and a torus we describe all such automorphisms.

Let G̃ = G_{s} × T be a product of a simply connected Lie group and a torus, and let G_{s} = G_{1} × ... × G_{r} be th splitting of G_{s} into simple simply connected Lie groups. Passing to the rationalization of the classifying spaces we can rewrite this as BG̃_{ℚ} = (BG_{1}_{ℚ})^{n_{1}} × ... × (BG_{s}_{ℚ})^{n_{s}} × BT_{ℚ}, where we collect homotopy equivalent factors. Among the simple simply connected Lie groups, Spin(2n + 1) and Sp(n), n ≥ 3, are the only nonisomorphic Lie groups, which rationally have homotopy equivalent classifying spaces, and which have isomorphic complex K–theory [A–M: 2.7].

We call an automorphism of

K(BG̃) ⊗ ℚ ≅ (K(BG_{1}) ⊗_{ℤ} ℚ) ⊗_{ℚ} ... ⊗_{ℚ} (K(BG_{r}) ⊗_{ℤ} ℚ) ⊗_{ℚ} (K(BT) ⊗_{ℤ} ℚ)

simple if it is of the form

(σ ∘ (⨂_{i=1}^{r} ψ^{a_{i}} ∘ Bα_{i}^{∗} ∘ ε_{i}^{k_{i}})) ⊗ A_{T} .

Here, σ ∈ Σ_{n_{1}} × ... × Σ_{n_{s}} permutes isomorphic factors of K(BG_{1}) ⊗ ... ⊗ K(BG_{r}). ψ^{a_{i}} is an Adams operation of degree a_{i} ∈ ℚ. α_{i} is an outer automorphism of G_{i}. ε_{i} is one of the exceptional maps described in [A–M; 2.15], which can only occur for Sp(2), F_{4}, G_{2}. k_{i} is 1 or 0. A_{T} : K(BT) ⊗ ℚ)⟶K(BT) ⊗ ℚ is an automorphism of K(BT) ⊗ ℚ.

3.1 Proposition. If θ̃: K(BG̃) ⊗ ℚ⟶K(BG̃) ⊗ ℚ is an isomorphism, then θ̃ fits into a commutative diagram

K(BTG̃) ⊗ ℚ \underrightarrow{θ̃_{T}} K(BTG̃) ⊗ ℚ
↑  ↑
K(BG̃) ⊗ ℚ \underrightarrow{θ̃} K(BG̃) ⊗ ℚ ,

if and only if θ̃ is simple.

Proof. For a simple isomorphism θ̃ = σ∘(⨂_{i=1}^{r} ψ^{a_{i}}∘Bα_{i}^{∗}∘ε_{i}^{k_{i}})⊗A_{T}, every factor ψ^{a_{i}}∘Bα_{i}^{∗}∘ ε_{i}^{k_{i}} lifts to an isomorphism of K(BT_{G_{i}}) ⊗ ℚ, and σ to an isomorphism of K(BT_{G_{s}}) ⊗ ℚ. This proves one part of the statement.

Now let θ̃ be an isomorphism, which has a lift θ̃_{T} : K(BTG̃) ⊗ ℚ⟶K(BTG̃) ⊗ ℚ. For the classifying space of a compact connected Lie group, the Chern character induces an isomorphism K( ) ⊗ ℚ ≅ H^{∗}( ; ℚ). This allows to pass to rational cohomology, where we have analogous diagrams and maps. We use the same notation for the maps.

In 2–dimensional rational cohomology, θ̃_{T} can be described by a matrix

M = ( \begin{matrix}θ_{s,s} & θ_{T,s} \\ θ_{s,T} & θ_{T,T} \end{matrix} ) ,

where

θ_{s,T} : H^{2}(BT_{G_{s}}; ℚ) ⟶H^{2}(BT_{G_{s}}; ℚ) ⊕ H^{2}(BT; ℚ)
\underrightarrow{θ̃_{T}} H^{2}(BT_{G_{s}}; ℚ) ⊕ H^{2}(BT; ℚ) ⟶ H^{2}(BT; ℚ) .

All the other entries are given by analogous compositions, and all entries are equivariant with respect to a suitable isomorphism of W_{G}. θ̃_{s,T} is a lift of H^{∗}(BG_{s}; ℚ)⟶H^{∗}(BT; ℚ). All maps can be realized over the rationals. Therefore, θ̃_{s,T} is trivial.

It is well known that the isomorphism

H^{2}(BT_{G_{s}}; ℚ) ≅ ⨁_{i} H^{2}(BT_{G_{i}}; ℚ)

is a splitting into irreducible W_{G_{s}}–modules. Thus, H^{2}(BT_{G_{s}}; ℚ) contains no submodul with trivial W_{G_{s}} action. This implies that θ_{T,s}, as an W_{G_{s}} equivariant map, is also trivial, that θ̃_{T} = θ_{s,s} ⊕ θ_{T,T} on H^{2}(BTG̃; ℚ) and θ̃_{T} = θ_{s,s} ⊗ θ_{T,T} on H^{∗}(BTG̃; ℚ), and that, by passing to the Weyl group invariats, θ̃ = θ_{G_{s}} ⊗ θ_{T,T}, where θ_{G_{s}} is an automorphism of H^{∗}(BG_{s}; ℚ).

The elements of positive degree in H^{∗}(BG_{s}; ℚ) generate an ideal J ⊂ H^{∗}(BT_{G_{s}}; ℚ). Since H^{∗}(G_{s}/T_{G_{s}}; ℚ) ≅ H^{∗}(BT_{G_{s}}, ℚ)/J, the above considerations establish an automorphism \overline{θ} of H^{∗}(G_{s}/T_{G_{s}}; ℚ). In [P], these automorphisms are calculated. They look like being induced by simple automorphisms of H^{∗}(BG_{s}; ℚ). The map G_{s}/T_{G_{s}}⟶BT_{G_{s}} induces an isomorphism in two dimensional rational cohomology. Therefore \overline{θ} determines θ̃_{T_{s}} as well as θ_{s}. Hence, θ_{G_{s}} and θ̃ are simple isomorphisms, which finishes the proof. □

3.2 Remark. In the case of a ℚ_{p}^{∧}–isomorphism K(BG̃) ⊗ ℚ_{p}^{∧}⟶K(BG̃) ⊗ ℚ_{p}^{∧}, all the arguments also work. Hence, the same statement is true over ℚ_{p}^{∧}. In this case the degree of the Adams operations are given by p–adic rationals.

For any compact connected Lie group G, we choose a finite universal covering K⟶(G̃ = G_{s} ×T)⟶G of G. This induces rational equivalences TG̃↪T_{G} and G̃↪G. for any λ–ring homomorphism θ : K(BG) ⊗ ℤ_{p}^{∧}⟶K(BG) ⊗ ℤ_{p}^{∧}, we can tensor the diagram (∗) with ℚ_{p}^{∧} yielding

K(BTG̃) ⊗ ℚ_{p}^{∧} \underrightarrow{θ̃_{T}} K(BTG̃) ⊗ ℚ_{p}^{∧}
↑  ↑
K(BG̃) ⊗ ℚ_{p}^{∧} \underrightarrow{θ̃} K(BG̃) ⊗ ℚ_{p}^{∧} .

We assume that, after tensoring with ℚ_{p}^{∧}, θ and θ̃ are isomorphisms. Proposition 3.1 tells us that θ̃ is of the form

θ̃ = (σ ∘ (⨂_{i=1}^{r} ψ^{a_{i}} ∘ Bα_{i}^{∗} ∘ ε_{i}^{k_{i}})) ⊗ A_{T} .

We would like to deduce more information about the exponenents of the Adams operations .

3.3 Lemma. a_{i} ∈ ℤ_{p}^{∧} for all i.

Proof. Let us start with some remarks. The image of

G_{1} × ... × G_{i−1} × G_{i+1} × ... × G_{r} × T ⟶ G

is normal in G and the quotient is denoted by \overline{G}_{i}. We introduce the diagram

\xymatrix{K(B\overline{G}_{i}) ⊗ ℤ_{p}^{∧} \ar[rr] \ar[d] &  & K(B\overline{G}_{i}) ⊗ ℚ_{p}^{∧} \ar[r]^{ϕ_{i}} & K(BG_{i}) ⊗ ℚ_{p}^{∧} \ar[d]^{ψ^{a_{i}} ∘Bα_{i}^{∗}∘ε_{i}^{k_{i}}} \\ K(BG) ⊗ ℤ_{p}^{∧} \ar[r]^{θ} & K(BG) ⊗ ℤ_{p}^{∧} \ar[d] &  & K(BG_{i}) ⊗ ℚ_{p}^{∧} \ar[d]^{β_{σ}} \\  & K(BG̃) ⊗ ℤ_{p}^{∧} \ar[r] & K(BG_{σ(i)}) ⊗ ℤ_{p}^{∧} \ar[r] & K(BG_{σ(i)}) ⊗ ℚ_{p}^{∧} & ,}

where ϕ_{i} is the map induced by the homomorphism G_{i}⟶\overline{G}_{i}, and β_{σ} the map induced by the permutation σ. In the case G_{i} ≠ G_{σ(i)}, i.e. one of the groups is Sp(n) and the other Spin(2n + 1), the inclusions

Sp(n) ↩ U(n)↪SO(2n)↪SO(2n + 1)

correspond to the identity on the maximal tori, which is equivariant with respect to the Weyl groups of Sp(n) and SO(2n + 1). For G_{i} = Sp(n) and G_{σ(i)} = Spin(2n + 1), β_{σ} is given by K(BSp(n)) ≅ K(BSO(2n + 1))⟶K(BSpin(2n + 1)) and vice versa by the rational inverse. In the case of G_{i} = G_{σ(i)}, β_{σ} is the identity.

We have similiar diagrams in terms of the maximal tori and K-theory as well as in terms of the maximal tori and cohomology with ℤ_{p}^{∧} or ℚ_{p}^{∧} as coefficients. In particular, if we look at the 2-dimensional cohomology of the maximal tori, all maps are described by matrices.

The above diagram implies that β_{σ} ∘ ψ^{a_{i}} ∘ Bα_{i}^{∗} ∘ ε_{i}^{k_{i}} ∘ ϕ_{i} is a ℤ_{p}^{∧}-matrix. Because Bα_{i}^{∗} is invertible over ℤ, we can drop it and the rest is still a ℤ_{p}^{∧}-matrix.

We have to consider several different cases. Let us start with those where the exceptional

map ε can occur. For G_{2}, Sp(2) and F_{4}, ε looks like ( \begin{matrix}1 & 2 \\ −2 & 1 \end{matrix} ), M or ( \begin{matrix}M & 0 \\ 0 & M \end{matrix} ), where

M is given by ( \begin{matrix}1 & 1 \\ 1 & −1 \end{matrix} ) [A–M; 2.5, 2.6]. For G_{2} and F_{4} the center is trivial, i.e. ϕ_{i} = id.

We can assume that ψ^{a_{i}} ∘ ε_{i}^{k_{i}} is a ℤ_{p}^{∧}-matrix, which implies that a_{i} ∈ ℤ_{p}^{∧}.

The center of Sp(2) is ℤ/2. Thus, for G_{i} = Sp(2), ϕ_{i} is either the identity or is

represented by ( \begin{matrix}2 & 0 \\ −1 & 1 \end{matrix} ). In both cases the product matrix ε_{i}^{k_{i}} ∘ ϕ_{i} has a 1 as an entry,

and, hence, a_{i} ∈ ℤ_{p}^{∧}.

For G_{i} = G_{σ(i)} = Spin(2n + 1) or Sp(n), β_{σ} is the identity, and we can assume that ψ^{a_{i}} ∘ϕ_{i} is a ℤ_{p}^{∧}-matrix. If the denominator of a_{i} contains a power of p, then every entry in the matrix ϕ_{i} must be divisible by p and the group ℤ/p × ... × ℤ/p, with rank G factors, is in the kernel of the homomorphism G_{i}⟶\overline{G}_{i}. This is a contradiction in view of the center of G_{i} (check all the cases) apart from the case G_{i} = SU(2) and p = 2. In this situation a_{i} may be of the form b/2, b a unit in ℤ_{2}^{∧}. Bigger powers of 2 cannot occur. But this implies that K(BSO(3)) ⊗ ℤ_{2}^{∧} and K(BSU(2)) ⊗ ℤ_{2}^{∧} are isomorphic as λ-rings, which is also a contradiction.

There is only one case left, namely G_{i} = Spin(2n + 1) and G_{σ(i)} = Sp(n) or vice versa, with n ≥ 3. We get det(ϕ_{i}) = 1 or 2, det(β_{σ}) = 2 or 1/2 and det(ψ^{a_{i}}) = a_{i}^{n}; in particuliar 4a_{i}^{n} ∈ ℤ_{p}^{∧}. Since n ≥ 3, a_{i} ∈ ℤ_{p}^{∧}. □

3.4 Proposition. If θ : K(BG) ⊗ ℤ_{p}^{∧}⟶K(BG) ⊗ ℤ_{p}^{∧} is a real λ-ring isomorphism, θ is realizable as a map BG_{p}^{∧}⟶BG_{p}^{∧}.

Proof. First we choose a finite universal cover q_{2} : G̃ = G_{s} × T ⟶G of G. By the proof of lemma 3.3, θ induces a map θ̃_{T} : H^{2}(BTG̃; ℤ_{p}^{∧})⟶H^{2}(BTG̃; ℤ_{p}^{∧}), which has the form (σ ∘ (⨁ ψ^{a_{i}} ∘ Bα_{i} ∘ ε_{i}^{k_{i}})) ⊕ A_{T} (notation as above, but we can write the map as a direct sum because H^{2}(BTG̃; ℤ_{p}^{∧}) splits into a direct sum). We will show that this map can be realized as a topological map BG̃⟶BG̃.

σ splits in a product of cycles, which, after renumbering the factors of G_{s} = G_{1}×…×G_{r}, can be represented by the symbol (l_{0}, …, l_{1})(l_{1} + 1, …, l_{2}) … (l_{j} − 1, …, l_{j}) with l_{0} = 1 and l_{j} = r. We compute the characteristic polynomial χ(θ̃_{T}). To do this, we define d_{i} := rank G_{i} and e_{i} := det(ε_{i}). Then,

χ(θ̃_{T}) = ∏_{n=1}^{j}(t^{(l_{n}−l_{n−1})d_{ln}} ± ( ∏_{m=l_{n−1}+1}^{l_{n}} a_{n}^{d_{m}}e_{n}^{k_{m}})) · χ(A_{T}) .

det(β_{σ}) does not occur, because the number of maps Spin⟶Sp is equal the number of maps Sp⟶Spin. The determinant of Bα_{i} is ±1.

Since θ̃_{T} and θ_{T} are conjugate they must have the same characteristic polynomial. Thus χ(θ̃_{T}) has ℤ_{p}^{∧}-coefficients. ℤ_{p}^{∧} is integrally closed in ℚ_{p}^{∧}. All polynomials have leading coefficient 1. a_{i} and e_{i} are p-adic integers. All together, this implies that χ(A_{T}) is also a p-adic integral polynomial.

From the assumptions follows that

det(θ_{T}) = det(θ̃_{T}) = (∏_{i} a_{i}^{d_{i}}e_{i}^{k_{i}})det(A)

is a p-adic unit. So all the factors must be p-adic units; in particuliar a_{i} and e_{i}^{k_{i}}. This shows that all Adams operations can be realized on BG̃ [W 1] and that the exeptional maps ε_{i} cannot occur for Sp(2) and F_{4} for p = 2, as well as for G_{2} for p = 3. They can be realized in all other cases [F].

Can we find a realisation of σ? For p odd there is no problem, because BSpin(2n + 1)_{p}^{∧} and BSp(n)_{p}^{∧} are equivalent.

For p = 2, we have to consider permutation of Spin and Sp. If there are any permutations of this type, we map Spin to Sp at least once. Let G_{i} = Spin(n + 1) and G_{σ(i)} = Sp(n). Because the exponents of all the Adams operations which occur in the splitting of θ̃ are p-adic integers, the big diagram of the proof of lemma 3.2 leads to

K(B\overline{G}_{i}) ⊗ ℤ_{2}^{∧} ⟶^{ϕ_{i}} K(BSpin(2n + 1)) ⊗ ℤ_{2}^{∧} ⟶^{β_{σ}} K(Sp(n)) ⊗ ℤ_{2}^{∧}
↓  ↑
K(BG) ⊗ ℤ_{2}^{∧} ⟶^{θ}  K(BG) ⊗ ℤ_{2}^{∧}  ⟶ K(BG̃) ⊗ ℤ_{2}^{∧} .

Outer automorphism don’t occur. For \overline{G}_{i} = SO(2n + 1), β_{σ} ∘ ϕ_{i} is a λ-ring isomorphism but not a real λ-ring map. This is a contradiction, because θ is a real λ-ring map by assumption. For \overline{G}_{i} = Spin(2n + 1), ϕ_{i} = id and det(β_{σ}) = det(β_{σ} ∘ ϕ_{i}) = 1/2 ∉ ℤ_{2}^{∧}, which is also a contradiction. Thus σ permutes only factors of the same 2-adic type and is induced by a topological map.

Now we choose a finite cover ρ : T ⟶T of T such that A_{T} ∘Bρ^{∗} has only p-adic integers as matrix entries, and is realizable. Using the finite universal covering q_{1} := (id × ρ) ∘ q_{2}, we can construct a map f̃: BG̃_{p}^{∧}⟶BG̃_{p}^{∧} which fits into a commutative diagram

K(BG̃) ⊗ ℤ_{p}^{∧} \underrightarrow{f̃^{∗}} K(BG̃) ⊗ ℤ_{p}^{∧}
Bq_{1}^{∗}↑  ^{Bq_{2}^{∗}}↑
K(BG) ⊗ ℤ_{p}^{∧} ⟶^{θ} K(BG) ⊗ ℤ_{p}^{∧} .

The map Bq_{2} ∘ f̃|_{BK}, where K is the kernel of q_{1}, induces the trivial map in K-theory and therefore, is homotopically trivial [Z]. By proposition 1.1, Bq_{1} ∘ f̃ factors over Bq_{2} and induces a map f : BG_{p}^{∧}⟶BG_{p}^{∧}, such that f^{∗} = θ. □

Theorem 1.5 is included in the following statement:

3.5 Corollary. Let R be ℤ, ℤ_{(p)} or ℤ_{p}^{∧}. Every real λ-ring isomorphism

θ : K(BG) ⊗ R⟶K(BG) ⊗ R

is induced from a topological map f : BG_{R}⟶BG_{R}.

BG_{R} means BG localized or completed, and BG_{ℤ} = BG.

Proof. We desribe the proof only for R = ℤ. Tensoring with ℤ_{p}^{∧} or ℚ yields λ-ring isomorphisms

θ_{p}^{∧} : K(BG) ⊗ ℤ_{p}^{∧}⟶K(BG) ⊗ ℤ_{p}^{∧}

for all primes p and

θ_{ℚ} : K(BG) ⊗ ℚ⟶K(BG) ⊗ ℚ .

θ_{p}^{∧} can be realized by propositiobn 5.2 and θ_{ℚ} can be realized because BG_{ℚ} is a product of Eilenberg-MacLane spaces. Glueing them together with the arithmetic square, we get a map f : BG⟶BG as desired. □

3.6 Remark. If the finite universal cover G̃⟶G of G does not contain both, Sp(n) and Spin(2n + 1), for any n ≥ 3, then there are no problems with the permutation in 3.2 and 3.3. In this case or if we know that Sp and Spin are not permuted, we do not have to assume anything about KO-theory.

4. Homotopy classification of rational self equivalences of BG.

Let f, g : BG⟶BG be two rational self equivalences of BG, where G is a compact connected Lie group, i.e. f and g induce isomorphisms in rational cohomology. In view of corollary 2.3, we can choose two finite universal covers q_{1}, q_{2} : G̃⟶G independently of f and g, such that f and g lift to rational equivalences f̃, g̃ : BG̃⟶BG̃. Moreover, f and g are homotopic if and only if f̃ and g̃ are homotopic.

G̃ = G_{1} × … × G_{r} × T is a product of simple simply connected Lie groups G_{i} and a torus T. In section 3, we described how f̃ and g̃ look like in terms of rational cohomology. If f and g induce the same map in K-theory, then they also do it in rational cohomology, as do f̃ and g̃. This means that in rational cohomology or K–theory both, f̃ and g̃, look like

(σ ∘ ⨂_{i}(ψ^{a_{i}} ∘ Bα_{i} ∘ ε^{k_{i}})) ⊗ A_{T} .

The notation is as in section 3.

Because f̃ is a map over the integers, σ can only permute factors of the same integral type. In particuliar σ is realizable by a map, which we also call σ. To show that f̃ and g̃ are homotopic it is sufficient to consider (σ^{−1} × id) ∘ f̃ and (σ^{−1} × id) ∘ g̃. Both of these maps look in rational cohomology like a product of maps. The main step in the proof of theorem 1.7 is to show that both these maps split topologically into a product of maps.

4.1 Lemma. If G_{1}, G_{2} and H are compact connected Lie groups and c : BG_{2}⟶BH is the constant map, the map

ad_{∗} : map(BG_{1}, BH)⟶map(BG_{1}, map(BG_{2}, BH)_{c})

is a mod-p equivalence for every prime p.

Proof. We introduce the following fiber sequences

[figure: map_{∗}(BG_{2}, H); H; ad; map(BG_{2}, H); X; BH; ad; map(BG_{2}, BH)_{c}; H]

where X is the homotopy fiber of ad, map_{∗}( , ) denotes the mapping space of pointed maps and e is the evaluation map. In this situation ad is nothing but the right inverse of the evaluation map. Therefore map_{∗}(BG_{2}, H)⟶X is an equivalence.

The homotopy groups of map_{∗}(BG_{2}, H) are computed in [F–M]. The result is that X is a product of Eilenberg-McLane spaces and that all homotopy groups are rational vector spaces, concentrated only in even dimensions. Hence BH⟶map(BG_{2}, BH)_{c} is a mod-p equivalence for all primes p.

Let us look at the lifting problem

\xymatrix{ & BH \ar[d] \\ BG_{1} \ar[r] \ar@{=>}[ur] & map(BG_{2}, BH) & .}

Since the obstruction groups H^{∗+1}(BG_{1}; π_{∗}(X)) vanish, ad_{∗} is a π_{0}-surjection. To prove that ad_{∗} is a π_{0}-injection, we use the diagram

[BG_{1}, BH]  ⟶  [BG_{1}, BH^{∧}]
↓  ↓
[BG_{1}, map(BG_{2}, BH)_{c}] ⟶ [BG_{1}, map(BG_{2}, BH)_{c}^{∧}] .

By obstruction theory the upper horizontal arrow is injective, and the right column is a bijection.

Now, we can argue anologously as in the proof of proposition 1.4 and get

map(BG_{1}, BH)_{f} _{p}^{∧} ≃ map(BG_{1}, map(BG_{2}, BH)_{c})_{ad^{∗}(f)}_{p}^{∧} .

This completes the proof. □

For a map f : BG_{1} × BG_{2}⟶BH_{1} × BH_{2} we define the maps f_{kl} to be the canonical compositions

f_{kl} : BG_{k} ⟶^{i_{k}} BG_{1} × BG_{2}⟶BH_{1} × BH_{2} ⟶^{pr_{l}} BH_{l} .

4.2 Lemma. If for f : BG_{1}×BG_{2}⟶BH_{1}×BH_{2} the maps f_{12} and f_{21} are homotopically trivial, then f ≃ f_{11} × f_{22}, and

map(BG_{1} × BG_{2}, BH_{1} × BH_{2})_{f}⟶map(BG_{1}, BH_{1})_{f_{11}} × map(BG_{2}, BH_{2})_{f_{22}}

is a mod-p equivalence for every prime p.

Proof. For the first statement we have to check the equivalence of pr_{k} ∘ f and pr_{k} ∘ (f_{11} × f_{22}), k = 1, 2. pr_{1} ∘ f ∘ i_{2} is homotopically trivial. Thus, by lemma 4.1, we get pr_{1} ∘ f ≃ ad_{∗}(f_{11}) ≃ pr_{1} ∘ (f_{11} × f_{22}), and the analogous for pr_{2} ∘ f.

Also, by lemma 4.1, map(BG_{1}, BH_{1})_{f_{11}}⟶map(BG_{1}, map(BG_{2}, BH_{1})_{pr_{1}∘f} is a mod-p equivalence. This establishes the second statement. □

We can now complete the proof of theorem 1.7.

Proof of theorem 1.7. Let f, g ∈ HE_{ℚ}(BG) be two rational self equivalences with K(f) = K(g). At the beginning of this section we chose finite universal coverings q_{1}, q_{2} : G̃ = ∏ G_{i} × T ⟶G of G and found maps f̃, g̃ ∈ HE_{ℚ}(BG̃) which are lifts of f and g. f̃ and g̃ contain the same permutation σ. This implies that f ≃ g iff f̃_{σ} := (σ^{−1} × id) ∘ f̃ ≃ (σ^{−1} × id) ∘ g̃ =: g̃_{σ}. In K-theory both maps look like a tensor product of maps. A map between classifying spaces is homotopically trivial if and only if the induced map in Ktheory is trivial [Z]. Hence, by corollary 4.2, f̃_{σ} ≃ ∏_{i} f × f_{T} and g̃_{σ} ≃ ∏ g_{i} × g_{T} split into a product of maps f_{i}, g_{i} : BG_{i}⟶BG_{i} and f_{T} , g_{T} : BT ⟶BT. All factors look equally in K-theory. Because G_{i} is simple, the maps must be equivalent, by theorem 1.6. This shows that f ≃ g and finishes the proof of the first part.

Using theorem 1.6 and corollary 4.2 again we get a sequence of equivalences or mod-p equivalences

BC(G̃) ⟶ ∏ BC(G_{i}) × BT ⟶ ∏ map(BG_{i}, BG_{i})_{id} × map(BT, BT)_{id} ⟶

∏ map(BG_{i}, BG_{i})_{f_{i}} × map(BT, BT)_{f_{T}} ⟶ map(BG̃, BG̃)f̃_{σ} ⟶ map(BG̃, BG̃)f̃ ,

which fit into a commutative diagram

\xymatrix{BC(G̃) \ar[r] \ar[dd]_{Bq_{2}|_{C(_{G})}} & map(BG̃, BG̃)_{id} \ar[r] & map(BG̃, BG̃)f̃ \ar[d]^{Bq_{2∗}} \\  &  & map(BG̃, BG)_{Bq_{2}∘f̃} \\ BC(G) \ar[r] & map(BG, BG)_{id} \ar[r] & map(BG, BG)_{f} \ar[u]_{Bq_{1}^{∗}} & .}

Let K be the kernel of q_{2}. By proposition 1.1 Bq_{1}^{∗} is an equivalence. BK is the fiber of Bq_{2} and, by proposition 1.2, the fiber of Bq_{2∗}. This shows that the bottom arrow is a mod-p equivalence for all primes p. □

5. Fake Lie Groups and the Genus of BG.

First we proof theorem 1.8.

Proof of theorem 1.8. Let θ : K(Y_{2})⟶K(Y_{1}) be a real λ–ring isomorphism, where Y_{1} and Y_{2} are in the adic genus of BG. Because Y_{1} and Y_{2} are Eilenberg-McLane spaces over the rationals, θ ⊗ ℚ can be realized by a map f_{ℚ} : Y_{1ℚ}⟶Y_{2ℚ}. For every p we choose homotopy equivalences g_{1}, g_{2} : BG_{p}^{∧}⟶Y_{1,2}. Then

K(g_{1}) ∘ θ ⊗ ℤ_{p}^{∧} ∘ K(g_{2}^{−1}) : K(BG) ⊗ ℤ_{p}^{∧} ⟶ K(BG) ⊗ ℤ_{p}^{∧}

is a real λ-ring isomorphism and can be realized by a topological map. This gives a realization f_{p}^{∧} : Y_{1}_{p}^{∧}⟶Y_{2}_{p}^{∧} of θ ⊗ ℤ_{p}^{∧}. The rational and the p–adic realization, f_{ℚ} and f_{p}^{∧}, must agree over ℚ_{p}^{∧}. So the arithmetic square produces a map f : Y_{1}⟶Y_{2} with K(f) = θ.

The next lemma will show that the maps f_{p}^{∧} are homotopy equivalences, and hence that f is a homotopy equivalence, which completes the proof. □

5.1 Lemma. A map g : BG_{p}^{∧}⟶BG_{p}^{∧} is a homotopy equivalence if and only if

K(g) : K(BG) ⊗ ℤ_{p}^{∧}⟶K(BG) ⊗ ℤ_{p}^{∧}

is an isomorphism.

Proof. Let g : BG_{p}^{∧}⟶BG_{p}^{∧} be a map which induces an isomorphism in K-theory. Analogously to the second proof of [N-S; 2.1], there exists a self homotopy equivalence g_{T} : BT_{G}_{p}^{∧}⟶BT_{G}_{p}^{∧} of the maximal torus of G and a commutative diagram

G/T_{G}_{p}^{∧} ⟶ BT_{G}_{p}^{∧} ⟶ BG_{p}^{∧}
g_{G/T} ↓  ^{g_{T}} ↓  ^{g}↓
G/T_{G}_{p}^{∧} ⟶ BT_{G}_{p}^{∧} ⟶ BG_{p}^{∧} .

In [ N-S 1; 2.1] it is assumed that g is a homotopy equivalence, but the proof works also in the present case; it is based on depends only on K-theory information.

H^{∗}(g_{G/T}; ℚ_{p}^{∧}) is an isomorhism, because g is an equivalence over ℚ_{p}^{∧}. H^{∗}(G/T_{G}; ℤ_{p}^{∧}) is torsion free. Therefore the kernel of H^{∗}(g_{G/T}; ℤ_{p}^{∧}) is trivial. Now let us look at the diagram

H^{∗}(BT_{G}_{p}^{∧}; ℤ_{p}^{∧}) ⟶ H^{∗}(G/T_{G}; ℤ_{p}^{∧}) ⟶ coker ⟶ 1
g_{T}^{∗} ↓  ^{g_{G/T}^{∗}} ↓  \overline{g}^{∗}↓
H^{∗}(BT_{G}; ℤ_{p}^{∧}) ⟶ H^{∗}(G/T_{G}; ℤ_{p}^{∧}) ⟶ coker ⟶ 1 .

coker is finite, because G/T_{G}_{p}^{∧}⟶BT_{G}_{p}^{∧} induces a surjection in rational cohomology. Since g_{G/T}^{∗} is one to one, \overline{g}^{∗} an injection . This implies that \overline{g}^{∗} is an isomorphism as well as g_{G/T}^{∗}. Thus, g_{G/T} and g are homotopy equivalences. □

To study fake Lie groups and to distinguish them from the genuine article, the concept of a maximal torus is used in [N-S 1,2,3 ]. We will recall the definitions and some of the results. For R = ℤ_{(p)}, (ℤ_{p}^{∧}) we denote by X_{R} the p-localisation or p-completion of X, for R = ℤ, X_{R} := X.

Definition. Let X be a fake Lie group of type G. A maximal torus over R is a map

f : (BT_{X})_{R}⟶BX_{R} ,

where T_{X} is a torus, such that (1) rank(T_{X}) = rank(G)

(2) the homology H^{∗}(hofib(f); R) is a finitely generated R-module.

The set of homotopy classes of maps W_{X} := {[w : BT_{R}→BT_{R}] | f ∘ w ≃ f} is called the Weyl group of X over R.

Remark. This definition is slightly different from the one in [N–S 1]. Instead of (2), a sharper condition is used. For R = ℤ it is supposed that hofib(f) has the homotopy type of a finite CW -complex. Also it is assumed that BX is in the genus of BG, not in the adic genus. The genus is defined using localisations instead of completions. But it turns out that even under these weaker conditions, all the statements of [N-S 1] remain true, as the proofs show. Using [N–S; 2.1,3.9], one can show that hofib(f) is simply connected. Hence hofib(f) has the homotopy type of a finite CW -complex for R = ℤ.

Now we will compare the adic genus and the normal genus of BG. Examples, due to A.K.Bousfield and G.Mislin (see [N–S 2,appendix]), show that these two genera are different. But we can offer 5.2 Proposition. Let BX be in the adic genus of BG. The following statements are equivalent:

(1) For every prime p, BX has p–locally a maximal torus (BT_{X})_{(p)}⟶BX_{(p)}.

(2) BX is in the genus of BG.

Proof. Let BX be in the adic genus of BG, and let (BT_{X})_{(p)}⟶BX_{(p)} be a maximal torus of X over ℤ_{(p)} with Weyl group W_{X}. By [N–S; 2.1] these maps fit into a commutative diagram

BT_{X} _{p}^{∧} ⟶ BT_{G}_{p}^{∧}
↓  ↓
BX_{p}^{∧} ⟶^{\overline{g}} BG_{p}^{∧} ,

where the right column is given by a classical maximal torus T_{G}⟶G. The rows are equivalences. The upper row induces a conjugation of the two representations

W_{X}, W_{G}⟶Gl(n, ℤ_{p}^{∧}) ,

n = rank(G), given by the action of W_{X} and W_{G} on the 2–dimesional (p)-local cohomology of the maximal tori. This conjugation establishes a homomorphism α : W_{X}⟶W_{G}. By [C–R; 30.17] α can be represented by a conjugation in GL(n, ℤ_{(p)}). That is to say that there exists a p-local equivalence g′ : BT_{(p)}⟶BT_{(p)} which is equivariant with respect to α. Because all the spaces are rational Eilenberg-MacLane spaces, this produces an diagram

BT_{ℚ} ⟶^{g′} BT_{ℚ}
↓  ↓
BX_{ℚ} ⟶^{\overline{g}′} BG_{ℚ} ,

where the rows are equivalences. The left column is a maximal torus of X over ℚ, whose Weyl group is isomorphic to W X [N–S; 3.6].

Of course, we will use an arithmetic square argument to construct a p-local equivalence. But the maps \overline{g} and \overline{g}′ may not be homotopic over ℚ_{p}^{∧}. We have to change the map \overline{g}. To do this we take the map h := g′ ∘ g^{−1} : BT_{G}_{p}^{∧}⟶BT_{G}_{p}^{∧}, which is equivariant with respect to the identity on W G. Restricting this map to the invariants of the W G-action on K(BT_{G}) ⊗ ℤ_{p}^{∧}, we get a λ-ring isomorphism θ : K(BG) ⊗ ℤ_{p}^{∧}⟶K(BG) ⊗ ℤ_{p}^{∧}.

If we pass to to a finite universal cover of G, θ̃ still commutes with the Weyl group (notation as in section 5) and hence cannot contain any permutation. By remark 3.6 and theorem 5.3, θ is realizable by a map \overline{h} : BG_{p}^{∧}⟶BG_{p}^{∧}. The composition \overline{h} ∘ \overline{g} agrees with \overline{g}′ over ℚ_{p}^{∧}. The arithmetic square argument is now applicable and establishes an equivalence BX_{(p)} ≃ BG_{(p)}.

If BX is in the genus of BG, then there is nothing to prove. □

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Inv. Math. 35 (1976), 1-41.

[B-K]  A.K. Bousfield and D.M. Kan. Homotopy limits,completions and

localizations, LNMS 304, Springer 1972.

[B-tD]  T. Bröcker and T. tom Dieck. Representations of compact Lie

groups, Springer 1985.

[C-R]  C.W. Curtis and I. Reiner. Methods of representation theory I, II,

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