A COMPARISON CRITERION FOR CERTAIN LOOP SPACES

RAN LEVI

Abstract. We study a comparison criterion for loop spaces on p-localized classifying spaces of certain finite p-perfect groups G. In particular we show that, under certain hypotheses, the homotopy type of those spaces is determined by the mod-p cohomology of G together with a finite Postnikov system.

1. Introduction and Statement of Results

Let p be a fixed prime. Recall that a group G is said to be p-perfect if H_{1}(BG; 𝔽_{p}) = 0. Let (−)_{(p)} denote the ℤ_{(p)}-completion functor of Bousfield and Kan, where ℤ_{(p)} is the ring of p-local integers. In this paper we study a comparison criterion for the homotopy type of spaces of the form Ω(BG)_{(p)}, where G belongs to a certain family of finite p-perfect groups. Since G is assumed finite, the completion functors with respect to ℤ_{(p)} and the prime field 𝔽_{p} coincide [3]. Furthermore, classifying spaces of p-perfect groups are “p-good” in the sense described in [3] and thus the ℤ_{(p)}-completion of BG can be thought of as a functorial mod-p analogue of the Quillen “plus” construction for perfect groups. In [5] F. Cohen conjectured that spaces of the form ΩBG^{+}, where G is a finite perfect group, have the property that their 1-connected cover is resolvable by finitely many fibrations over spheres and loop spaces on spheres. The conjecture takes a more general form in [9], where the term “perfect” is replaced by p-perfect and the “plus” construction is replaced by the 𝔽_{p}-completion functor. Cohen’s conjecture suggests a partial motivation for our study. We refer the reader to [5, 9] for details and examples. The method we employ here is a generalization of the techniques of [7].

Our main result is a theorem, which claims that, under certain hypotheses on the group G, the homotopy type of ΩBG_{(p)} is determined by H^{∗}(BG; 𝔽_{p}) together with a finite stage of the Postnikov tower for BG_{(p)}.

In special cases this result can be applied to show that the homotopy type of ΩBG_{(p)} depends only on the mod-p cohomology algebra of BG together with the action of the higher Bockstein operators.

We apply our methods to study certain examples of ΩBG_{(p)}, where G is a finite p-perfect group. In particular we produce explicit examples where the homotopy type of ΩBG_{(p)} is completely determined by the cohomology of BG.

To state our results, we need to set some conventions and terminology. Unless otherwise specified, all spaces are assumed to have the homotopy type of a CW-complex and to be 1connected, p-local and of finite type. Let H^{∗}(−) denote H^{∗}(−; 𝔽_{p}). Let A_{p} denote the mod-p Steenrod algebra. A p-local space X is said to be finite dimensional if H^{∗}(X) vanishes above a certain dimension. The top dimension in which the mod-p cohomology of X is non-trivial is said to be the dimension of X.

Date: January 1, 1994.

1991 Mathematics Subject Classification. Primary 55R35, Secondary 55R40, 55Q52.

The author is supported by a DFG grant.

Recall that a ring R is said to be Cohen-Macaulay if there exists a polynomial subring P ⊂ R such that R is a finitely generated free P-module [11]. Let CMG_{p} denote the family of all groups G such that H^{∗}(BG) is Cohen-Macaulay.

We say that a space K is mod-p polynomial if H^{∗}(K) is a finitely generated polynomial algebra. Let G ∈ CMG_{p} and let K be a mod-p polynomial space. A map f : BG_{(p)} ⟶ K_{(p)} is said to be admissible if the image of f^{∗} in H^{∗}(BG) is a polynomial subring, over which H^{∗}(BG) is a finitely generated free module.

Let X and Y be simply-connected, p-local spaces. We say that H^{∗}(X) ≅ H^{∗}(Y) as A_{p}- algebras together with the operation of the higher Bocksteins, if there exists an isomorphism of A_{p}-algebras

ϕ : H^{∗}(X) ⟶ H^{∗}(Y)

which induces an isomorphism of differential graded algebras on each stage of the corresponding Bockstein spectral sequences.

For any space X and a positive integer d, let X[d] denote the d-th stage of the Postnikov tower for X. We say that X and Y have the same d-th Postnikov stage if there is a map l_{Y} : Y ⟶ X[d], inducing an isomorphism on π_{i}(−) for 0 ≤ i ≤ d.

Definition 1.1. Two spaces X and Y are said to be comparable of degree d if 1. X and Y have the same d-th Postnikov stage X[d].

2. H^{∗}(X) ≅ H^{∗}(Y) as A_{p}-algebras together with the operation of the higher Bocksteins.

3. There is an isomorphism ϕ : H^{∗}(X) ⟶ H^{∗}(Y) realizing (ii), such that ϕ ∘ l_{X}^{∗} = l_{Y}^{∗} in mod-p cohomology, up to dimension d.

Let G ∈ CMG_{p} be a finite p-perfect group. A comparison system for G is a triple (C, β, K), where K is a mod-p polynomial space, β : BG_{(p)} ⟶ K_{(p)} is an admissible map and C is the homotopy theoretic fibre of β. By [12] both ΩK_{(p)} and C are finite dimensional. Furthermore, since the space BG_{(p)} is rationally contractible, it follows that the map ΩK_{(p)} ⟶ C is a rational equivalence. As we will observe, H^{∗}(C) satisfies Poincaré duality with respect to any ring of coefficients. Thus the top dimensional class in H^{∗}(C) is rationally non-trivial and so dimC = dimΩK_{(p)}. Let d = dimC. The number d is called the dimension of the system (C, β, K).

Our main result claims that if G ∈ CMG_{p} is a finite p-perfect group, which admits a d-dimensional comparison system, then the homotopy type of ΩBG_{(p)} is determined by H^{∗}(BG) together with d-th stage of the Postnikov tower for BG_{(p)}.

Theorem 1.1. Let G be a finite p-perfect group in CMG_{p}, which admits a d-dimensional comparison system. Suppose that X and BG_{(p)} are comparable of degree d. Then ΩBG_{(p)} ≃ ΩX.

A 2-stage Postnikov system is a principal fibration

ΩK_{2} ⟶ E ⟶ K_{1}

where K_{1} and K_{2} are generalized Eilenberg-MacLane spaces. Suppose that G is a finite p-perfect group in CMG_{p}, which admits a d-dimensional comparison system. If the d-th stage of the Postnikov tower for BG_{(p)} can be inscribed in a 2-stage Postnikov system then theorem 1.1 implies a stronger result.

Theorem 1.2. Let G be a finite p-perfect group in CMG_{p}, which admits a d-dimensional comparison system. Let X denote BG_{(p)} and suppose that there is a 2-stage postnikov system

ΩK_{2} ⟶^{j_{1}} X[d] ⟶^{p_{1}} K_{1}

such that p_{1}^{∗} is an epimorphism in mod-p cohomology up to dimension d + 1. Let Y be a space such that H^{∗}(Y) ≅ H^{∗}(X) as an A_{p}-algebra together with the operation of the higher Bocksteins. Then ΩY ≃ ΩX.

There are many examples of finite p-perfect groups, which admit comparison systems. In particular a theorem of E. Friedlander [8] implies that finite groups of Lie type do so at most primes. Let 𝔽_{q} be the finite field of q elements. Recall that if G is a finite group then BG_{p}^{∧} ≃ BG_{(p)}, where (−)_{p}^{∧} denotes the 𝔽_{p}-completion functor [9].

Theorem 1.3. Let G(ℂ) be a complex connected reductive Lie group, whose maximal compact subgroup is d-dimensional. Let G(q) be an associated finite Chevalley group over 𝔽_{q}. Let p be a prime such that H^{∗}(G(ℂ); ℤ) has no p-torsion and p ≠ char(𝔽_{q}). Then there is a map

ψ_{q} : BG(q)_{p}^{∧} ⟶ BG(ℂ)_{p}^{∧}

such that the triple (G(ℂ)_{p}^{∧}, ψ_{q}, BG(ℂ)_{p}^{∧}) is a d-dimensional comparison system for G(q).

If G(q) is a group satisfying the conditions of theorem 1.3, with respect to some prime p then it follows that the homotopy type of ΩBG(q)_{(p)} is determined by the mod-p cohomology of BG(q) together with the d-th stage of the Postnikov tower for BG(q)_{(p)}. We next apply our theory to study ΩBG_{(p)}, where G = SL_{3}(q) at odd primes p ≠ char(𝔽_{q}).

Theorem 1.4. Let p be an odd prime and let l be any prime different from p. Let X be a space such that H^{∗}(X) ≅ H^{∗}(BSL_{3}(l^{k})_{(p)}) as algebras over the Steenrod algebra A_{p} with the same Bockstein operators. Then ΩX_{(p)} ≃ ΩBSL_{3}(l^{k})_{(p)}.

This example is in some sense generic, as one can obtain similar results for various finite groups of Lie type, if one is willing to restrict attention to sufficiently large primes. An account of the group SL_{3}(q), q ≡ 3 mod 4 at the prime 2 is given in [7]. The case q ≡ 1 mod 4 will be considered elsewhere.

Many examples of finite p-perfect groups G, studied in [9, II.5-7], turn out to admit comparison systems. There however we study the existence of finite resolutions over spheres and loop spaces on spheres for ΩBG_{(p)}. In view of [9], theorem 1.4 yields

Corollary 1.1. Let p be an odd prime and let l be any prime different from p. Let X be a space such that H^{∗}(X) ≅ H^{∗}(BSL_{3}(l^{k})_{(p)}) as algebras over the Steenrod algebra A_{p} with the same Bockstein operators. Then ΩX is spherically resolvable of weight 4.

The paper is organized as follows. In section 2 we state some facts preliminary to our study. In section 3 we prove theorem 1.1. Section 4 contains the proof of theorem 1.2. In section 5 we recall a theorem of E. Friedlander and prove theorem 1.3. In sections 6 we apply our techniques to prove theorem 1.4.

The author takes the pleasure to thank Fred Cohen for many invaluable conversations. We are also grateful to the University of Heidelberg for its kind hospitality and support.

2. Preliminaries

Let X be a space. The n-th stage of the Postnikov tower for X is a space X[n] together with a map l_{X} : X ⟶ X[n] such that

1. π_{i}X[n] = 0 for i > n and

2. l_{X} induces an isomorphism on π_{k}(−) for 0 ≤ k ≤ n. The following lemma is elementary.

Lemma 2.1. Let X[n] be the n-th stage of the Postnikov tower for a space X. Then for any space such that dimC ≤ n + 1, the canonical map l : X ⟶ X[n] induces an epimorphism of sets

l_{∗} : [C; X] ⟶ [C; X[n]].

Furthermore, if dimC ≤ n, then l_{∗} is an isomorphism.

Next we state a fact about comparison systems. The following lemma is an immediate consequence of the big collapse theorem of L. Smith [12].

Lemma 2.2. Let G be a finite p-perfect group in CMG_{p}, which admits a comparison system (C, β, K). Let P_{G} = Im(β^{∗}) ⊂ H^{∗}(BG). Then

H^{∗}(C) ≅ H^{∗}(BG)/(P_{G}) ⊗ E

as an H^{∗}(BG)-module, where E is a trivial H^{∗}(BG)-module and is isomorphic to an exterior algebra on generators corresponding to a regular sequence generating Ker(β^{∗}). Moreover, if p ≠ 2 then this isomorphism is an isomorphism of algebras.

We remark that by [1], the algebra H^{∗}(BG)/(P_{G}) satisfies Poincaré duality. Thus, under the hypotheses of lemma 2.2, so does H^{∗}(C).

Let G be a fixed finite p-perfect group in CMG_{p} and let (C, β, K) be a d-dimensional comparison system for G. Let B[d] denote the d-th stage of the Postnikov tower for BG_{(p)}. Let X be a space comparable to BG_{(p)} of degree d.

The map β gives rise to a principal fibration

ΩK_{(p)} ⟶^{γ} C ⟶^{g} BG_{(p)}.

By lemma 2.1 there are isomorphisms of sets

[C; BG_{(p)}] ≅ [C; B[d]] ≅ [C; X].

Thus there exists a map f : C ⟶ X, unique up to homotopy, such that the following square homotopy commutes.

[figure: C; f; BG_{(p)}; l_{G}; g; X; B[d]; l_{X}]  (2.1)

By [6] diagram 2.1 can be extended to give a commutative diagram of fibrations

[figure: ΩF_{X}; ΩX; ΩB[d]; ΩF_{G}; L; S; ψ; F_{G}; ΩBG_{(p)}; ΩK_{(p)}; C; f; BG_{(p)}; l_{G}; γ; g; ΩB[d]; F_{X}; X; B[d]; l_{X}]  (2.2)

The map l_{X} induces an isomorphism on homotopy groups up to dimension d and π_{i}B[d] = 0 for i > d. Thus the fibre F_{X} is d-connected. Since dimΩK_{(p)} = d, it follows that the composite f ∘ γ is null homotopic. Hence there is a lifting h : ΩK_{(p)} ⟶ S such that ψ ∘ h = γ and we get a commutative diagram of fibrations

[figure: ΩBG_{(p)}; ϵ; ΩK_{(p)}; h; C; =; γ; ΩX; S; C; ψ]  (2.3)

In particular, the map ϵ : ΩBG_{(p)} ⟶ ΩX is an equivalence if and only if h is.

3. Proof of Theorem 1.1

Theorem 1.1 follows at once by showing that any map

h : ΩK_{(p)} ⟶ S

such that ψ ∘ h ≃ γ is a homotopy equivalence. We prove this in two steps. First we show that any such map induces a ℤ_{(p)}-homology isomorphism up to dimension d − 1 and a split monomorphism in dimension d. Then we show that H_{i}(S) = 0, for i > d and argue that h must therefore be a homotopy equivalence.

Consider the homotopy commutative diagram

[figure: F; j; S; ψ; δ; ΩK_{(p)}; α; C; f; π; BG_{(p)}; l_{G}; γ; g; X; B[d]; l_{X}]  (3.1)

where the map π is given by the composite l_{G} ∘ g and F is the homotopy fibre of π. There are three fibrations in the diagram, each of which having a simply-connected rationally contractible base space. Thus one immediately observes that all maps in the top square are rational equivalences.

Lemma 3.1. Let h : ΩK_{(p)} ⟶ S be any map such that ψ ∘ h = γ. Then

h_{∗} : H_{i}(ΩK_{(p)}; ℤ_{(p)}) ⟶ H_{i}(S; ℤ_{(p)})

is an isomorphism for 0 ≤ i ≤ d − 1 and a split monomorphism for i = d.

Proof. The maps l_{G} and l_{X} induce an isomorphism on homotopy groups up to dimension d and an epimorphism in dimension d + 1. Thus, by the 5-lemma both α and δ induce an isomorphism on homotopy groups up to dimension d − 1 and an epimorphism in dimension d. Consequently both α and δ induce an isomorphism on homology with ℤ_{(p)} coefficients, up to dimension d − 1 and an epimorphism in dimension d. Furthermore, since α is a rational equivalence and H_{d}(ΩK_{(p)}; ℤ_{(p)}) is torsion free, it follows that α_{∗} is an isomorphism through dimension d. In particular H_{i}(F; ℤ_{(p)}) is torsion free for 0 ≤ i ≤ d. A similar argument applied to δ yields that δ_{∗} maps the torsion-free part of H_{d}(S; ℤ_{(p)}) isomorphically to H_{d}(F; ℤ_{(p)}).

Next let h : ΩK_{(p)} ⟶ S be any map such that ψ ∘ h = γ. Then

j ∘ δ ∘ h = ψ ∘ h = γ = j ∘ α.

Since the map j : F ⟶ C is a principal fibration, up to homotopy, with fibre ΩB[d], it follows that there exists a map θ : ΩK_{(p)} ⟶ ΩB[d] such that the following diagram homotopy commutes.

[figure: ΩB[d] × F; F; μ; ΩK_{(p)}; θ⊤α; δ∘h]

Here μ denotes the principal action map. Since ΩB[d] is a rationally contractible space and H_{∗}(F; ℤ_{(p)}) is torsion free, up to dimension d, it follows that

(μ ∘ (θ⊤α))_{∗} = α_{∗}

up to dimension d. Thus α_{∗} = (δ ∘ h)_{∗} up to dimension d, whence the result.

We next argue that dimS ≤ d. Indeed, let P_{G} = Im(β^{∗}) ⊂ H^{∗}(BG), as before. By lemma 2.2, H^{∗}(C) ≅ H^{∗}(BG)/(P_{G})⊗E as an H^{∗}(BG)-module, where E is isomorphic to an exterior algebra on generators x_{1}, ⋯, x_{r} corresponding to a regular sequence generating Ker(β^{∗}). Let m denote the rank of H^{∗}(K_{(p)}) and let k denote the rank of P_{G}, then r = m − k. Notice that dim_{𝔽_{p}}(E) = 2^{r}.

Consider the Eilenberg-Moore spectral sequence for the fibration

ΩK_{(p)} ⟶ C ⟶ BG_{(p)}.

The E_{2} term is given by

E_{2} = Tor_{H^{∗}(BG)}(H^{∗}(C); 𝔽_{p}).

We compute E_{2}, using the Cartan-Eilenberg change of rings spectral sequence [4], with respect to the algebra extension

P_{G} ⟶ H^{∗}(BG) ⟶ H^{∗}(BG)/(P_{G}).

Here the E_{2} term is given by

E_{2} = Tor_{H^{∗}(BG)/(P_{G})}(H^{∗}(C); Tor_{P_{G}}(𝔽_{p}; 𝔽_{p})) ≅
Tor_{H^{∗}(BG)/(P_{G})}(H^{∗}(BG)/(P_{G}) ⊗ E; Tor_{P_{G}}(𝔽_{p}; 𝔽_{p})) ≅
E ⊗ Tor_{P_{G}}(𝔽_{p}; 𝔽_{p}).

Thus the spectral sequence collapses at the E_{2} term. Now

dim_{𝔽_{p}}(E ⊗ Tor_{P_{G}}(𝔽_{p}; 𝔽_{p})) = dim_{𝔽_{p}}(E) · dim_{𝔽_{p}}(Tor_{P_{G}}(𝔽_{p}; 𝔽_{p})) =
2^{r} · 2^{k} = 2^{m} = dim_{𝔽_{p}}(H^{∗}(ΩK_{(p)})).

Hence the Eilenberg-Moore spectral sequence collapses at the E_{2} term as well. Finally, the E_{2} term of the Eilenberg-Moore spectral sequence for the fibration

S ⟶^{ψ} C ⟶^{f} X  (3.2)

depends only on the H^{∗}(X)-module structure of H^{∗}(C). By assumption BG_{(p)} and X are comparable of degree d. Thus there is an isomorphism of algebras ϕ : H^{∗}(BG) ⟶ H^{∗}(X), such that ϕ ∘ l_{X}^{∗} = l_{G}^{∗} up to dimension d. Since dimC = d it follows that g^{∗} = f^{∗} ∘ ϕ. Thus ϕ induces an isomorphism of bigraded modules (not necessarily differential)

Tor_{H^{∗}(X)}(H^{∗}(C); 𝔽_{p}) ≅ Tor_{H^{∗}(BG)}(H^{∗}(C); 𝔽_{p}).

But Tor_{H^{∗}(BG)}(H^{∗}(C); 𝔽_{p}) ≅ H^{∗}(ΩK_{(p)}) as graded modules. Hence S is finite dimensional of dimension less than or equal to d. On the other hand S is rationally equivalent to ΩK_{(p)} and so dimS ≥ d. Thus dimS = d. In particular H_{d}(S) ≅ ℤ/pℤ. By lemma 3.1 it follows that the map h : ΩK_{(p)} ⟶ S induces an isomorphism on homology and thus is a homotopy equivalence. This completes the proof of theorem 1.1.

4. 2-Stage Postnikov Systems and Proof of Theorem 1.2

Let X and Y be two simply-connected p-local spaces such that there exists an isomorphism

ϕ : H^{∗}(X) ⟶ H^{∗}(Y)

of A_{p}-algebras together with the operation of the higher Bocksteins. Under this hypothesis, all primary cohomology operations on X and Y coincide. More precisely if v : X ⟶ K_{1} is a cohomology class and θ : K_{1} ⟶ K_{2} is a cohomology operation, where both K_{1} and K_{2} are p-local generalized Eilenberg-MacLane spaces of finite type, then the class ϕ(v) : Y ⟶ K_{1} satisfies the equality

ϕ(θv) ≃ θϕ(v).

Hence the composite θ ∘ v is null-homotopic if and only if the composite θ ∘ ϕ(v) is.

Let

[figure: ΩK_{2}; E; p_{1}; j_{1}; K_{1}; K_{2}; k_{1}]

be a 2-stage Postnikov system. Let X and Y be as above and suppose that there exist maps X ⟶^{v} K_{1} and X ⟶^{l} E such that p_{1} ∘ l ≃ v. Then the map Y ⟶^{ϕ(v)} K_{1} lifts to map Y ⟶^{h} E with p_{1} ∘ h ≃ ϕ(v). The question, whether or not one can compare the lifts l and h so that ϕ ∘ l^{∗} = h^{∗}, depends of course on secondary cohomology operations. However notice that

ϕ ∘ l^{∗} ∘ p_{1}^{∗} = ϕ ∘ v^{∗} = ϕ(v)^{∗} = h^{∗} ∘ p_{1}^{∗}.

Thus if p_{1}^{∗} is an epimorphism, then ϕ ∘ l^{∗} = h^{∗}.

Let G be a group satisfying the conditions of theorem 1.2. Let X denote BG_{(p)} and let X[d] denote the d-th stage of the Postnikov tower for X. By hypothesis, X[d] can be defined as a 2-stage Postnikov system

[figure: ΩK_{2}; X[d]; p_{1}; K_{1}; K_{2}; k_{1}]

such that p_{1}^{∗} is an epimorphism in mod-p cohomology up to dimension d + 1. Let Y be simply-connected p-local space such that H^{∗}(X) ≅ H^{∗}(Y) as algebras over the Steenrod algebra with the same secondary Bockstein operators. Let

ϕ : H^{∗}(X) ⟶ H^{∗}(Y)

be an isomorphism, as above.

There is a map l_{X} : X ⟶ X[d] inducing an isomorphism in mod-p cohomology up to dimension d + 1. Let v : X ⟶ K_{1} denote the composite p_{1} ∘ l_{X}. Then the composite k_{1} ∘ v is null homotopic and hence so is

Y ⟶^{ϕ(v)} K_{1} ⟶^{k_{1}} K_{2}.

Let l_{Y} : Y ⟶ X[d] be a map lifting ϕ(v). Thus p_{1} ∘ l_{Y} ≃ ϕ(v). By the argument above one has ϕ ∘ l_{X}^{∗} = l_{Y}^{∗} up to dimension d + 1. Hence l_{Y}^{∗} is an isomorphism in mod-p cohomology up to dimension d + 1. By assumption all spaces are simply-connected, p-local, rationally contractible and of finite type. Thus l_{Y} induces an isomorphism on homotopy groups up to dimension d. Theorem 1.2 now follows at once from theorem 1.1.

5. Finite Groups of Lie Type

Let G(ℂ) be a complex connected reductive Lie group. Let G(q) be an associated finite group of Lie type over the field of q elements 𝔽_{q}. We start by recalling a theorem of E. Friedlander [8].

Theorem 5.1 (Friedlander). Let G(ℂ) be a complex connected reductive Lie group. Let q = l^{k} for some prime l. Then for every prime p ≠ l there is a homotopy pull back square

[figure: BG(q)_{p}^{∧}; ψ_{q}; BG(ℂ)_{p}^{∧}; Δ; BG(ℂ)_{p}^{∧}; (BG(ℂ)_{p}^{∧})^{×2}; 1⊤ϕ^{q}]  (5.1)

where Δ is the diagonal.

The map ψ_{q} in 5.1 above has G(ℂ)_{p}^{∧} as its homotopy fibre. Theorem 1.3 follows by observing that, under our hypotheses, the triple (G(ℂ)_{p}^{∧}, ψ_{q}, BG(ℂ)_{p}^{∧}) is a comparison system for BG(q)_{p}^{∧}.

If H^{∗}(G(ℂ); ℤ) has no p-torsion than it is well known [2] that

H^{∗}(BG(ℂ)) ≅ P [v_{1}, v_{2}, ⋯, v_{n}]

where v_{i} is even dimensional for all i. Thus it suffices to show that the map ψ_{q} : BG(q)_{p}^{∧} ⟶ BG(ℂ)_{p}^{∧} is admissible. But this follows at once from the following

Lemma 5.1. Let G(ℂ) be a complex connected reductive Lie group. Let p be a prime such that H^{∗}(BG(ℂ); ℤ) has no p-torsion. Let q = l^{k} for some prime l ≠ q. Let G(q) be an associated finite Chevalley group over the field 𝔽_{q}. Then the mod-p Eilenberg-Moore spectral sequence for the fibre square 5.1 above collapses at the E_{2} term. Thus

H^{∗}(BG(q)_{p}^{∧}) ≅ H^{∗}(BG(ℂ))/(Im(1⊤ϕ^{q})^{∗}) ⊗ E

as H^{∗}(BG(ℂ))-modules, where E is a finitely generated exterior algebra on odd-dimensional generators. Moreover, if p ≠ 2 then this is an isomorphism of algebras.

Proof. This follows at once from [9, II Lemma 5.3.1] and the big collapse theorem of L. Smith

6. Example; The Spaces ΩBSL_{3}(q)_{(p)}, p odd

Suppose p be an odd prime and let q = l^{k} for some positive integer k, where l is a prime different from p. The space BSL_{3}(q)_{(p)} can be obtained by means of theorem 5.1 from BSL_{3}(ℂ)_{p}^{∧} ≃ BSU(3)_{p}^{∧}.

By lemma 5.1 the mod-p Eilenberg-Moore spectral sequence associated with the corresponding fibre square collapses at the E_{2} term. Thus write

H^{∗}(BSU(3)) ≅ P [u_{4}, u_{6}].

Then the ideal (Im(1⊤ϕ^{q})^{∗}) is generated by (q^{2} − 1)u_{4} and (q^{3} − 1)u_{6}. Thus

H^{∗}(BSL_{3}(q)_{(3)}) = \{ \begin{matrix}P [u_{4}, u_{6}] ⊗ E[v_{3}, v_{5}] & if p|q − 1 \\ P [u_{4}] ⊗ E[v_{3}] & if p|q + 1 \\ P [u_{6}] ⊗ E[v_{5}] & if p|q^{2} + q + 1. \end{matrix}

The last two cases are periodic and dealt with in [5]. We restrict attention to the case p|q − 1, where

H^{∗}(BSL_{3}(q)_{p}^{∧}) ≅ P [u_{4}, u_{6}] ⊗ E[v_{3}, v_{5}]

as an algebra. Let r and s be integers such that β_{r}v_{3} = u_{4} and β_{s}v_{5} = u_{6}, where β_{j} denotes the j-th Bockstein operator. The precise values of r and s can be worked out explicitly for each case but the calculation is irrelevant for our purpose.

Let X be a space such that H^{∗}(X) ≅ H^{∗}(BSL_{3}(q)_{(p)}) as algebras over the Steenrod algebra with the same Bockstein operators. We first consider the case p ≥ 5.

Indeed there is a map

α : BSL_{3}(q)_{(p)} ⟶ K(ℤ/p^{r}ℤ, 3) × K(ℤ/p^{s}ℤ, 5)

taking ι_{3} to v_{3} and ι_{5} to v_{5}. Since p ≥ 5, one immediately observes that α induces an isomorphism in mod-p cohomology through dimension 10. Thus the corresponding map

ϕ(α) : X ⟶ K(ℤ/p^{r}ℤ, 3) × K(ℤ/p^{s}ℤ, 5)

has the same cohomological effect and theorem 1.4 follows at once from 1.1 and 1.3 for the case p ≥ 5.

The case p = 3 is almost equally easy in view of theorem 1.2. A straight forward computation, using the fibration

SU(3)_{3}^{∧} ⟶ BSL_{3}(q)_{(3)} ⟶ BSU(3)_{3}^{∧}

and the known low dimensional homotopy of SU(3), yields the following:

π_{i}BSL_{3}(q)_{(3)} = \{ \begin{matrix}ℤ/3^{r}ℤ & i = 3 \\ ℤ/3^{s}ℤ & i = 5 \\ ℤ/3ℤ & 6 ≤ i ≤ 8 \\ 0 & i ≤ 2 and i = 4. \end{matrix}

Using only the Adem relations and the fact that P^{1}v_{3} = P^{1}v_{5} = 0 in H^{∗}(SU(3)), one can show that

P^{1}v_{3} = P^{1}u_{4} = P^{1}u_{6} = 0, and P^{1}v_{5} = av_{3}u_{6}

in H^{∗}(BSL_{3}(q)), for some a ∈ 𝔽_{3}. As the precise value of a does not make any difference in our calculation, we do not attempt to compute it.

Start with a map

α : BSL_{3}(q)_{(3)} ⟶ K(ℤ/3^{r}ℤ, 3) × K(ℤ/3^{s}ℤ, 5)

defined as above. One observes immediately that α^{∗} is an epimorphism in mod-3 cohomology up to dimension 9 with kernel, generated with respect to the primary Bockstein operators, by P^{1}ι_{3}, P^{1}β_{r}ι_{3} and the class P^{1}ι_{5} + aι_{3}β_{s}ι_{5}. Thus define

k_{1} : K(ℤ/3^{r}ℤ, 3) × K(ℤ/3^{s}ℤ, 5) ⟶ K_{7} × K_{8} × K_{9}

where K_{i} = K(ℤ/3ℤ, i), to be the map taking ι_{7} to P^{1}ι_{3}, ι_{8} to P^{1}β_{r}ι_{3} and ι_{9} to P^{1}ι_{5}+aι_{3}β_{s}ι_{5}. Let E denote the homotopy fibre of k_{1}. It is easy to verify that E is homotopy equivalent to the 8-th stage of the Postnikov tower for BSL_{3}(q)_{p}^{∧}. Thus α lifts to a map

l : BSL_{3}(q)_{(3)} ⟶ E

inducing an isomorphism in mod-3 cohomology up to dimension 9. Moreover, the projection

p_{1} : E ⟶ K(ℤ/3^{r}ℤ, 3) × K(ℤ/3^{s}ℤ, 5)

induces an epimorphism in mod-3 cohomology up to dimension 9. Thus the hypotheses of theorem 1.2 are satisfied and the proof of theorem 1.4 for p = 3 is complete.

References

[1] D. J. Benson and J. F. Carlson; Projective Resolutions and Poincaré Duality Complexes; to appear in the Transactions of the A.M.S.

[2] A. Borel; Sur l’Homologie et la Cohomologie des Groupes de Lie Compact Connexe; Amr. Jour. Math., 76 (1954), 273-342.

[3] A. Bousfield and D. Kan; Homotopy Limits Completions and Localizations; Lec. Notes Math., 304 Springer Verlag (1972).

[4] H. Cartan and S. Eilenberg; Homological Algebra; Princeton Univ. Press, (1956).

[5] F. R. Cohen; Remarks on the Homotopy Theory Associated with Finite Perfect groups; Lec. Notes Math., 1509, Springer Verlag (1992).

[6] F. R. Cohen, J. C. Moore and J. Neisendorfer Exponents in Homotopy Theory; Ann. Math. Stud., 113, Princeton Univ. Press, (1987), 3-34.

[7] F. R. Cohen, J. R. Harper and R. Levi On the Homotopy Theory associated to Certain Finite Groups of 2-rank 2; to appear.

[8] E. Friedlander Etale Homotopy of Simplicial Schemes; Ann. Math. Stud., 104, Princeton Univ. Press, (1982).

[9] R. Levi On Finite Groups and Homotopy Theory; to appear.

[10] J. Martino; Stable Splittings and the Sylow 2-Subgroups of SL_{3}(𝔽_{q}), q odd; Thesis, Northwestern University (1988).

[11] Matsumura Commutative Algebra; W. A. Benjamin Inc. (1970).

[12] L. Smith Homological Algebra and the Eilenberg-Moore Spectral Sequence; Amer. Math. Soc. Transl., 129, (1967), 58-93.

Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany

E-mail address: rlvi@vogon.mathi.uni-heidelberg.de