UNSTABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧}

John Martino and Stewart Priddy

1. Introduction and statement of results.

For nilpotent spaces p-completion is well behaved and reasonably well understood. By p-completion we mean Bousfield-Kan completion with respect to the field F_{p} [BK]. For non-nilpotent spaces the completion process often has a chaotic effect, this is true even for small spaces. One knows however that the classifying space of a compact Lie group is F_{p}-good even though it is usually non-nilpotent.

In this paper we consider BG_{p}^{∧}, the p-completion of BG the classifying space of a finite group G. We prove the following classification result which determines BG_{p}^{∧} up to homotopy equivalence from its p-local data, that is, conjugacy classes of p-subgroups of G and their Weyl groups. Before stating this result we recall the Frobenius category P(G) which has as objects the p-subgroups of G and as morphisms the homomorphisms induced by conjugation by elements of G.

Theorem 1.1. (Unstable classification) For two finite groups G and G′ the following are equivalent:

(1) BG_{p}^{∧} and BG′_{p}^{∧} are homotopy equivalent.

(2) For every p-group Q, there is a bijection

Rep(Q, G) → Rep(Q, G′)

The first author is partially supported by NSF Grant DMS-9007361, the second by NSF Grant DMS-9101415.

2 JOHN MARTINO AND STEWART PRIDDY

natural with respect to Q. Rep(Q, G) = Hom(Q, G)/G with G acting by conjugation.

(3) For every p-group Q, there is a bijection

Inj(Q, G) → Inj(Q, G′)

natural with respect to Q. Inj(Q, G) ⊆ Rep(Q, G) consists of conjugacy classes of injective homomorphisms.

(4) There is an equivalence of Frobenius categories F : P(G) → P(G′) such that F (Q) ≈ Q for every p-subgroup of G.

We call an equivalence F of Frobenius categories isotypical equivalence if Q ≈ F (Q) as groups. Remark. By natural with respect to Q we mean if ϕ : R → Q is a homomorphism then the following diagram commutes.

Rep(Q, G) ⟶ Rep(Q, G′)
ϕ^{∗}↓  ^{ϕ^{∗}}↓
Rep(R, G) ⟶ Rep(R, G′)

Note it follows that Rep(Q, G) and Rep(Q, G′) must be isomorphic as Out(Q)-sets, where Out(Q) is the outer automorphism group of Q. Naturality is defined similarly for Inj(Q, G) using injective homomorphisms. Remark. For the reader interested in group cohomology, we recall that (1) is equivalent to the existence of a map BG_{p}^{∧} ⟶ BG′_{p}^{∧} (not necessarily derived from a homomorphism) inducing an isomorphism in cohomology with trivial coefficients in F_{p}. In this case G and G′ have isomorphic cohomology algebras (over the Steenrod algebra).

Let W_{G}(Q) denote the Weyl group of Q ≤ G, i.e.

W_{G}(Q) = N_{G}(Q)/Q · C_{G}(Q)

where N_{G}(Q) is the normalizer and C_{G}(Q) is the centralizer of Q in G. Then W_{G}(Q) ≤ Out(Q). Frequently Weyl groups are defined without quotienting by Q; however, since we are interested in homotopy classes of maps, inner automorphisms are trivial so the present definition is better. It is easily seen that

Inj(Q, G) ≈ ⨆ Out(Q_{i})/W_{G}(Q_{i})

where the disjoint union runs over conjugacy classes of subgroups Q_{i} ≤ G such that Q_{i} ≈ Q. We also remark that hom_{P(G)}(Q, Q) ≈ N_{G}(Q)/C_{G}(Q). Thus the homotopy type of BG_{p}^{∧} depends essentially on the p-local structure of G.

A necessary condition for BG_{p}^{∧} and BG′_{p}^{∧} to be homotopy equivalent is for G and G′ to have isomorphic Sylow p-subgroups. The corresponding sufficient condition is detailed in the following corollary.

Corollary 1.2. Let G and G′ be two finite groups with Sylow p-subgroups P and P′ respectively. Then BG_{p}^{∧} and BG′_{p}^{∧} are homotopy equivalent if and only if there exists an isomorphism f : P → P′ such that,

(i) if Q_{1} and Q_{2} are subgroups of P then Q_{1} and Q_{2} are G-conjugate if and only if f(Q_{1}) and f(Q_{2}) are G′-conjugate, and

(ii) W_{G}(Q) and f^{−1}(W_{G′}(f(Q)))f are conjugate in Out(Q) for every subgroup Q of P such that Out(Q) is not a p-group,

Remark. For a p-group Q, the outer automorphism group Out(Q) is a p-group if and only if the automorphism group Aut(Q) is a p-group. For almost all p-groups Q, Out(Q) is a p-group [Ma]. By “almost all” we mean that for a nilpotency class of p-groups the percentage of p-groups Q for which Out(Q) is not a p-group decreases to zero as the number of generators of Q increases.

Theorem 1.1 was suggested by an analogous theorem of G. Mislin [Ms] for a homomorphism inducing a mod-p cohomology isomorphism. Such a homomorphism would induce a homotopy equivalence between the p-completions of the classifying spaces. Theorem (Mislin) [Ms]. Let ρ : G → G′ be a homomorphism of compact Lie groups and let p be a prime. Then the following are equivalent:

(1) ρ^{∗} : H^{∗}(BG′; F_{p}) → H^{∗}(BG; F_{p}) is an isomorphism.

(2) ρ_{∗} : Rep(Q, G) → Rep(Q, G′) is a bijection for every finite p-group Q.

(3) ρ_{∗} : P(G) → P(G′) is an equivalence of categories.

Mislin uses the term Quillen category for the Frobenius category P(G).

It is worthwhile pointing out that there are maps BG_{p}^{∧} → BG_{p}^{′∧} not induced by homomorphisms. This is true even for equivalences, consequently several people have raised the question as to whether a homotopy equivalence BG_{p}^{∧} ≃ BG′_{p}^{∧} can be realized on the group level as an alternating sequence of homomorphisms each inducing an equivalence. We give examples showing this is false.

A major step in Mislin’s work is Theorem [Ms]. The natural map BG → BG_{p}^{∧} induces a bijection

[BQ, BG] ⟶^{≈} [BQ, BG_{p}^{∧}]

for all p-groups Q.

By [X, Y] we mean homotopy classes of unpointed maps between CW-complexes. Since Rep(Q, G) ≈ [BQ, BG] we have a bijection

(1.3)  Rep(Q, G) ⟶^{≈} [BQ, BG_{p}^{∧}]

UNSTABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧} 5

natural in both Q and G. It follows from (1.3) that a map f : BG_{p}^{∧} → BG′_{p}^{∧} induces maps

f_{∗} : Rep(Q, G) → Rep(Q, G′)

and

f_{∗} : Inj(Q, G) → Inj(Q, G′)

For such a map we have the following variation of Theorem 1.1: Theorem 1.4. Let G and G′ be finite groups and f : BG_{p}^{∧} → BG′_{p}^{∧} be a map. Then the following are equivalent:

(i) f_{∗} : Rep(Q, G) → Rep(Q, G′) is a bijection for all p-groups Q.

(ii) f_{∗} : Inj(Q, G) → Inj(Q, G′) is a bijection for all p-groups Q.

(iii) f^{∗} : H^{∗}(BG′; F_{p}) → H^{∗}(BG; F_{p}) is an isomorphism.

(iv) f is a homotopy equivalence.

In a previous paper the authors proved a classification result for the stable homotopy type of classifying spaces. For a set S, let F_{p}S denote the F_{p} vector space with basis S. Theorem [MP2,MP3]. (Stable classification) For two finite groups G and G′ the following are equivalent:

(1) BG_{p}^{∧} and BG′_{p}^{∧} are stably homotopy equivalent.

(2) For every p-group Q,

F_{p}Rep(Q, G) ≈ F_{p}Rep(Q, G′)

as F_{p}Out(Q) modules.

(3) For every p-group Q,

F_{p}Inj(Q, G) ≈ F_{p}Inj(Q, G′)

as F_{p}Out(Q) modules.

6 JOHN MARTINO AND STEWART PRIDDY

Remarks. There are several points of comparison between the stable and unstable case.

i) The stable classification is a “linearization” of the unstable classification. Notice however that Theorem 1.1 requires naturality with respect to Q while in the stable case only compatibility with the Out(Q) action is required. In general naturality does not hold in the stable case since (in the notation of [MP2,MP3])

F_{p}Rep(Q, G) ≈ {BQ_{+}, BG_{+}}/J_{τ} (Q) ⊗ F_{p}

is natural in G but not in Q. ii) The authors are not aware of a stable analogue to condition (4). The obvious linearization of the Frobenius categories does not yield the desired result. iii) For finite G, the p-localization and p-completion of BG are equivalent. Our stable classification could equally well have been proved using localization. In the unstable case however our proofs using hocolim’s seem to require completions. iv) In [MP3] examples are given of groups G, G′ such that BG_{p}^{∧} ≃ BG′_{p}^{∧} stably but not unstably. In these examples the mod-p cohomology algebras H^{∗}(BG; F_{p}) and H^{∗}(BG′; F_{p}) differ as algebras over the Steenrod algebra. Other examples have been found where these cohomology algebras differ simply as rings [L].

This paper is organized as follows: In Section 2 we show how to construct BG_{p}^{∧} as a homotopy colimit using the Frobenius category. These results are used in Section 3 to give a proof of Theorem 1.1. Section 4 is devoted to the proof of Corollary 1.2. Section 5 contains the proof of Theorem 1.4. Finally in Section 6 we use the groups P SL_{2}(F_{q}) to show the impossibility of constructing all homotopy equivalences between classifying spaces as alternating sequences of homotopy equivalences induced from homomorphisms.

UNSTABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧} 7

The authors would like to thank Hans-Werner Henn, Stefan Jackowski, Nick Kuhn, Jim McClure, and Bob Oliver for useful comments on this material. We especially thank Bill Dwyer for suggestions relating the proof of Theorem 2.2 and the use of homotopy colimits in Section 2.

2. Constructing BG from subgroups using hocolim.

In this section we prepare some preliminary results needed in the proof of Theorem 1.1. We wish to construct BG as a homotopy colimit from subgroups. For a discussion of hocolim see [BK], [JM], and [JMO]; for convenience we recall the definition. Let CW denote the category of CW-complexes. If I is a small category and F : I ⟶ CW is a functor then hocolim_{I}F is defined as the geometric realization

hocolim_{I}F = |I_{F} |

where I_{F} is the category whose objects are pairs (i, x), i ∈ I, x ∈ F (i) and whose morphisms (i, x) → (i′, x′) are morphisms β : i → i′ of I such that F (β)(x) = x′.

There are two superficially similar categories which describe how the subgroups of G relate to the full group. One is the Frobenius category P(G), introduced by Puig [P; p.8], where the objects are p-subgroups of G and the morphisms are induced by conjugation. This includes inclusions Q ≤ P of subgroups since these are induced by conjugation by the identity. The other is the orbit category O(G), where the objects are subgroups of G, and the morphisms are right multiplication by an element of G on orbit spaces G/H ⟶^{g} G/K where H, K ≤ G. Note that the objects in the orbit category are all the subgroups of G not just the p-subgroups. To illustrate the difference between these categories we state the endomorphisms of a p-subgroup Q: morph_{P(G)}(Q, Q) = N_{G}(Q)/C_{G}(Q) while morph_{O(G)}(Q, Q) = N_{G}(Q)/Q. The orbit category is better suited to the use of homotopy colimits because the following diagram commutes:

G/G
↗  ↖
G/H ⟶_{g} G/K,

where H, K are subgroups of G and g ∈ morph_{O(G)}(H, K). By applying the Borel

UNSTABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧} 9

construction EG ×_{G} −, which commutes with homotopy colimits. The above diagram yields a map

hocolim_{I}EG ×_{G} (G/H) ⟶ EG ×_{G} (G/G) = BG,

where I is a subcategory of O(G). Jackowski and McClure have found a way to translate information from the Frobenius category to the orbit category. Let A(G) be the full subcategory of P(G) where the objects are non-trivial elementary abelian p-subgroups. Then C_{G} : A(G)^{op} ⟶ O(G), E ↦ C_{G}(E), is a functor on A(G)^{op} since taking centralizers reverses variance. Jackowski and McClure have shown Theorem 2.1 [JM]. The projections EG ×_{G} (G/Q) ⟶ EG ×_{G} (G/G) = BG induce a natural map

π : hocolim_{A(G)^{op}}EG ×_{G} (G/C_{G}(E)) ⟶ BG

which is an isomorphism in mod-p homology.

Let CW′ be a modified subcategory of CW-complexes. Define CW′ to be the image of I_{G} in the theorem below. By “modified” we mean a redefinition of the identity morphism. Theorem 1.1 follows from Theorem 2.2. Let G be a finite group. There exists a functor I_{G} : O(G) → CW′ such that the following hold.

(1) There exists a natural transformation ϕ : I_{G} → (EG/−)_{p}^{∧} such that ϕ_{H} : I_{G}(H) → (EG/H)_{p}^{∧} is a deformation retraction.

(2) If ψ : P(G) → P(G′) is an equivalence such that Q ≈ ψ(Q) for Q ∈ obj(P(G)), then I_{G}(G) and I_{G′}(G′) are homotopy equivalent.

Remark. The strategy of the proof is straightforward: We proceed by induction on the order of G. If G does not have a p-component to its center then we use Theorem

10 JOHN MARTINO AND STEWART PRIDDY

2.1 to construct I_{G}(G). Otherwise, we repeatedly quotient by Z = Ω_{1}(Z(G)), the subgroup generated by elements of order p in Z(G), the center of G, until the resulting group no longer has a p-component to its center. Then we use Theorem 2.1 and repeat the process until we reach p′-groups at which point the result is trivial. The rather lengthy details are given after the following two lemmas which we use to lift an equivalence from quotient groups to total groups.

Let Π : F̃ ⟶ F be a natural transformation of functors F, F̃ : I ⟶ CW such that for each object i ∈ I, there is a quasifibration [DL]

W (i) ⟶ F̃(i) ⟶^{Π(i)} F (i)

such that for each morphism α : i → i′, F̃(α) : F̃(i) → F̃(i′) is a homeomorphism when restricted to the fiber Π(i)^{−1}(x), for every x ∈ F (i). If F (i) is path-connected then the fibers of F̃(i) ⟶^{Π(i)} F (i) are homotopy equivalent. Let W (i) represent the homotopy class of the fibers. For each morphism α : i → i′, there is a map of quasifibrations

W (i) ⟶^{≅} W (i′)
↓ ↓
F̃(i) ⟶^{F̃(α)} F̃(i′)
↓ ↓
F (i) ⟶^{F (α)} F (i′).

Then Π induces a functor

π : IF̃ ⟶ I_{F}

given by (i, x̃) → (i, Π(i)(x̃)). Lemma 2.3. With the above hypotheses π induces a quasifibration

W ⟶ hocolim_{I} F̃ ⟶ hocolim_{I}F,

To prove Lemma 2.3 we use the

UNSTABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧} 11

Corollary to Quillen’s Theorem B [Q]. Suppose τ : C → C′ is a prefibered functor and for every morphism u : Y → Y′ the base-change functor u^{∗} : τ^{−1}(Y) → τ^{−1}(Y′) is a homotopy equivalence. Then for any Y in C′, the category τ^{−1}(Y) is homotopy equivalent to the homotopy fiber of τ over Y.

Proof of Lemma 2.3. We will show π satisfies the hypotheses of the Corollary of Quillen’s Theorem B [Q], and thus induces a quasifibration, since hocolim_{I}F is the geometric realization of I_{F} and W is the homotopy fiber of π. For Y = (i, x) ∈ I_{F}, let π^{−1}(Y) be the subcategory of IF̃ consisting of arrows which map to the identity of Y. Thus π^{−1}(Y) consists of objects (i, x̃), x̃ ∈ π^{−1}(x) and their identity maps. We require π to be a prefibered functor and for every map v : Y_{1} ⟶ Y_{2}, Y_{j} = (i_{j}, x_{j}), the induced base change functor v^{∗} : π^{−1}(Y_{2}) ⟶ π^{−1}(Y_{1}) is an isomorphism. Thus for any object Y ∈ IF̃, the category π^{−1}(Y) is homotopy equivalent to the homotopy fiber of π over Y.

Let Y \π be the category of pairs (X, v) where v : Y → π(X) is a map of I_{F} and maps (X, v) → (X′, v′) are maps w : X → X′ of IF̃ such that π(w)v = v′. To see that π is prefibered we must first show that for all Y, the functor T : π^{−1}(Y) → Y \π given by X ↦ (X, id_{Y}) has a right adjoint T′. We define T′ as follows: if (X_{1}, v_{1}) ∈ Y \π where X_{1} = (i_{1}, x̃_{1}) and v_{1} : (i, x) → (i_{1}, x_{1}) corresponds to an α : i → i_{1} then let T′(X_{1}, v_{1}) = (i, x̃) where F̃(α)(x̃) = x̃_{1}. The existence and uniqueness of x̃ follows since F̃(α)|_{Π(i)^{−1}(x)} is a homeomorphism. On maps we necessarily have T′(w) = id. It is easily seen that T′ is a right adjoint to T. That base change is an isomorphism also follows directly and will be left to the reader. □

Let Z = Ω_{1}(Z(G)).

Lemma 2.4. An equivalence of categories F : P(G) ⟶ P(G′) induces equiva-

12 JOHN MARTINO AND STEWART PRIDDY

lences

i)

\overline{F} : P(\overline{G}) ⟶ P(\overline{G′})

where \overline{G} = G/Z, \overline{G′} = G′/Z′, Z′ = F (Z). ii)

F : P(C_{G}(E)) ⟶ P(C_{G′}(E′))

where E ∈ A(G), E′ = F (E′).

Proof. It suffices to assume F : SP(G) ⟶^{≈} SP(G′) is an isomorphism of skeletal categories, i.e. SP(G) is a full subcategory of P(G) with obj(SP(G)) a set of representatives of the conjugacy classes of p-subgroups of G.

i) On objects F is induced by restriction from an isomorphism α : P ≈ P′ of Sylow p-subgroups of G, G′ respectively. Thus \overline{F} is also a bijection on objects since obj(SP(\overline{G})) = \{Q ∈ obj(SP(G)) : Z ≤ Q\} and Z′ = α(Z). It is clear that \overline{F} is surjective on morphisms since ck̃ : Q̃_{1} ⟶ Q̃_{2} where Q̃ = π^{−1}(Q), k̃ ∈ π^{−1}(k).

Now suppose \overline{F} (c_{k_{1}}) = \overline{F} (c_{k_{2}}) then F (ck̃_{1}) = F (ck̃_{2}) mod Z′. The following diagram defines F (ck̃_{i})

Q_{1} ⟶^{ck̃_{i}} Q_{2}
α↓  ^{α}↓
F (Q_{1}) ⟶^{F (ck̃_{i})} F (Q_{2})

Thus αck̃_{1}α^{−1} = αck̃_{2}α^{−1} mod Z′ hence ck̃_{1} = ck̃_{2} mod α^{−1}Z′α and so c_{k_{1}} = c_{k_{2}} since α^{−1}Z′α = Z. ii) Q ≤ C_{G}(E) iff [Q, E] = e iff [α(Q), α(E)] = e iff F (Q) ≤ C_{G′}(E′). It follows that F : SP(C_{G′}(E)) ⟶ SP(C_{G′}(E′)) is a bijection on objects. To see that F is also a bijection on morphisms c_{k} : Q_{1} ⟶ Q_{2} of SP(C_{G}(E)) we consider the diagram which defines F (c_{k}) uniquely. The only point to check is whether F (c_{k})

UNSTABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧} 13

is a morphism in SP(C_{G′}(E′)), i.e., whether F (c_{k}) restricts to the identity on E′. Since Q_{1} · E, Q_{2} · E are p-groups we have the commutative diagram

Q_{1} · E ⟶^{c_{k}} Q_{2} · E
α↓  ^{α}↓
F (Q_{1} · E) ⟶^{F (c_{k})} F (Q_{2} · E)

On E, c_{k} restricts to id_{E}, therefore F (c_{k}) restricts to id_{E′} by functorality. □

Proof of Theorem 2.2.

We adopt the notation H, K for subgroups and Q, P for p-subgroups. \underline{Part 1}

If we have constructed an I_{G}(H) such that (EG/H)_{p}^{∧} is a deformation retract of I_{G}(H), then for f : G/H → G/K a morphism in O(G) define

I_{G}(f) : I_{G}(H) ⟶^{ϕ_{H}} (EG/H)_{p}^{∧} ⟶^{f_{∗}} (EG/K)_{p}^{∧} ⟶^{i} I_{G}(K).

Define ϕ_{f} to be the commutative square:

I_{G}(H) ⟶^{ϕ_{H}} (EG/H)_{p}^{∧}
I_{G}(f)↓  ^{f_{∗}}↓
I_{G}(K) ⟶^{ϕ_{K}} (EG/K)_{p}^{∧}.

While I_{G}(id) is not the identity map, I_{G}(id) does satisfy the categorical definition of an identity in CW′. Also we have I_{G}(gf) = I_{G}(g)I_{G}(f), so I_{G} is a functor. With this reinterpretation of the identity ϕ can also be viewed as a natural transformation, ϕ_{gf} = ϕ_{g}ϕ_{f}. By the above we may assume I_{G} is natural for all groups of order less than |H|.

To construct I_{G}(H) we induct on the order of H. If |H| = 1 there is nothing to prove. Let Z = Ω_{1}(Z(H)), where Z(H) is the center of H and Ω_{1}(.) denotes the

14 JOHN MARTINO AND STEWART PRIDDY

subgroup generated by elements of order p. We distinguish two cases according to whether Z is trivial or not. Case 1: Z = e.

For a finite group H such that Z = e we construct a space I_{G}(H) having the desired properties. I_{G}(H) is the mapping cylinder of a homotopy colimit. We use the mapping cylinder to insure I_{G}(H) is natural, as explained below. We call the homotopy colimit I′_{G}(H). It is a homotopy colimit of I_{G}(K)’s. We construct I_{G} and I′_{G} by induction so the I_{G}(K)’s are already defined. In a sense I_{G}(H) is an iterated sequence of homotopy colimits. We will call the spaces from which I_{G}(H) is built by repeated homotopy colimits the initial spaces of I_{G}(H).

If H is a p′-group then define I_{G}(H) to be (EG/H)_{p}^{∧}. If p divides the order of H then |C_{H}(E)| < |H| for E ∈ A(H)^{op}. Thus by induction we may assume the result for C_{H}(E), i.e., there exists a deformation retraction

ϕ_{C_{H}(E)} : I_{G}(C_{H}(E)) ⟶ EG/C_{H}(E)_{p}^{∧}.

Define χ_{H} as the composite

χ_{H} : I′_{G}(H) = hocolim I_{G}(C_{H}(E))

⟶ hocolim (EG/C_{H}(E))_{p}^{∧} ⟶^{π} (EG/H)_{p}^{∧}

where the hocolim’s are taken over E ∈ A(H)^{op} and the first arrow is hocolim (ϕ_{C_{H}(E)}). By Theorem 2.1 χ_{H} is a mod-p cohomology equivalence. Hence after p-completion χ_{H} is a homotopy equivalence. We define I_{G}(H) as a mapping cylinder so that (EG/H)_{p}^{∧} is a deformation retract of I_{G}(H). Let θ_{H} : (EG/H)_{p}^{∧} → I′_{G}(H)_{p}^{∧} be a homotopy inverse of χ_{H}. Let I_{G}(H) be the mapping cylinder of θ_{H}. Then I_{G}(H) is homotopy equivalent to I′_{G}(H)_{p}^{∧}. The inclusion i : (EG/H)_{p}^{∧} ⟶ I_{G}(H) is a cofibration. We use the homotopy extension property to extend a homotopy from χ_{H}θ_{H} : (EG/H)_{p}^{∧} → (EG/H)_{p}^{∧} to the identity to a homotopy h : I_{G}(H) × I → (EG/H)_{p}^{∧}

UNSTABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧} 15

such that h(−, 0) is projection from I_{G}(H) onto the bottom of the cylinder I′_{G}(H)_{p}^{∧} followed by the p-completion of χ_{H}. Thus the top of the cylinder (EG/H)_{p}^{∧} is a retract of I_{G}(H). Define

ϕ_{H} = h(−, 1) : I_{G}(H) ⟶ (EG/H)_{p}^{∧}

which is a deformation retraction since i is a homotopy equivalence. Case 2: Z ≠ e.

This case is a little more complicated. We construct a quasifibration I_{G}(H) → I_{H/Z}(H/Z). Since |H/Z| ≨ |H|, I_{H/Z}(H/Z) is defined by induction as an iterated sequence of homotopy colimits, hocolim_{K}I_{H/Z}(K), where K is a smaller group than H/Z. We would like to lift K by the projection H → H/Z to K̃ and define I_{G}(H) = hocolimK̃I_{G}(K̃). Unfortunately, we do not have enough control over the fiber of I_{G}(K̃) → I_{G}(K). To control the fiber we define constructions Ĩ_{G} and Ĩ′_{G} so that Ĩ_{G}(K̃) is the mapping cylinder of Ĩ′_{G}(K̃) to insure naturality of Ĩ_{G}. The space Ĩ_{G}(K̃) is the same iterated sequence of homotopy colimits as I_{G}(K̃) but with modified, although homotopy equivalent, initial spaces. The initial spaces are altered to insure proper behavior of the fibers, but since the new initial spaces are homotopy equivalent to the old we have Ĩ_{G}(K̃) ≃ I_{G}(K̃). We then use Ĩ_{G}(K̃) to define I_{G}(H).

For a finite group K let ρ : K → \overline{K} be an epimorphism with a central kernel. Let C(\overline{K}) be a subcategory of P(\overline{K}). Define C(K) = ρ^{−1}C(\overline{K}); that is an object of C(K) consists of a p-group Q = ρ^{−1}\overline{Q}, \overline{Q} ∈ C(\overline{K}) and a morphism Q → R is induced by conjugation c_{v}, v ∈ K such that c_{ρ(v)} induces a morphism ρ(Q) → ρ(R) of C(\overline{K}). If ρ(v) = ρ(v′) then c_{v} = c_{v′}; hence ρ_{∗} : C(K) → C(\overline{K}) is an isomorphism of categories.

Define π : H_{0} = H → H_{1} = H/Z to be the projection. If Z_{1} = Ω_{1}(Z(H_{1})) ≠ e then define π_{1} : H_{1} → H_{2} = H_{1}/Z_{1} to be the projection. Recursively define

16 JOHN MARTINO AND STEWART PRIDDY

Z_{i} = Ω_{1}(Z(H_{i})) and π_{i} : H_{i} → H_{i+1} = H_{i}/Z_{i}, until Z_{k} = e. This procedure terminates when H_{k} satisfies Case 1. Define C(H_{k}) = A(H_{k}) unless H_{k} is a p′ group in which case we set C(H_{k}) = {e ⟶^{id} e}. Let C(H_{i}) = π_{i}^{−1}C(H_{i+1}). By induction from Case 1 we can define χ_{H_{k}} as the composite

χ_{H_{k}} : I′_{H_{k}}(H_{k}) = hocolim (I_{H_{k}}(C_{H_{k}}(E))

⟶ hocolim (EH_{k}/C_{H_{k}}(E))_{p}^{∧} ⟶^{π} (EH_{k}/H_{k})_{p}^{∧}.

where the hocolim’s are taken over E ∈ C(H_{k}) and the first arrow is hocolim (ϕ_{C_{H_{k}} (E)}). Let θ_{H_{k}} be a homotopy inverse of χ_{H_{k}} after p-completion. Let I_{H_{k}}(H_{k}) be the mapping cylinder of θ_{H_{k}}. Let F_{H_{k}} be the functor C_{H_{k}} : C(H_{k})^{op} → O(H_{k}). Define a functor F_{H_{i}} : C(H_{i})^{op} → O(H_{i}) by F_{H_{i}}(Q) = π_{i}^{−1}F_{H_{i+1}}(π_{i}(Q)).

The functor p-completion preserves fibrations. If the fiber is connected and the fundamental group of the base acts nilpotently on the homology of the fiber, then the fiber of the p-completed fibration is homotopy equivalent to the p-completion of the original fiber [BK]. If F → E → B is a fibration then by F′_{∞} we mean the fiber of the p-completed fibration.

For any subgroup K of H_{i} which contains Z_{i}, we shall construct a quasifibration

(2.5)  (EH_{i}/Z_{i})′_{∞} → Ĩ_{H_{i}}(K) → I_{H_{i+1}}(π_{i}(K)),

where Ĩ_{H_{i}}(K) is also a mapping cylinder with (EH_{i}/K)_{p}^{∧} as deformation retract. As in case 1 we define Ĩ_{H_{i}}(K) by induction on the order of K using I′_{H_{i}}(K) where

Ĩ′_{H_{i}}(K) = hocolim_{Q∈C(K)^{op}} Ĩ_{H_{i}}(F_{K}(Q)),

where the functor F_{K} : C(K)^{op} → O(K) is defined in an analogous manner to F_{H_{i}} by repeatedly quotienting by central subgroups until K_{k} satisfies Case 1 and then lifting the centralizer functor from K_{k} to K. Since Z_{i} is a central elementary abelian p-subgroup of K then by the definition of F_{K} we have Z_{i} ≤ F_{K}(Q) and

UNSTABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧} 17

F_{K}(Q) ≨ K unless C(K) = {P ⟶^{id} P }, where P is a Sylow p-subgroup of K. This last case occurs when K_{k} is a p′-group. To begin the induction we need to define Ĩ_{H_{i}}(K) where C(K) = {P ⟶^{id} P }. Let 𝔅K to be the pullback

𝔅K  ⟶ ((EH_{i} × EH_{i+1})/H_{i})_{p}^{∧}
↓  ^{π_{i}}↓
(EH_{i+1}/π_{i}(K))_{p}^{∧} ⟶  (BH_{i+1})_{p}^{∧}.

where (EH_{i}/Z_{i})′_{∞} is the common fiber. Let Ĩ′_{H_{i}}(K) to be the pullback

Ĩ′_{H_{i}}(K) ⟶  𝔅K
↓ ↓
I′_{H_{i+1}}(π_{i}(K)) ⟶^{χ_{π_{i}(K)}} (EH_{i+1}/π_{i}(K))_{p}^{∧}.

Define χ̃_{K} : Ĩ′_{H_{i}}(K) → 𝔅K to be the obvious map. If C(K) ≠ {P ⟶^{id} P }, define χ̃_{K} : Ĩ′_{H_{i}}(K) → 𝔅K by projection maps onto (EH_{i+1}/π_{i}(K))_{p}^{∧} and ((EH_{i} × EH_{i+1})/H_{i})_{p}^{∧}.

By induction on the order of H and Lemma 2.3, there is a map of quasifibrations

(EH_{i}/Z_{i})′_{∞} ⟶^{id} (EH_{i}/Z_{i})′_{∞}
↓ ↓
Ĩ′_{H_{i}}(K) \underrightarrow{χ̃_{K}}  𝔅K
↓ ↓
I′_{H_{i+1}}(π_{i}(K)) ⟶^{χ_{π_{i}(K)}} (EH_{i+1}/π_{i}(K))_{p}^{∧}.

Define Ĩ_{H_{i}}(K) and ϕ̃_{K} : Ĩ_{H_{i}}(K) ⟶ EH_{i}/H_{i} in an analogous manner to I_{G}(H) and ϕ_{H} in Case 1. The mapping cylinder is a homotopy colimit over the category with two objects and one morphism between the objects. Thus by applying Lemma 2.3 we can construct the desired quasifibration

(EH_{i}/Z_{i})′_{∞} ⟶ Ĩ_{H_{i}}(K) ⟶ I_{H_{i+1}}(π_{i}(K)).

of (2.5).

Similarly define

(2.6)  I′_{H_{i}}(H_{i}) = hocolim_{Q∈C(H_{i})^{op}} Ĩ_{H_{i}}(F_{H_{i}}(Q));

χ_{H_{i}} : I_{H_{i}}(H_{i}) ⟶ (BH_{i})_{p}^{∧} is defined since the χ’s are constructed from projection maps. Then, by again using Lemma 2.3, we have a quasifibration

(EH_{i}/Z_{i})′_{∞} ⟶ I′_{H_{i}}(H_{i}) ⟶ I′_{H_{i+1}}(H_{i+1}).

Finally, by (2.6) and Lemma 2.3 we have a quasifibration

(EH_{i}/Z_{i})′_{∞} ⟶ I_{H_{i}}(H_{i}) ⟶ I_{H_{i+1}}(H_{i+1}).

Since χ_{π_{i}(K)}^{∗} is an isomorphism in mod-p cohomology and Z_{i} is central it follows that χ_{K}^{∗} is also an isomorphism, so after p-completion χ̃_{K} is a homotopy equivalence. That 𝔅 is a functor follows from the universal property of pullbacks. Then ϕ̃_{K} : Ĩ_{H_{i}}(K) → 𝔅K is a deformation retraction, as with ϕ_{H} in Case 1. Similarly, χ_{H_{i}} : I_{H_{i}}(H_{i}) ⟶ (BH_{i})_{p}^{∧} is a homotopy equivalence after p-completion, and ϕ_{H_{i}} : I_{H_{i}}(H_{i}) → (BH_{i})_{p}^{∧} is a deformation retraction. I′_{G}(H) is defined using the fibration

(EG/Z)′_{∞} ⟶ ((EG × EH_{1})/H_{0})_{p}^{∧} ⟶ (EH_{1}/H_{1})_{p}^{∧}

where H = H_{0}. I_{G}(H) and ϕ_{H} are defined in an analogous manner as for H_{k}.

We use the pullback to define Ĩ′_{H_{i}}(K) when C(K) = {P ⟶^{id} P } so that Ĩ_{G}(H), and hence I_{G}(H), will be an iterated sequence of homotopy colimits over subcategories of Frobenius categories. This will be crucial below in concluding that an isotypical equivalence of Frobenius categories for G and G′ induces a homotopy equivalence of I_{G}(G) and I_{G′}(G′). This concludes the proof of part (1). \underline{Part 2}

UNSTABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧} 19

Suppose we have an isotypical equivalence ψ : P(G) → P(G′). For a finite group G one can determine from the Frobenius category whether Ω_{1}(Z(G)) is trivial or not since a p-subgroup of G is the source of only the identity morphism iff it is central. Thus it follows from Lemma 2.4 that ψ induces an equivalence ψ : C(H) → C(H′). Therefore I_{G}(G) and I_{G′}(G′) are the same iterated sequence of homotopy colimits, but with possibly different initial spaces. The sequences stop for groups H with H_{k} a p′-group. In this situation C(H) = {P ⟶^{id} P }, where P is a Sylow p-subgroup of H. Moreover, H ≈ P ×W, where W is a p′-group. This follows since W stabilizes a central series of P and hence acts trivially by [G; Th. 5.3.2]. Under the hypothesis on ψ it follows P ≈ P′, where P′ is a Sylow p-subgroup of H′. Let H′ ≈ P′ × W′, where W′ is a p′-group.

The following maps are homotopy equivalences after p-completion

EG/H ⟵ (EG × EG′)/Δ ⟶ EG′/H′,

where Δ is the subgroup of G × G′ where Δ = (W × e)Δ(P, P′)(e × W′) and Δ(P, P′) = {(p, α(p))|p ∈ P }, α : P → P′ the isomorphism induced by ψ. These equivalences induce equivalences of I_{G}(G) and I_{G′}(G′) by the following maps. If c_{k} ∈ morph(C(H)) for some step in the iterated homotopy colimit, then for c_{k} : Q → R define EG/F_{H}(R) → EG/F_{H}(Q) by right multiplication by k, for EG × EG′, (resp. EG′) by right multiplication by (k, k′), (resp. k′), where c_{k′} = ψ(c_{k}). These maps are well-defined since the ambiguity in k and k′ have been quotiented. Thus by [BK;XII,4.2] we have a homotopy equivalence between the diagram for I_{G}(G) and the diagram for I_{G′}(G′). Therefore, an isotypical equivalence of Frobenius categories induces a homotopy equivalence I_{G}(G) ⟶^{≃} I_{G′}(G′). □

3. Proof of Theorem 1.1.

(1) ⇒ (2). (1.3).

(2) ⇒ (3). This follows by upward induction from the following lemma, the case Q = e being trivial. Let R denote a set of representatives of the isomorphism classes of finite groups; let Surj(Q, R) ⊆ Rep(Q, R) be the set of conjugacy classes of surjective homomorphisms.

Lemma 3.1.

i) There is a natural isomorphism

ϕ : ⨆_{R∈R}Surj(Q, R)×_{Out(R)}Inj(R, G) ⟶ Rep(Q, G)

given by ϕ([β] × [α]) = [αβ]. ii) If Q is a p-group then R can be taken to be a set of representatives for the

isomorphism classes of finite p-groups.

Proof. [MP3; Lemma 4.7].

(3) ⇒ (4). It suffices to give an isomorphism of skeletal categories F : SP(G) → SP(G′). Suppose Φ : Inj(−, G) → Inj(−, G′) is a natural bijection and Q ∈ obj(SP(G)) with inclusion map i : Q ⟶ G. Then define F (Q) = α(Q) where α is a representative of Φ(Q ⟶^{i} G) chosen so that α(Q) ∈ obj(SP(G′)). F (Q) is well defined since the different choices of α have conjugate images. Given a map c_{g} : Q_{1} → Q_{2} in morph(SP(G)) write

Q_{2}
c_{g} ↗ ↘ ^{i}
Q_{1} ⟶_{c_{g}} G.

UNSTABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧} 21

Then by the naturality of Φ

Q_{2}
c̃_{g} ↗ ↘ ^{β}
Q_{1} ⟶_{α} G′

commutes up to conjugation in G′ where β ∈ Φ(i) and c̃_{g} = c_{g}, although it is not necessarily conjugation by an element of G′. Thus

β(Q_{2})

βc̃_{g}α^{−1}↗  ↘ ^{i}
α(Q_{1}) ⟶_{i}  G′

commutes up to conjugation c_{g′} by some g′ ∈ G′. We define F (c_{g}) = c_{g′} and note that it is a well defined map of SP(G′). Clearly F (id) = id. To check functorality suppose c_{g}, c_{h} ∈ morph(SP(G)), then we have commutative diagrams

Q_{1} ⟶^{c̃_{g}} Q_{2} ⟶^{c̃_{h}} Q_{3}
α↓ ^{β}↓ ^{γ}↓
α(Q_{1}) ⟶^{c_{g′}} β(Q_{2}) ⟶^{c_{h′}} γ(Q_{3})

Since c_{gh} = c_{g}c_{h} implies c̃_{gh} = c̃_{g}c̃_{h} we have

F (c_{gh}) = γc̃_{gh}α^{−1} = γc̃_{g}β^{−1}βc̃_{h}α^{−1} = c_{g′}c_{h′} = F (c_{g})F (c_{h}).

It remains to show F is bijective. On objects this is clear; on morphisms the argument is similar to those above and will be left to the reader. □

(4) ⇒ (1) From Theorem 2.2 we have a composite of homotopy equivalences

BG_{p}^{∧} ⟶ I_{G}(G) ⟶ I_{G′}(G′) ⟶ BG′_{p}^{∧}.

□

22 JOHN MARTINO AND STEWART PRIDDY

Example 3.2. For the alternating groups A_{2n} and A_{2n+1}, all the copies of A_{2n} contained in A_{2n+1} are conjugate. Therefore, at the prime 2 the Frobenius categories P(A_{2n}) and P(A_{2n+1}) are equivalent although they are not isomorphic. Hence we can conclude that the 2-completions of BA_{2n} and BA_{2n+1} are homotopy equivalent, for all n. However, this example follows from Mislin’s theorem.

Consider the alternating group A_{7} and the projective special linear group P SL_{2}(F_{q}), where q ≡ 7 or 9 mod 16. Either group has Sylow 2-subgroups isomorphic to D_{8}. Thus if Q is a 2-subgroup then Q is isomorphic to either e, Z/2, Z/4, Z/2 × Z/2, or D_{8}. For all proper subgroups the Out(Q)-set Out(Q)/W (Q) is trivial [M], where W (Q) is the Weyl group defined in Section 1. In either case W (P) = e, if P is a Sylow 2-subgroup. Thus the naturality condition of Theorem 1.1 is trivially satisfied, so there exists a homotopy equivalence (BA_{7})_{2}^{∧} ⟶^{≃} BP SL_{2}(F_{q})_{2}^{∧}. If the characteristic of F_{q} is large enough then this homotopy equivalence cannot be induced by a homomorphism. We will show in Section 5 that there are homotopy equivalences which are not alternating sequences of homotopy equivalences induced by homomorphisms.

Remark. It is clear from the proof of Theorem 2.2 that to conclude BG_{p}^{∧} and BG′_{p}^{∧} are homotopy equivalent one need only verify condition (2), (3), or (4) for p-subgroups Q of G (and G′) which contain the p-component of the center of the group.

UNSTABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧} 23

4. Proof of Corollary 1.2.

Assume that f : P ⟶^{≈} P′ exists. We want to show BG_{p}^{∧} ≃ BG′_{p}^{∧}. We first remark that f_{∗}(Inj(Q, P)) = Inj(Q, P′) since f_{∗} maps injections to injections.

We will show condition (3) of Theorem 1.1 is satisfied. Note if Q is not isomorphic to a subgroup of P then Inj(Q, P) and Inj(Q, G) are both empty. So we can restrict our attention to subgroups of P. Inj(Q, G) is a disjoint union of transitive Out(Q)- sets indexed over a set of conjugacy classes of subgroups isomorphic to Q. Let R(Q) be a set of G-conjugacy class representatives of p-subgroups isomorphic to Q, Then

Inj(Q, G) ≈ ⨆_{Q_{i}∈R(Q)} Out(Q_{i})/W_{G}(Q_{i})

Choose R(Q) as follows, for each G-conjugacy class choose a representative Q_{i} such that a Sylow p-subgroup of N_{G}(Q_{i}) is contained in P. Thus N_{P} (Q_{i}) is a Sylow p-subgroup of N_{G}(Q_{i}), and W_{P} (Q_{i}) is a Sylow p-subgroup of W_{G}(Q_{i}). Let π : Inj(Q, P) → Inj(Q, G) be the obvious surjection and π′ the corresponding function for P′ and G′.

We first examine p-groups Q where Out(Q) is a p-group. In this case W_{P} (Q_{i}) = W_{G}(Q_{i}) for all Q_{i} ∈ R(Q). Since by condition (i) f maps G-conjugacy classes to G′-conjugacy classes, we have a bijection ϕ : Inj(Q, G) → Inj(Q, G′) induced by f on each orbit.

Now suppose Out(Q) is not a p-group. Note the preimage of W_{G}(Q_{i}) ∈ Inj(Q, G) is the set

π^{−1}(Q_{i}) = ⨆_{j} W_{G}(Q_{j})/W_{P} (Q_{j}),

where j runs over P -conjugacy classes of subgroups conjugate to Q_{i} in G. Since by condition (ii) W_{G}(Q_{i}) and f^{−1}(W_{G′}(f(Q_{i})))f are conjugate in Out(Q_{i}) and f maps G-conjugacy classes to G′-conjugacy classes, we again have a bijection ϕ : Inj(Q, G) → Inj(Q, G′).

24 JOHN MARTINO AND STEWART PRIDDY

In either case the bijections ϕ can be chosen so that the diagram commutes

Inj(Q, P) ⟶^{f_{∗}} Inj(Q, P′)
π↓  _{π′}↓
Inj(Q, G) ⟶^{ϕ} Inj(Q, G′)

for every p-group Q isomorphic to a subgroup of P. Let π : Inj(Q, P) → Inj(Q, G) be the obvious surjection and π′ the corresponding function for P′ and G′. The diagram implies for α ∈ Inj(Q, P), ϕ(π(α)) = π′(f_{∗}(α)).

Thus we need only check the naturality condition of Theorem 1.1. Let i : R → Q be an inclusion monomorphism. For π(α) ∈ Inj(Q, G), we need to show that ϕ(π(α)∘i) = ϕ(π(α))∘i, but this is equivalent to requiring π′∘f∘(α∘i) = π′∘(f∘α)∘i. Hence we have established the naturality requirement of condition (3).

If g : BG_{p}^{∧} →^{≃} BG′_{p}^{∧} then P and P′ are isomorphic. Consider the composite

Inj(P, P) ⟶^{incl_{∗}} Inj(P, G) ⟶_{≈}^{g_{∗}} Inj(P, G′).

Let f be the image of the identity under the above composite, and define P′ = im f. If Q isomorphic to a subgroup of P then by the naturality of g_{∗} we have the following commutative diagram,

Inj(Q, P)
incl_{∗} ↙  ↘ ^{f_{∗}}
Inj(Q, G)  ⟶^{≈}  Inj(Q, G′).

Let Q_{1} and Q_{2} be two subgroups of P isomorphic to Q. Suppose Q_{1} and Q_{2} are not P -conjugate. Let ι_{Q_{i}} : Q_{i} → G denote inclusion. Then Q_{1} and Q_{2} are G-conjugate if and only if incl_{∗}(ι_{Q_{1}}) = incl_{∗}(ι_{Q_{2}}) if and only if f_{∗}(ι_{Q_{1}}) = f_{∗}(ι_{Q_{2}}) if and only if f(Q_{1}) and f(Q_{2}) are G′-conjugate. Hence f satisfies (i).

For Q_{1} a subgroup of P isomorphic to Q, we have incl_{∗}(Out(Q_{1})/W_{P} (Q_{1})) = Out(Q_{1})/W_{G}(Q_{1}) and f_{∗}(Out(Q_{1})/W_{P} (Q_{1})) = Out(f(Q_{1}))/W_{G′}(f(Q_{1})). Thus

UNSTABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧} 25

W_{G}(Q_{1}) and W_{G′}(f(Q_{1})) are conjugate in Out(Q). Hence f satisfies condition (ii). □

Example 4.1. Consider the semidihedral group of order 16, SD_{16}. Up to conjugation there are only two subgroups Q where Out(Q) is not a 2-group. One is isomorphic to Z/2×Z/2 and the other is isomorphic to the quaternion group Q_{8}. In both cases Out(Q) ≈ Σ_{3}, the symmetric group on 3 letters.

If we examine the Mathieu group M_{11} and the special linear group SL_{3}(F_{3}) then both have Sylow 2-subgroups isomorphic to SD_{16} with W_{G}(Q) ≈ Σ_{3} for G ≈ M_{11} or SL_{3}(F_{3}) and Q ≈ Z/2 × Z/2 or Q_{8}. Furthermore, in each group there is only one conjugacy class for each isomorphism class of p-subgroups. Therefore, (BM_{11})_{2}^{∧} and BSL_{3}(F_{3})_{2}^{∧} are homotopy equivalent. This example answers a question of Fred Cohen.

Remark 4.2. The relatively uncomplicated local structure of such 2-groups as the dihedral, generalized quaternion, and semidihedral groups force the conditions of Corollary 1.2 to be satisfied even if BG_{2}^{∧} and BG′_{2}^{∧} are only known to be stably homotopy equivalent, see [M] for details. Therefore, if G and G′ are two finite groups with isomorphic Sylow 2-subgroups and these Sylow 2- subgroups are isomorphic to a dihedral, a generalized quaternion, or a semidihedral group, then BG_{2}^{∧} and BG′_{2}^{∧} are unstably homotopy equivalent if and only if they are stably homotopy equivalent. In general this is not true; see [MP3] for an example.

26 JOHN MARTINO AND STEWART PRIDDY

5. Proof of Theorem 1.4.

(i) ⇒ (ii). Let NI(Q, G) = {non-injective homomorphisms h : Q → G}/G. Then Rep(Q, G) = Inj(Q, G) ⊔ NI(Q, G). From (1.3), f_{∗} : Rep(Q, G) → Rep(Q, G′) restricts to an injection f_{∗} : NI(Q, G) → NI(Q, G′). It suffices to show this map is bijective. Suppose h′ : Q → G′ belongs to NI(Q, G′) and let R = h′(Q) < G′. By hypothesis f_{∗} : Rep(R, G) → Rep(R, G′) is bijective, hence there exists k : R → G such that f_{∗}(k) = (R ↪ G′). Hence f_{∗}(Q ↪ R ⟶^{k} G) = h′.

(ii) ⇒ (iii). This is a corollary of Theorem 4.9 in [MP3].

A more direct proof may be obtained by showing that f^{∗} induces a bijection of stable elements. Consider the inclusion i_{Q} : Q ↪ G. By (ii), f_{∗}(i_{Q}) is a conjugacy class of monomorphisms Q → G′. Thus for Q = P, a Sylow p-subgroup of G, we can choose an isomorphism h_{P} : P → P′, P′ a Sylow p-subgroup of G′ such that

BP ⟶^{Bh_{P}} BP′
i_{P} ↓  ^{i_{P′}} ↓
BG_{p}^{∧} ⟶^{f} BG_{p}^{′∧}

commutes up to homotopy. For a general Q < P we set h_{Q} = h_{P} |_{Q} obtaining an isomorphism h_{Q} : Q → Q′.

By (ii) the conjugacy classes of p-subgroups of G correspond bijectively to those of G′. Thus it suffices for each Q < P to show h_{Q} induces a bijection of Weyl groups. For this we consider the diagram:

BQ ⟶^{Bh_{Q}} BQ′ ⟶^{Bi_{Q′}} BG_{p}^{′∧}
Bc_{g} ↓ ^{r}↓ ^{id}↓
BQ ⟶^{Bh_{Q}} BQ′ ⟶^{Bi_{Q′}} BG_{p}^{′∧}

where g ∈ N_{G}(Q) and r = B(h_{Q}c_{g}h_{Q}^{−1}). Since the first diagram commutes by definition and the outer diagram commutes up to homotopy, the second also commutes

UNSTABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧} 27

up to homotopy. Thus r ≃ Bc_{g′} for some g′ ∈ N_{G′}(Q′). A similar argument shows that W_{G}(Q) corresponds to W_{G′}(Q′) under h_{Q}.

(iii) ⇒ (iv). This is a basic property of completion [BK; I,5.5].

(iv) ⇒ (i). (1.3). □

6. On the nature of homotopy equivalences.

As an application of Theorem 1.1 we answer in the negative a question concerning the nature of homotopy equivalences between classifying spaces. Let G and H be two finite groups such that BG_{p}^{∧} and BH_{p}^{∧} are homotopy equivalent. The most obvious way to construct a homotopy equivalence between two classifying spaces is an alternating sequence of homotopy equivalences induced from homomorphisms.

BG_{p}^{∧} ⟶^{Bf_{0}} BG_{1}_{p}^{∧} ⟵^{Bf_{1}} BG_{2}_{p}^{∧} ⟶^{Bf_{2}} BG_{3}_{p}^{∧} ⟵^{Bf_{3}} ⋯ ⟶^{Bf_{n}} BH_{p}^{∧}

In this section we will use the groups P SL_{2}(F_{q}) to produce a homotopy equivalence which is not of this form. We recall |P SL_{2}(F_{q})| = q(q^{2} − 1)/2.

Theorem 6.1. Let q = p^{m} and q′ = (p′)^{n}, where p, p′ are distinct odd primes, such that 16 is the largest power of 2 dividing |P SL_{2}(F_{q})| and |P SL_{2}(F_{q′})|. Then there is a homotopy equivalence

BP SL_{2}(F_{q})_{2}^{∧} ≃ BP SL_{2}(F_{q′})_{2}^{∧}

but there does not exist an alternating sequence of homotopy equivalences induced by homomorphisms between the two classifying spaces.

Before we proceed with our analysis we state two theorems, one concerning the subgroups of P SL_{2}(F_{q}) and the other concerning groups with dihedral Sylow 2subgroups.

Theorem 6.2 (Dickson) [H]. The group P SL_{2}(F_{q}), q = p^{n}, p prime, possesses only the following subgroups:

(1) Elementary abelian p-groups.

(2) Cyclic groups of order m, with m | (q ± 1)/k, where k = (q − 1, 2).

UNSTABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧} 29

(3) Dihedral groups of order 2m with m as in (2).

(4) Alternating groups A_{4} for p > 2 or p = 2 and n ≡ 0 mod 2.

(5) Symmetric groups Σ_{4} for q^{2} − 1 ≡ 0 mod 16.

(6) Alternating groups A_{5} for p = 5 or q^{2} − 1 ≡ 0 mod 5.

(7) Semidirect products of elementary abelian groups of order p^{r} by cyclic groups of order s, where s | p^{r} − 1 and s | q − 1.

(8) Groups P SL_{2}(F_{p^{t}}) for t | n and P GL_{2}(F_{p^{t}}) for 2t | n.

Let O(G) be the largest odd-order normal subgroup of G.

Theorem 6.3 (Gorenstein-Walter) [G]. If G is a finite group with dihedral Sylow 2-subgroups, then G/O(G) is isomorphic to either

(1) a subgroup of P ΓL_{2}(F_{q}) containing P SL_{2}(F_{q}), q odd,

(2) the alternating group A_{7}, or

(3) a Sylow 2-subgroup of G. In particular if G is simple, it is isomorphic to either P SL_{2}(F_{q}), q odd, q > 3, or to A_{7}.

Remark. P ΓL_{2}(F_{q}) = Aut(P SL_{2}(F_{q})). It is isomorphic to the semidirect product of P GL_{2}(F_{q}) by a cyclic group A of order n, where q = p^{n}, p prime. The automorphisms determined by A arise from corresponding ones of SL_{2}(F_{q}) that are induced from automorphisms of the field F_{q}.

The only finite groups G with BG_{2}^{∧} homotopy equivalent to BP SL_{2}(F_{q})_{2}^{∧} are either ones where G/O(G) ≈ P SL_{2}(F_{r}) ⋊ B, with B ≤ A, B of odd order, and A as in the above remark, or G/O(G) ≈ A_{7}. This follows from Theorem 1.1, Theorem 6.3, the requirement that H^{1}(BG; F_{2}) = 0, and [MP1,Thm. 1]. The only subgroups of P SL_{2}(F_{q}) with classifying spaces homotopy equivalent to BP SL_{2}(F_{q})_{2}^{∧}

30 JOHN MARTINO AND STEWART PRIDDY

are P SL_{2}(F_{r}), where F_{r} ⊆ F_{q}. This follows from Theorem 1.1, Theorem 6.2, and [HK;MiP;MP1,Thm. 1].

Definition 6.4. If G/O(G) ≈ P SL_{2}(F_{q}) ⋊ B, B an odd cyclic group, then call G P SL_{2}-like. Let the characteristic of G, char(G), be the characteristic of the field F_{q}.

Clearly char(G) is well defined since any normal subgroup of G/O(G) isomorphic to P SL_{2}(F_{r}) must intersect the above P SL_{2}(F_{q}) and hence be contained therein. By Theorem 6.2 the characteristic of F_{r} equals the characteristic of F_{q}.

Lemma 6.5. If G is P SL_{2}-like and there is a non-zero homomorphism

ϕ : P SL_{2}(F_{q}) → G,

q > 3, then char(G) equals the characteristic of F_{q}.

Proof. P SL_{2}(F_{q}) simple implies that ϕ is a monomorphism. If G ≈ P SL_{2}(F_{r}) ⋊ B then (im ϕ) ∩ P SL_{2}(F_{r}) is a normal subgroup of im ϕ, hence equal. Theorem 6.2 now implies the lemma. □

We now state a theorem due independently to L. Evens and S. Jackowski, which we will use to deal with the problem of O(G) in Theorem 6.3.

Theorem 6.6 (Evens, Jackowski) [J]. For two finite groups G and H and a homomorphism f : G → H such that f induces an isomorphism f^{∗} : H^{∗}(BH; F_{p}) → H^{∗}(BG; F_{p}), then ker f is a p′-subgroup and im f has index prime to p in H.

Remark. If f induces an isomorphism of mod-p cohomology then f induces a homotopy equivalence BG_{p}^{∧} → BH_{p}^{∧}. It follows from Theorem 6.6 that BG_{p}^{∧} is homotopy equivalent to B(im f)_{p}^{∧} since ker f is a p′-group.

UNSTABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧} 31

We may therefore assume that a homomorphism inducing a homotopy equivalence is either an epimorphism or a monomorphism. Let ϕ be such a homomorphism of P SL_{2}-like groups. If ϕ is an epimorphism then ϕ induces an isomorphism \overline{ϕ} : G/O(G) → H/O(H) since ker ϕ ≤ O(G) by Theorem 6.6, thus char(G) = char(H). If ϕ is a monomorphism, then ϕ induces a monomorphism ϕ̂ : G/(G ∩ O(H)) → H/O(H). If H/O(H) ≈ P SL_{2}(F_{q}) ⋊ B, then the inclusion G/(G∩O(H))∩P SL_{2}(F_{q}) ↪ H/O(H) induces a homotopy equivalence, since they have isomorphic Sylow 2-subgroups and the first cohomology groups are both zero [MP1]. By Theorem 6.2 G/(G ∩ O(H)) ∩ P SL_{2}(F_{q}) ≈ P SL_{2}(F_{r}), F_{r} ⊆ F_{q}. Thus we have

P SL_{2}(F_{r}) ↪ G/(G ∩ O(H)) ↪ H/O(H).

Lemma 6.5 implies that if ϕ is a monomorphism then char(G) = char(H). Therefore, we have established that an alternating sequence of homomorphisms of P SL_{2}- like groups inducing homotopy equivalences forces all the groups in the sequence to have the same characteristic.

Proof of Theorem 6.1. Either group has Sylow 2-subgroups isomorphic to D_{16}. Thus if Q is a 2-subgroup then Q is isomorphic to either e, Z/2, Z/4, Z/8, Z/2 × Z/2, D_{8}, or D_{16}. For either group and all proper 2-subgroups Q ≉ Z/8 the Out(Q)-set Out(Q)/W (Q) is trivial [M], where W (Q) is the Weyl group defined in Section 1. Since the Sylow 2-subgroups are isomorphic to D_{16} and Out(D_{16}) is a 2-group it follows that for either group W (P) = e, where P is a Sylow 2-subgroup. If Q ≈ Z/8 then Out(Z/8) ≈ Z/4 and W (Q) ≈ Z/2. For any monomorphism Z/8 → D_{16} the resulting function Out(D_{16}) → Out(Z/8)/(Z/2) is the same. From which it follows

32 JOHN MARTINO AND STEWART PRIDDY

that the diagram below commutes,

Out(D_{16})  ⟶  Out(D_{16})
↓  ↓
Out(Z/8)/(Z/2) ⟶ Out(Z/8)/(Z/2).

Thus the naturality condition of Theorem 1.1 is satisfied, so there exists a homotopy equivalence BP SL_{2}(F_{q})_{2}^{∧} ⟶^{≃} BP SL_{2}(F_{q′})_{2}^{∧}. However, char(P SL_{2}(F_{q})) = p ≠ p′ = char(P SL_{2}(F_{q′})). □

Remark. One can also show that there exist homotopy equivalent classifying spaces of finite groups with Sylow 2-subgroups isomorphic to D_{8} such that there does not exist an alternating sequence homomorphism-induced homotopy equivalences. One shows this by choosing the characteristics large enough so that there cannot be a homomorphism in or out of A_{7} which induces a homotopy equivalence, and then proceeding as in the case of D_{16}. For example, there is a homotopy equivalence between BP SL_{2}(F_{41})_{2}^{∧} and BP SL_{2}(F_{89})_{2}^{∧} not induced by an alternating sequence of homomorphism-induced homotopy equivalences.

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John Martino  Stewart Priddy
Department of Mathematics  Department of Mathematics
University of Virginia  Northwestern University
Charlottesville, VA 22903  Evanston, IL 60208