ON SPACES OF SELF HOMOTOPY EQUIVALENCES OF p-COMPLETED CLASSIFYING SPACES OF FINITE GROUPS AND

HOMOTOPY GROUP EXTENSIONS

CARLOS BROTO AND RAN LEVI

1. introduction

Let G and π be groups and let p be a prime number. A mod-p homotopy group extension of π by G is a fibration with base space Bπ_{p}^{∧} and fibre BG_{p}^{∧}, where (−)_{p}^{∧} denotes the Bousfield-Kan p-completion functor.

In this paper we study homotopy extensions of finite groups. Thus if X is the total space in a mod-p homotopy group extension of π by G, where both π and G are finite, we shall call X a finite homotopy group extension. As one would expect from a classification problem of a class of fibrations, the project involves studying spaces of homotopy equivalences of the fibres under consideration. This problem forms one of the main cores of the paper and is of independent interest. Indeed, spaces of self homotopy equivalences for compact connected simple Lie groups are well understood as a byproduct of [8]. Our results are partially the finite group analogue.

Let p be a prime and let G be a finite group. Then O_{p′}G is defined to be the maximal normal subgroup of G of order prime to p. Define the mod-p′ reduction of G to be the quotient G/O_{p′}G. If O_{p′}G is trivial then G is said to be p′-reduced. Notice that the natural projection from G to G/O_{p′}G induces a mod-p homology isomorphism and hence a p-local homotopy equivalence on classifying spaces.

For an arbitrary space X, let Aut(X) denote topological monoid of all self homotopy equivalences of X. Let SAut(X) denote the identity component of Aut(X) and let Out(X) denote the group of components π_{0}(Aut(X)). By general theory, fibre homotopy equivalence classes of Mod-p homotopy group extensions of π by G are in 1-1 correspondence with the set of homotopy classes of maps from Bπ_{p}^{∧} to B Aut(BG_{p}^{∧}). The next theorem describes the homotopy type of the identity component SAut(BG_{p}^{∧}).

Theorem 1.1. Let G be a finite group. Then SAut(BG_{p}^{∧}) ≃ BZ(G/O_{p′}G).

Recall that a group G is said to be p-perfect if its first mod-p homology group is trivial and p-superperfect if in addition its second mod-p homology vanishes. For every discrete group G there exists a maximal normal p-perfect subgroup O^{p}G < G and BO^{p}G_{p}^{∧} is the 1-connected cover of BG_{p}^{∧}. Furthermore, if G is itself p-perfect then there is a central extension U^{p}G of G by H_{2}(BG; ℤ_{(p)}) and BU^{p}G_{p}^{∧} is the 2-connected cover of BG_{p}^{∧}. For simplicity of notation we shall denote U^{p}(O^{p}G) by U^{p}G, even if G itself is not p-prefect. As corollaries of Theorem 1.1 we obtain

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

Key words and phrases. Classifying spaces, self equivalences, finite groups.

C. Broto is partially supported by DGICYT grant PB97-0203.

Corollary 1.2. Let X be a mod-p homotopy extension of π by G. Then there is a homotopy equivalence

X⟨2⟩ ≃ BU^{p}π_{p}^{∧} × BU^{p}G_{p}^{∧},

where X⟨2⟩ is the 2-connected cover of X.

Corollary 1.3. Assume π is p-perfect and let X be a mod-p homotopy extension of π by G. Then, there is a homotopy equivalence

X ≃ BH_{p}^{∧}

where H is an extension of π by G with trivial action π ^{}Out(G).

Corollary 1.4. Assume π is p-superperfect and let G be any finite group. Then every mod-p homotopy extension of π by G is trivial.

The general case might be easily reduced to the case in which the base is the classifying space of a finite p-group π. In fact, if π is not a p group, then there is a fibration BO^{p}(π)_{p}^{∧} ^{}Bπ_{p}^{∧} ^{}Bπ/O^{p}(π), where π/O^{p}(π) = π_{1}(Bπ_{p}^{∧}) is a pgroup. Then, if X is a mod-p homotopy extension of π by G, one has a diagram of fibrations

BG_{p}^{∧} = = = = BG_{p}^{∧}

[figure: Y; X; Bπ/O^{p}(π)]

[figure: BO^{p}(π)_{p}^{∧}; Bπ_{p}^{∧}; Bπ/O^{p}(π); ‖]

Now, Corollary 1.3 applies to the fibration in the left column and Y ≃ BH_{p}^{∧}, hence X is expressed in the middle row fibration as a mod-p homotopy groups extension of a p-group, π/O^{p}(π), by a group H.

We now turn to the group of components Out(BG_{p}^{∧}). There is an obvious group homomorphism

γ_{G} : Out(G) ^{}Out(BG_{p}^{∧}),

which, as we shall see below, is not generally an isomorphism. A different approximation of the group Out(BG_{p}^{∧}) is given as follows. Let C_{p} denote the category whose objects are finite p-groups and whose morphisms are conjugacy classes of monomorphisms π ^{}π′. For a finite group G, let I_{G} : C_{p}^{op} ^{}Sets denote the functor which takes a finite p-group π to the set Inj(π, G) of conjugacy classes of monomorphism π ^{}G and which takes a morphism π′ ^{ρ}^{}π in C_{p} to the induced map Inj(π, G) ^{}Inj(π′, G). The set of all natural equivalences of I_{G} forms a group under composition, which we denote by Aut(I_{G}). Let O_{p}^{c}(G) denote the orbit category of p-centric subgroups of G. Let Z : O_{p}^{c}(G) ^{}Ab denote the contravariant functor which associate with an orbit G/P the center Z(P).

Theorem 1.5. Let G be a finite group. Then there is an exact sequence

0 ^{}lim^{1}_{O_{p}^{c}(G)^{op}} Z ^{}Out(BG_{p}^{∧}) ^{ψ_{G}}^{}Aut(I_{G}).

Furthermore, the obstruction for ψ_{G} being onto is an element of lim^{2}_{O_{p}^{c}(G)^{op}} Z. In partic-

ular if both higher limits vanish then ψ_{G} is an isomorphism.

The general behavior of the maps γ_{G} and ψ_{G} is not well understood. One class of examples where γ_{G} is always an isomorphism is given by groups G, whose Sylow p-subgroup is normal. However, different groups with the same mod-p homotopy type may well have different outer automorphism groups (e.g. (BM_{11})_{2}^{∧} ≃ BSL_{3}(3)_{2}^{∧} but ℤ/2 ≅ Out(SL_{2}(3)) ≠ Out(M_{11}) = {1}) and so γ_{G} cannot be an isomorphism in general. The map ψ_{G} is known not to be a monomorphism in general as the next theorem demonstrates, but there is no known example where ψ_{G} fails to be an epimorphism.

Theorem 1.6. Let q = 3^{2^{s}}, s ≥ 1. Let G = SL_{2}(q) and let H = P SL_{2}(q). Then

1. There is a commutative diagram of isomorphisms

Out(G) \overline{^{≅}} Out(H)

[figure: Out(BG_{2}^{∧}); ≅; ≅; Out(BH_{2}^{∧}); ≅]

and Out(G) ≅ ℤ/2ℤ × ℤ/2^{s}ℤ.

2. lim^{i}_{O_{p}^{c}(G)} Z = 0 for all i > 0. Thus Out(BH_{2}^{∧}) ≅ Aut(I_{H}).

3. lim^{i}_{O_{p}^{c}(H)} Z ≅ ℤ/2ℤ if i = 1, but vanishes otherwise. Thus there is a non-split short

exact sequence

0 ^{}ℤ/2ℤ ^{}Out(BH_{2}^{∧}) ^{}Aut(I_{H}) ^{}0.

A complete identification of Out(BG_{p}^{∧}) for an arbitrary finite group G will appear in a future paper joint with R. Oliver.

The authors would like to express their warm gratitude to Bill Dwyer for sharing with them his insightful observations. We also thank Bob Oliver for many useful discussions and for his continuing interest in the project. Finally both authors are grateful to the Centre de Recerca Matemtica in Barcelona for allowing them to meet frequently enough to complete this work.

2. The Space Map(Bπ, BG_{p}^{∧})

In this section we record the following

Proposition 2.1. Let π be a finite p-group and let G be a finite group. Then Map(Bπ, BG_{p}^{∧}) is p-complete and the natural map

Map(Bπ, BG) ^{}Map(Bπ, BG_{p}^{∧}) , (1)

induces a mod-p homology isomorphism.

The proposition is known to the experts, but as we are not aware of a suitable reference, a proof is included here.

Proof. The case where π is an elementary abelian p-group is due to Lannes [9]. By induction we may thus assume the statement of the proposition for every p-group κ of order less than p^{n}.

Let π a group of order p^{n}. There is an extension

κ ^{}π ^{}V

where V is a non-trivial elementary abelian p-group. The group V acts on B̃κ = Eπ/κ and one has a homotopy equivalence

Map(Bπ, BG_{p}^{∧}) ≃ Map(B̃κ, BG_{p}^{∧})^{hV} .

Recall that for a general V -space X, the homotopy fixed points space is defined as the space of V -equivariant maps Map_{V} (EV, X) and this can be reinterpreted as the space of sections of the projection X_{hV} ^{}BV. In fact, using the action of BV on Map(BV, X_{hV}) one obtains in general a homotopy equivalence over BV

BV × X^{hV} ^{≃}^{}Map(BV, X_{hV})¯1

where Map(BV, X_{hV})¯1 denotes the set of components of Map(BV, X_{hV}) over the component of the identity Map(BV, BV)_{1} ≃ BV.

In our case we take X to be either Map(B̃κ, BG_{p}^{∧}) or Map(B̃κ, BG). Both spaces are related by the induction hypothesis, and after taking homotopy quotients, we have

(2) Map(B̃κ, BG_{p}^{∧})_{hV} ≃ ( Map(B̃κ, BG)_{p}^{∧})_{hV} ≃ ( Map(B̃κ, BG)_{hV} )_{p}^{∧} .

Now, notice that Map(B̃κ, BG)_{hV} has the homotopy type of a union of a finite number of classifying spaces of finite groups. Hence we have homotopy equivalences over BV

(3) BV × Map(B̃κ, BG_{p}^{∧})^{hV} ≃ Map (BV, Map(B̃κ, BG_{p}^{∧})_{hV} )¯1 ≃

[ Map (BV, Map(B̃κ, BG)_{hV} )¯1]_{p}^{∧} ≃ [BV × Map(B̃κ, BG)^{hV} ]_{p}^{∧} ,

from which we obtain Map(Bπ, BG_{p}^{∧}) ≃ Map(Bπ, BG)_{p}^{∧}.

Notice that Equation (2) implies that Map(B̃κ, BG_{p}^{∧})_{hV} ≃ (Map(B̃κ, BG_{p}^{∧})_{hV})_{p}^{∧} and the same arguments as above yield

Map(Bπ, BG_{p}^{∧}) ≃ Map(Bπ, BG_{p}^{∧})_{p}^{∧}.

Hence Map(Bπ, BG_{p}^{∧}) is p-complete. This finishes the proof of the proposition.

3. The Topological Monoid SAut(BG_{p}^{∧})

In this section we prove the following

Theorem 3.1. Let G be a finite group. Then

SAut(BG_{p}^{∧}) ≃ BZ(G/O_{p′}G),

where O_{p′}G is the maximal normal subgroup of G of order prime to p.

Proof. Notice that since G/O_{p′}G is p′-reduced, its center is automatically an abelian p-group. Also BG_{p}^{∧} ≃ B(G/O_{p′}G)_{p}^{∧} and so SAut(BG_{p}^{∧}) ≃ SAut(B(G/O_{p′}G)_{p}^{∧}). Thus to prove the theorem it suffices to show that if G is p′-reduced then SAut(BG_{p}^{∧}) ≃ BZ(G).

This is done in three steps. In Proposition 3.6 below it is shown that SAut(BG_{p}^{∧}) has at most one non-trivial homotopy group

π_{1}(SAut(BG_{p}^{∧}), id) = L ,

where L is a finite p-group and furthermore, that there is a monomorphism i : L ^{}G such that Z(G) ≤ i(L) ≤ Z(π), where π is any Sylow p-subgroup of G containing L. Next, identify L with its image i(L) < G. In Proposition 3.7 it is proved that the inclusion of the centralizer of L in G, C_{G}(L) ≤ G, induces an isomorphism in mod-p cohomology; that is to say L is p-cohomologically central in G. Finally one uses a theorem of Mislin, stated below, which shows that L is central in G. Since L ≥ Z(G), the two subgroups have to coincide, and therefore SAut(BG_{p}^{∧}) ≃ BZ(G).

Theorem 3.2 (Mislin [12]). Let G be a finite p′-reduced group. Then, every p-coho-mologically central subgroup of G is central.

Remark 3.3. This statement is not explicit in [12] but the necessary arguments are contained in there. We sketch here the proof for convenience of the reader. By definition, if P is p-cohomologically central, then the inclusion of its centralizer C_{G}(P) in G induces a mod-p cohomology isomorphism and therefore an equivalence of the corresponding Frobenius categories of p-subgroups. In particular C_{G}(P) controls pfusion in G and according to the Z^{∗}-theorem this implies C_{G}(P) = G, because G is p′-reduced. It might be worth pointing out that the Z^{∗}-theorem for finite groups depends on the classification of finite simple groups [3]

For a finite group G and a collection C of subgroups of G, which is closed under conjugacy, the C-orbit category is the category whose objects are the G-sets given by left cosets G/H with H ∈ C and whose morphisms are given by all G-maps. Notice that the action of G on a coset is given by g′(gH) := g′gH. Thus any morphism φ from G/H to G/K in the orbit category is determined by φ(H) = gK for some g ∈ G, which conjugates H into K. Moreover this element g is unique up to a right shift by an element of K. Thus each map φ : G/H ^{}G/K gives rise to a unique faithful representation of H in K (a conjugacy class of monomorphisms). If g ∈ G conjugates H to K then g^{−1} conjugates C_{G}(K) to C_{G}(H). Furthermore, if g_{1} and g_{2} represent the same morphism G/H ^{}G/K then conjugation by g_{1}^{−1} and g_{2}^{−1} coincides on C_{G}(K). Recall the following

Definition 3.4. A p-subgroup π of a finite group G is said to be p-centric if

C_{G}(π) ≅ Z(π) × W,

where W is a group of order prime to p. Let O_{p}^{c}(G) denote the full subcategory of the orbit category whose objects are orbits of p-centric subgroups of G.

Our main tool is a homology decomposition theorem which expresses the mod-p homotopy type of BG by means of classifying spaces of p-centric subgroups. The technique is originally due to Jackowski, McClure and Oliver [7, 8]. A very coherent account of these methods by Dwyer, with some significant improvements and unification of methods appears in [4, 5]. The following is a particular case, which is all we need here. Let ϕ_{G} : O_{p}^{c}(G) ^{}G − Sets be the functor associating with an orbit G/H the G-set given by G/H itself. For a G-space X let X_{hG} denote the homotopy orbit space X ×_{G} EG, where EG is a free contractible G space, thus, ϕ_{G}(G/H)_{hG} = G/H ×_{G} EG ≃ BH

Theorem 3.5. Let G be a finite group. Then the natural map

Φ_{G}: hocolim_{O_{p}^{c}(G)}(ϕ_{G})_{hG} ^{}BG

induces a mod-p homology isomorphism.

Thus one has for any finite group G a homotopy equivalence

(4) Map(BG_{p}^{∧}, BG_{p}^{∧}) ≃ holim_{O_{p}^{c}(G)} Map ((ϕ_{G})_{hG}, BG_{p}^{∧}) .

Proposition 3.6. Let G be a finite p′-reduced group and π a p-Sylow subgroup of G. Then

π_{i}(SAut(BG_{p}^{∧})) = \begin{cases}L & i=1 \\ 0 & otherwise \end{cases}

where L is an abelian p-group which can be identified with a subgroup L′ < G such that Z(G) ≤ L′ ≤ Z(π).

Proof. We write SAut(BG_{p}^{∧}) = Map(BG_{p}^{∧}, BG_{p}^{∧})_{id} and then use the Bousfield-Kan spectral sequence for computing the homotopy groups of a homotopy limit associated to equation (4)

E_{2}^{s,t} = lim^{s}_{O_{p}^{c}(G)} π_{t}(Map((G/H)_{hG}, BG_{p}^{∧})) ⇒ π_{t−s}(Map(BG_{p}^{∧}, BG_{p}^{∧})).

The differentials in this spectral sequence have the form d_{r} : E_{r}^{s,t} ^{}E_{r}^{s+r,t+r−1}. Take the identity map of BG_{p}^{∧} as a base point. It restricts to the standard inclusion

inc: (G/H)_{hG} = BH ^{}BG_{p}^{∧}

for any centric p-subgroup H < G. One has

Map((G/H)_{hG}, BG_{p}^{∧})_{inc} ≃ Map(BH, BG_{p}^{∧})_{inc} ≃ (Map(BH, BG)_{inc})_{p}^{∧} ≃ BC_{G}(H)_{p}^{∧} ≃ BZ(H).

Let Z : O_{p}^{c}(G) ^{}Ab denote the contravariant functor which associates to each orbit G/H the center Z(H). Then, for our choice of base point, the E_{2} term of the spectral sequence has the form

E_{2}^{s,t} ≅ lim^{s}_{O_{p}^{c}(G)} π_{t}(Map((G/H)_{hG}, BG_{p}^{∧})_{inc}) = \begin{cases}lim^{s}_{O_{p}^{c}(G)} Z , & t = 1 \\ 0 , & otherwise. \end{cases}

Hence the only relevant value is obtained in the spectral sequence for s = 0. Consequently π_{i}(SAut(BG_{p}^{∧})) = 0 if i > 1 and

π_{1}(SAut(BG_{p}^{∧})) = L := lim^{0}_{O_{p}^{c}(G)} Z.

We proceed by proving the properties of L claimed in the proposition. By definition the inverse limit is the kernel of the map

ϕ : ∏_{G/H∈Obj(O_{p}^{c}(G))} Z(H) ^{}∏_{φ∈Mor(O_{p}^{c}(G))} Z(range(φ))

defined by its projection to the φ component as x_{P} − BC_{G}(φ)(x_{Q}), where φ : P ^{}Q, x_{P} ∈ Z(P) and x_{Q} ∈ Z(Q). Notice that for every G/P ∈ O_{p}^{c}(G), Z(P) > Z(G), since Z(G) is a p-group. Since all maps in O_{p}^{c}(G) are induced by translation in G, the induced maps on centralizers are given by sub conjugation and it follows at once that Z(G) is a subgroup of L. An obvious embeding is given by including Z(G) diagonally in the domain of ϕ.

Finally, notice that if x̄ ∈ L is represented by a sequence of elements of G indexed by objects of O_{p}^{c}(G) and x_{P} is the projection of x̄ to the component of P, then x_{P} is invariant under the action of the normalizer N_{G}(P) and hence is central in P. Let π be a Sylow p-subgroup of G containing P. Then x_{P} = x_{π} and so x_{π} is central in π. Notice that projection from L to the component of each one of the groups involved is a monomorphism. Thus L can be embedded in the center of each Sylow p-subgroup. In particular L is an abelian p-group and the proof is complete.

The inclusion of L in G obtained in the above Proposition factors through a p-Sylow subgroup π of G. This is represented up to conjugation by the composition

BL ^{}BZ(π) ≃ Map((G/π)_{hG}, BG_{p}^{∧}) ^{ev}^{}BG_{p}^{∧}

and therefore by

BL ≃ Map(BG_{p}^{∧}, BG_{p}^{∧})_{id} ^{ev}^{}BG_{p}^{∧}

This last map is canonically defined. However a representing homomorphism is only well defined up to conjugation. The next Proposition however shows that L is pcohomologically central in G and hence central in G, by Mislin theorem 3.2. Thus L coincides with all its conjugates. Hence we conclude that the identification of L = π_{1} Map(BG_{p}^{∧}, BG_{p}^{∧})_{id} with a subgroup of G is canonically induced by evaluation.

In the next Proposition identify the group L with its image L′ under some appropriate embedding.

Proposition 3.7. Fix an embeding of L in G and identify L with its image. Then the mod-p cohomology restriction

Res : H^{∗}(BG) ^{}H^{∗}(BC_{G}(L))

is an isomorphism.

Proof. Let π be a Sylow p-subgroup of G containing L. Then L < Z(π) and so π < C_{G}(Z(π)) < C_{G}(L). Thus the restriction map is a monomorphism.

Next notice that there is a homotopy equivalence

ϕ : BL ^{}SAut(BG_{p}^{∧}),

and a homotopy commutative diagram

BL \overline{^{ϕ}} SAut(BG_{p}^{∧}) = Map(BG, BG_{p}^{∧})_{id}

[figure]

BZ(π) ≃ Map(Bπ, BG)_{Bi}^{^{Bi^{∗}}}

where i : π ^{}G is the inclusion. Taking adjoints we get a commutative diagram

BL × BG \overline{}BG_{p}^{∧}

[figure]

BL × Bπ^{^{1×Bi} }

Taking adjoints the other way and writing BC_{G}(L)_{p}^{∧} for Map(BL, BG_{p}^{∧}) gives a commutative diagram

BG \overline{}BC_{G}(L)_{p}^{∧}

[figure]

Bπ^{^{Bi} }

Let ψ : BG ^{}BC_{G}(L)_{p}^{∧} denote the resulting map. By commutativity this map induces a monomorphism on mod-p cohomology. Composed with the restriction BC_{G}(L) ^{}BG we get a self map of BC_{G}(L) which induces a cohomology monomorphism. Since H^{∗}(BC_{G}(L)) is a graded group of finite type any monomorphism is an isomorphism. It follows then that the restriction map is an epimorphism and thus an isomorphism and the proof is complete.

Corollary 3.8. For every finite group G the space B Aut(BG_{p}^{∧}) is a 2-stage Postnikov tower with homotopy groups in dimensions 1 and 2. In particular

π_{2}(B Aut(BG_{p}^{∧})) = Z(G/O_{p′}G).

Let a mod-p homotopy group extension of π by G be given. Then one has the following commutative diagram of fibrations, where the bottom square is a homotopy pull-back.

BG_{p}^{∧} \overline{}BG_{p}^{∧}

[figure: Y; X]

[figure: BU^{p}π_{p}^{∧}; Bπ_{p}^{∧}]

But the left column is also a mod-p homotopy group extension, where the base space is 2-connected. Since B Aut(BG_{p}^{∧}) is a 2-stage Postnikov tower with homotopy in dimension 1 and 2, the classifying map for this extension is trivial. Hence the extension is split, namely

Y ≃ BU^{p}π_{p}^{∧} × BG_{p}^{∧}.

But Y is the fibre of the composition

X ^{}Bπ_{p}^{∧} ^{}Bπ_{p}^{∧}[2],

where Bπ_{p}^{∧}[2] denotes the second Postnikov section of Bπ_{p}^{∧}. Hence Y and X have the same two connected covers and Corollary 1.2 follows.

Let a finite mod-p homotopy extension of π by G be given, with π a p-perfect group and assume without loss of generality that G is p′-reduced. Then Bπ_{p}^{∧} is simply-connected and hence the classifying map Bπ_{p}^{∧} ^{}B Aut(BG_{p}^{∧}) is trivial on fundamental groups. It follows that the action of π_{1}(Bπ_{p}^{∧}) on BG_{p}^{∧} is trivial and so there exists at least one mod-p homotopy extension of π by G with this action, namely the trivial extension. Hence fibre homotopy equivalence classes of mod-p homotopy extensions of π by G with a trivial action of π_{1}(Bπ_{p}^{∧}) are in 1-1 correspondence with H^{2}(Bπ_{p}^{∧}, Z(G)). Now, ordinary group extensions of π by G with trivial outer action of π on G are classified by H^{2}(π, Z(G)) ≅ H^{2}(Bπ_{p}^{∧}, Z(G)), since π is finite and Z(G) is a p-group. But given any such ordinary group extension, p-completion of the fibration resulting from applying the classifying space construction to the extension remains a fibration. Corollary 1.3 follows.

Finally assume that π is p-superperfect. Then the classifying map

Bπ_{p}^{∧} ^{}B Aut(BG_{p}^{∧})

is trivial. Corollary 1.4 follows.

4. The Group of Components Out(BG_{p}^{∧})

Let G be a finite group. As before we shall assume that G is p′-reduced. In this section we give an approximation for Out(BG_{p}^{∧}). We shall essentially reduce the calculation to the computation of two higher limits. In the case where these limits vanish we shall obtain an identification of Out(BG_{p}^{∧}) with the group of natural equivalences of the functor I_{G} defined in the introduction.

Recall that the category C_{p} has all finite p-groups as objects and conjugacy classes of monomorphisms π ^{}π′ as morphisms. The functor I_{G} : C_{p}^{op} ^{}Sets takes a finite p-group π to the set Inj(π, G) of conjugacy classes of monomorphism π ^{}G and a morphism π ^{ρ}^{}π′ in C_{p}^{op} to the induced map Inj(π, G) ^{}Inj(π′, G).

Lemma 4.1. For any finite group G, there are homomorphisms of groups

δ_{G} : Out(G) ^{}Aut(I_{G})

and

γ_{G} : Out(G) ^{}Out(BG_{p}^{∧}).

Proof. For [α] ∈ Out(G) and a finite p-group π define

δ_{G}([α])_{π} : I_{G}(π) ^{}I_{G}(π)

to be the map taking [f] ∈ I_{G}(π) to [α ∘ f], where α is any automorphism of G representing its class. That δ_{G} is a homomorphism is clear from the definition and naturality follows from functoriality of Inj(π, −).

The definition of γ_{G} is as the map induced by a representative for an outer automorphism by applying it to the second variable in the respective mapping space.

Let G be a finite group and let π be a finite p-group. By Proposition 2.1 there are isomorphisms of sets

π_{0}(Map(Bπ, BG_{p}^{∧})) ≅ π_{0}(Map(Bπ, BG)) ≅ Rep(π, G) .

The following lemma is well known.

Lemma 4.2. Let f : π ^{}G be a homomorphism. Then the following are equivalent:

1. f is a monomorphism

2. H^{∗}Bπ is a finitely generated module over Bf^{∗}(H^{∗}(BG)).

Lemma 4.3. There is a group homomorphism

ψ_{G} : Out(BG_{p}^{∧}) ^{}Aut(I_{G})

Proof. Let [f] ∈ Out(BG_{p}^{∧}) be the homotopy class of some self equivalence

f : BG_{p}^{∧} ^{}BG_{p}^{∧} .

We must present a natural equivalence ψ_{G}(f) of I_{G}. Thus for each π ∈ C_{p} let

ψ_{G}(f)_{π} : I_{G}(π) ^{}I_{G}(π)

be the function which takes the conjugacy class of a monomorphism φ : π ^{}G to the conjugacy class of a homomorphism inducing the composition

Bπ ^{Bφ}^{}BG_{p}^{∧} ^{f}^{}BG_{p}^{∧}

up to homotopy. By the Proposition 2.1 such a class exists. Furthermore, by the Lemma 4.2 it follows that ψ_{G}(f)_{π}[φ] is the conjugacy class of a monomorphism. The definition does not depend on the choice of f or φ, but rather only on their homotopy classes and so ψ_{G}(f)_{π} is well defined.

Naturality of ψ_{G}(f) follows at once by similar considerations. Also, to see that ψ_{G} is a group homomorphism observe that multiplication in both domain and range is given by composition.

Definition 4.4. Let G be a finite group and π a finite p-group. Let Bπ denote any model for the classifying space for π. Define Inj(Bπ, BG_{p}^{∧}) to be the subset of the mapping space Map(Bπ, BG_{p}^{∧}) consisting of all maps which are homotopic to Bf for some monomorphism f : π ^{}G.

Let C_{p}^{G} denote the full subcategory of C_{p} whose objects are all p-subgroups of G. Consider the functor

K_{G} : C_{p}^{G} ^{}Sets

taking a p-subgroup π of G to the set π_{0}(Inj((G/π)_{hG}, BG_{p}^{∧})) and a morphism in C_{p}^{G} to the induced map. Then there is a natural transformation of functors

β : I_{G} ^{}K_{G},

which by Proposition 2.1 is a natural equivalence. Here I_{G} is considered as a functor defined on C_{p}^{G}.

Consider the homology decomposition of Theorem 3.5

Φ_{G} : hocolim_{O_{p}^{c}(G)}(ϕ_{G})_{hG} ^{}BG.

Thus Φ_{G} induces a mod-p homology equivalence and so its p-completion is a homotopy equivalence. Thus one has an isomorphism of monoids

[(hocolim_{O_{p}^{c}(G)^{→}}(ϕ_{G})_{hG})_{p}^{∧}, (hocolim_{O_{p}^{c}(G)^{→}}(ϕ_{G})_{hG})_{p}^{∧}] ≅ [BG_{p}^{∧}, BG_{p}^{∧}] .

The proof of Theorem 1.5 thus amounts to showing that given any natural equivalence α ∈ Aut(I_{G}), the obstruction to existence of a lift of α to a self map α̃ of BG_{p}^{∧} such that ψ_{G}(α̃) = α is an element of lim^{2}_{O_{p}^{c}(G)} Z and if a lift exists then any two differ

by an element of lim^{1}_{O_{p}^{c}(G)} Z.

Lemma 4.5. Let G be a finite p′-reduced group. For any p-subgroup π of G, the following are equivalent

1. π is a p-centric subgroup of G.

2. π is a p-centric subgroup of every p-Sylow subgroup of G that contains π.

Proof. Assume first that π is p-centric in G. That is, C_{G}(π) ≅ Z(π) × W, where W is a p′-group. For any subgroup H of G containing π, Z(π) < C_{H}(π) < C_{G}(π), hence C_{H}(π) ≅ Z(π) × W′, where W′ = C_{H}(π) ∩ W is a p′-groups and so π is p-centric in H.

Conversely, assume that π is p-centric in every Sylow p subgroup of G containing it. Let R be a Sylow p subgroup of C_{G}(π). We must show that R = Z(π). Notice that π · R < G is a p-subgroup. Let S be a Sylow p-subgroup of G containing π · R. Since π is p-centric in S, one has C_{S}(π) = Z(π). Then R < C_{S}(π), since it is contained in S and centralizes π. On the other hand C_{S}(π) = Z(π) is contained in every p-Sylow of C_{G}(π), in particular, it is contained in R and so R = Z(π) as claimed.

Lemma 4.6. Let π be a finite p group and assume that there exists a monomorphism f : π ^{}G, including π in G as a p-centric subgroup. Let [f] denote the conjugacy class of f. Then for any natural equivalence φ ∈ Aut(I_{G}) any representative of the class φ_{π}[f] is a monomorphism t : π ^{}G such that t(π) is p-centric in G.

Proof. First note that if t : π ^{}G is a monomorphism such that t(π) < G is p-centric, then any conjugate of t also includes π in G as a p-centric subgroup. Let t : π ^{}G be any representative for φ_{π}[f]. It suffices to show that t(π) < G is p-centric.

Let S < G be a Sylow p-subgroup containing t(π). Let ι_{S} : S ^{}G and t̄ : π ^{}S denote the respective inclusions, such that t = ι_{S} ∘ t̄.

Since φ is a natural equivalence, there is a commutative diagram

[figure: Inj(S, G); φ_{S}; Inj(S, G); Inj(π, G); t̄^{∗}; φ_{π}; Inj(π, G); t̄^{∗}]

Thus t̄^{∗}[ι_{S}] = [t]. By commutativity and the fact that φ is an equivalence one has t̄^{∗}φ_{S}^{−1}[ι_{S}] = [f]. Since S is a Sylow p-subgroup in G, there exists an automorphism α of S, such that φ_{S}^{−1}[ι_{S}] = [ι_{S} ∘ α] and it follows that ι_{S} ∘ α ∘ t̄ is conjugate to f in G.

Since f includes π in G as a p-centric subgroup, it follows by Lemma 4.5 that α ∘ t̄ includes π as a p-centric subgroup of S. Since α is an automorphism, t̄ includes π in S as a p-centric subgroup. Since this applies to any Sylow subgroup S containing t(π), it follows, again by Lemma 4.5, that t(π) is p-centric in G.

Lemma 4.7. For each φ ∈ Aut(I_{G}) and each p-centric subgroup π < G there is a map

χ_{π}^{φ} : (G/π)_{hG} ^{}BG_{p}^{∧}

such that if f : G/π ^{}G/π′ is a morphism in O_{p}^{c}(G) then the diagram

[figure: (G/π)_{hG}; χ_{π}^{φ}; BG_{p}^{∧}; χ_{π′}^{φ}; (G/π′)_{hG}; f_{hG}] (5)

commutes up to homotopy.

Proof. By the remarks above, we have a natural equivalence of functors

β : I_{G} ^{}K_{G},

where I_{G} is restricted to C_{p}^{G}. Thus each natural equivalence φ of I_{G} induces a natural equivalence of K_{G}, which by abuse of notation, we also denote by φ.

For two p-subgroups π, π′ < G, the set of morphisms Mor_{O_{p}^{c}(G)}(G/π, G/π′) consists of all elements of G conjugating π into π′ modulo the right action of π′. On the other hand the group of inner automorphisms of π′ operates from the right on the set of all injective homomorphisms from from π to π′ and the orbits of this action form the morphism set Mor_{C_{p}^{G}}(π, π′). In other words this is the set of those faithful representations of π in π′ induced by conjugation in G. Hence there is a functor α : O_{p}^{c}(G) ^{}C_{p}^{G}, which sends an orbit of a p-centric G/π to the group π and a morphism in O_{p}^{c}(G) to the induced representation. Let K̃_{G} denote the composition K_{G} ∘ α. Then every natural self equivalence φ of K_{G} induces a self equivalence of K̃_{G}, which again we denote by φ.

For each p-centric π < G let [ι_{π}] ∈ K̃_{G}(G/π) denote the class of the map induced by inclusion. Let

χ_{π}^{φ} : (G/π)_{hG} ^{}BG_{p}^{∧}

be any representative for the homotopy class of φ_{π}[ι_{π}].

Let f : G/π ^{}G/π′ be a morphism in O_{p}^{c}(G). Then, since φ is a natural equivalence we have K̃_{G}(f)∘φ_{π′} = φ_{π} ∘K̃_{G}(f). Homotopy commutativity of Diagram (5) is equivalent to the following equation holding:

K̃_{G}(f)[χ_{π′}^{φ}] = [χ_{π}^{φ}].

But since Diagram (5) commutes up to homotopy if χ_{π}^{φ} and χ_{π′}^{φ} are replaced by ι_{π} and ι_{π′} respectively, we have K̃_{G}(f)[ι_{π′}] = [ι_{π}]. Hence

K̃_{G}(f)[χ_{π′}^{φ}] = K̃_{G}(f)φ_{π′}[ι_{π′}] = φ_{π}K̃_{G}(f)[ι_{π′}] = φ[ι_{π}] = [χ_{π}^{φ}],

as required. This completes the proof.

Lemma 4.7 implies that each natural equivalence φ ∈ Aut(I_{G}) gives rise to a map

E_{φ}^{1} : hocolim_{O_{p}^{c}(G)} ^{(1)}(ϕ_{G})_{hG} ^{}BG_{p}^{∧},

where the superscript on the left hand side means the 1-skeleton of the homotopy colimit.

Let G be a finite group and let π be a p-centric subgroup. For a natural equivalence φ ∈ Aut(I_{G}) let t_{π}^{φ} : π ^{}G denote any choice of a monomorphism such that

B(t_{π}^{φ})_{p}^{∧} : Bπ ^{}BG_{p}^{∧}

is homotopic to χ_{π}^{φ}, under the usual identification of (G/π)_{hG} with Bπ. By Lemma 4.5, t_{π}^{φ} includes π in G as a p-centric subgroup.

For every j ≥ 1, define a functor

Π_{j}^{φ} : O_{p}^{c}(G) ^{}Ab

by

Π_{j}^{φ}(G/π) = π_{i}(Map((G/π)_{hG}, BG_{p}^{∧})_{χ_{π}^{φ}}).

Notice that

Map((G/π)_{hG}, BG_{p}^{∧})_{χ_{π}^{φ}} ≃ Map((G/π)_{hG}, BG_{p}^{∧})_{B(t_{π}^{φ})_{p}^{∧}} ≃ BC_{G}(t_{π}^{φ}(π))_{p}^{∧} ≃ BZ(π).

Thus Π_{j}^{φ} is the zero functor for all j except possibly for j = 1 and Π_{1}^{φ}(G/π) ≅ Z(π). General obstruction theory for maps out of a homotopy colimit [13] gives that if

E_{φ}^{n} : hocolim_{O_{p}^{c}(G)^{→}} ^{(n)}(ϕ_{G})_{hG} ^{}BG_{p}^{∧}

has been constructed, the obstructions to extending it to a map E_{φ}^{n+1} out of the (n + 1)-st skeleton of the homotopy colimit are in the group lim^{n+1}_{O_{p}^{c}(G)} Π_{n}^{φ} and if any

extension exists then any two differ by an element in lim^{n}_{O_{p}^{c}(G)} Π_{n}^{φ}. For a self equivalence

φ of the functor I_{G}, let

Θ_{j}^{i}(φ) = lim^{i}_{O_{p}^{c}(G)} Π_{j}^{φ}.

We proceed by analyzing the obstruction groups. For a finite (p-reduced) group G, let

Z: O_{p}^{c}(G)^{op} ^{}Ab

denote the functor, which associates to a p-centric subgroup π of G its center Z(π).

Definition 4.8. For any finite group G and i ≥ 0, define

J^{i}(G) = lim^{i}_{O_{p}^{c}(G)^{op}} Z

Notice in particular that the discussion in section 3 implies that J^{0}(G) = Z(G) if G is p′-reduced.

Proposition 4.9. For every natural equivalence φ ∈ Aut(I_{G}), we have Θ_{j}^{i}(φ) = 0 for j ≥ 2 and

J^{i}(G) ≅ Θ_{1}^{i}(φ)

for all i ≥ 0.

Proof. Let t_{π}^{φ} : π ^{}G be the homomorphisms chosen above. Let μ_{π}^{φ} denote the composition

Z(π) × π \overline{^{μ_{π}^{φ}}} G
[figure: μ; t_{π}^{φ}]
π

where μ denotes the multiplication map. Then the maps μ_{π}^{φ} induce homotopy equivalences

B̃μ_{π}^{φ}: BZ(π) ^{≃}^{}Map(Bπ, BG_{p}^{∧})_{(Bt_{π}^{φ})_{p}^{∧}}.

Furthermore, by Lemma 4.7, if f: G/π ^{}G/π′ is a morphism in O_{p}^{c}(G), which is determined by an inner automorphism c of G sub conjugating π to π′, then there is a homotopy commutative diagram

BZ(π) × Bπ \overline{^{Bμ}} Bπ
[figure: Bc^{−1}×1; B(t_{π}^{φ})_{p}^{∧}]
BZ(π′) × Bπ  BG_{p}^{∧}
[figure: 1×Bc; B(t_{π′}^{φ})_{p}^{∧}]
[figure: BZ(π′) × Bπ′; Bμ; Bπ′; Bc]

Taking respective adjoints one gets a homotopy commutative diagram

[figure: BZ(π); ≃; Map(Bπ, BG_{p}^{∧})_{(Bt_{π}^{φ})_{p}^{∧}}; BZ(π′); Bc^{−1}; ≃; Map(Bπ′, BG_{p}^{∧})_{(Bt_{π′}^{φ})_{p}^{∧}}; Bc^{∗}]

which in turn induces a commutative diagram of homotopy groups for every j ≥ 1

π_{j}(BZ(π)) \overline{^{≅}} π_{j}(Map(Bπ, BG_{p}^{∧})_{(Bt_{π}^{φ})_{p}^{∧}}) = Π_{j}^{φ}(G/π)

[figure: π_{j}(BZ(π′)); (Bc^{−1})_{#}; ≅; π_{j}(Map(Bπ′, BG_{p}^{∧})_{(Bt_{π′}^{φ})_{p}^{∧}}) = Π_{j}^{φ}(G/π′); (Bc^{∗})_{#}]

But Π_{j}^{φ}(G/π) = 0 for all π in O_{p}^{c}(G) and every j ≥ 2. Thus for j ≥ 2 one gets Θ_{j}^{i}(φ) = 0. On the other hand, for j = 1, one obtains a natural equivalence of functors on O_{p}^{c}(G)

Z ^{≅}^{}Π_{1}^{φ}

which induces the required isomorphism upon taking higher limits.

Let ψ_{G} : Out(BG_{p}^{∧}) ^{}Aut(I_{G}) denote the map constructed in Lemma 4.3. The discussion above now implies the following theorem, which in particular gives Theorem 1.5.

Theorem 4.10. Let G be a finite group and let φ ∈ Aut(I_{G}) be any natural equivalence. Let

E_{φ}^{1} : hocolim_{O_{p}^{c}(G)^{→}} ^{(1)}(−)_{hG} ^{}BG_{p}^{∧}

be a map constructed by the procedure described above. Then 1. The obstruction to lifting E_{φ}^{1} to a map

E_{φ} : hocolim_{O_{p}^{c}(G)}(ϕ_{G})_{hG} ^{}BG_{p}^{∧}

such that ψ_{G}([(E_{φ})_{p}^{∧}]) = φ is an element in J^{2}(G).

2. If some lifting E_{φ} of φ exists then all homotopy classes of lifts are in 1-1 corre-

spondence with J^{1}(G). In other words the sequence

0 ^{}J^{1}(G) ^{}Out(BG_{p}^{∧}) ^{ψ_{G}}^{}Aut(I_{G})

is exact and if J^{2}(G) = 0 then ψ_{G} is an epimorphism.

Remark 4.11. The group of automorphisms of the functor I_{G} is strongly related to self equivalences of the Frobenius category of G, namely, the category whose objects are all p-subgroups of G and whose morphisms are all homomorphisms induced by inclusions and conjugations. These equivalences are frequently quite easy to describe. Thus one may wonder about the case when the map ψ_{G} is in fact an isomorphism. By the theorem this is the case when the groups J^{i}(G) vanish for i = 1, 2. Below we compute an example where J^{1}(G) ≠ 0. This may suggest that J^{i}(G) is non-vanishing in general. However, it would be interesting to examine conditions on the group G which insure vanishing of the obstruction groups.

5. Sample Calculations

5.1. Normal Sylow p-subgroup. Let G be a finite p′-reduced group with a Sylow p-subgroup π and assume that π is normal in G.

Lemma 5.1. Let G be a p′-reduced finite group with a Sylow p-subgroup π ◁ G then Out(BG_{p}^{∧}) ≅ Aut(I_{G}).

Proof. Since π ◁ G, it is the only p-subgroup which is both p-centric and p-stubborn. Thus we may compute higher limits over the category containing π alone as an object and W = G/π as morphisms. But in this case all higher limits vanish and so Theorem 1.5 applies to give the result.

Proposition 5.2. Let G be a finite p′-reduced group with a normal Sylow p-subgroup π. Then

Out(BG_{p}^{∧}) ≃ Out(G).

Proof. Let [BG, BG_{p}^{∧}]^{∗} denote the classes of maps which correspond to equivalences of BG_{p}^{∧}. Let W = G/π. Then W operates on EG/π and one has

(6) [BG, BG_{p}^{∧}] = π_{0}(Map(BG, BG_{p}^{∧})) ≅

π_{0}(Map((EG/π)_{hW} , BG_{p}^{∧})) ≅ π_{0}((Map(EGπ, BG_{p}^{∧}))^{hW}).

To compute [BG, BG_{p}^{∧}]^{∗} we only need to consider components of inclusions of π in G. Let ī denote the collection of all faithful representations π ^{}G. Then for every i ∈ ī Map(EG/π, BG)_{Bi} ≃ BC_{G}(i(π)) ≃ BZ(π) is p-complete. Thus

Map(EG/π, BG_{p}^{∧})ī ≃ (Map(EG/π, BG)ī)_{p}^{∧} ≃ Map(EG/π, BG)ī

so

(7) π_{0}((Map(EG/π, BG_{p}^{∧})ī)^{hW}) ≃ π_{0}((Map(EG/π, BG)ī)^{hW}) ≃ π_{0}(Map(BG, BG)_{equiv}) = Out(BG) = Out(G).

This completes the proof.

5.2. The Groups SL_{2}(q) and P SL_{2}(q) at 2. Let SL_{2}(q) denote the special linear group over 𝔽_{q}, where q = p^{n} is an odd prime power. Then SL_{2}(q) < GL_{2}(q), the general linear group over 𝔽_{q} and the quotients of both by their respective centers give the projective groups P SL_{2}(q) and P GL_{2}(q) respectively. Diagrammatically one has the following diagram, where rows are group extensions and columns are central extensions.

[figure: ℤ/2; 𝔽_{q}^{∗}; ℤ/\frac{q−1}{2} ℤ; SL_{2}(q); GL_{2}(q); det; 𝔽_{q}^{∗}; P SL_{2}(q); P GL_{2}(q); ℤ/2] (8)

Observe that the composition 𝔽_{q}^{∗} ^{}GL_{2}(q) ^{det}^{}𝔽_{q}^{∗} is the squaring map. The outer automorphism group of P SL_{2}(q) is given by

Out(P SL_{2}(q)) ≅ ℤ/2 × ℤ/n

where ℤ/2 is generated by α given as the outer action defined by the extension in the bottom row of the above diagram, and ℤ/n ≅ Gal(𝔽_{q}|𝔽_{p}) if q = p^{n}, generated by the Frobenius automorphism ϕ, acts in the obvious way.

The Sylow 2-subgroup of P SL_{2}(q) is a dihedral 2-group

D_{2^{s}} = ⟨x, y | x^{2^{s−1}} = y^{2} = (xy)^{2} = 1⟩

of order 2^{s} depending on q. For s = 2, D_{4} is elementary abelian of rank two, hence Aut(D_{4}) = Out(D_{4}) ≅ Σ_{3}. The automorphism group of D_{2^{s}} is also easily described as the semidirect product

Aut(D_{2^{s}}) = ℤ/2^{s−1} ⋊ (ℤ/2^{s−1})^{∗},

where an element l ∈ ℤ/2^{s−1} corresponds to the automorphism g_{l} with g_{l}(x) = x and g_{l}(y) = x^{l}y, and an element a ∈ (ℤ/2^{s−1})^{∗} corresponds to the automorphism f_{a} such that f_{a}(x) = x^{a} and f_{a}(y) = y. One verifies that g_{l} with l even and f_{−1} are the inner automorphisms and since (ℤ/4)^{∗} ≅ (ℤ/2) and (ℤ/2^{s−1})^{∗} ≅ ℤ/2 × ℤ/2^{s−3}, s ≥ 4, generated by −1 and 3 modulo 2^{s−1} one obtains

Out(D_{4}) = Σ_{3}
Out(D_{8}) ≅ ℤ/2
Out(D_{2^{s}}) ≅ ℤ/2 × ℤ/2^{s−3}, s ≥ 4

generated by the classes ḡ_{1} (order 2) and f̄_{3} ( order 2^{s−3}) of g_{1} and f_{3} respectively.

Proposition 5.3. Assume that q ≡ 1 (mod 8), q = p^{n}, p an odd prime number, and let s be the largest integer such that 2^{s} | q − 1. There is a homomorphism defined by restriction

Out(P SL_{2}(q)) ^{}Out(D_{2^{s}})

that sends α to ḡ_{1} and ϕ to f̄_{p}.

In particular, if q = 3^{2^{s−2}}, s ≥ 3, then there is an extension

1 ^{}ℤ/2 ^{}Out(P SL_{2}(q)) ^{}Out(D_{2^{s}}) ^{}1

where ℤ/2 in the kernel is represented by ϕ^{2^{s−3}}.

Proof. Let ϵ be a 2^{s} root of unity in 𝔽_{q}. The 2-Sylow subgroup of P SL_{2}(q) is D_{2^{s}}, generated by the classes of the matrices X = ( \begin{matrix}ϵ & 0 \\ 0 & ϵ^{−1} \end{matrix} ) and Y = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ):

D_{2^{s}} ≅ ⟨X̄, Ȳ | X̄^{2^{s−1}} = Ȳ^{2} = (X̄Ȳ)^{2} = 1⟩ ⊂ P SL_{2}(𝔽_{q}) .

Since ϵ is not a square in 𝔽_{q}, the class matrix A = ( \begin{matrix}ϵ & 0 \\ 0 & 1 \end{matrix} ) provides a set theoretic section of the bottom extension in diagram (8), and therefore the outer automorphism α of P SL_{2}(𝔽_{q}) is described as conjugation by Ā in P GL_{2}(𝔽_{q}). One can now check that this conjugation leaves the given 2-Sylow subgroup stable and induces the outer automorphism ḡ_{1} of D_{2^{s}}. On the other hand the Frobenius ϕ is defined as p-th power on 𝔽_{q} and then it induces f̄_{p} on D_{2^{s}}.

In case q = 3^{2^{s−2}}, s ≥ 3, f̄_{3} is a generator of ℤ/2^{s−3} ⊂ Out(D_{2^{s}}), so the restriction is an epimorphism with kernel clearly generated by ϕ^{2^{s−3}} that has order 2.

Lemma 5.4. There are isomorphism

Aut(SL_{2}(q)) ^{≅}^{}Aut(P SL_{2}(q))

and

Out(SL_{2}(q)) ^{≅}^{}Out(P SL_{2}(q)) .

Proof. Since P SL_{2}(q) is the quotient of SL_{2}(q) by its center, any automorphism of SL_{2}(q) induces an automorphism of P SL_{2}(q). Conversely, any automorphism of P SL_{2}(q) preserves the extension class for SL_{2}(q) and thus induces an automorphism of SL_{2}(q). This proves the first statement. For the second statement, notice that

Inn(SL_{2}(q)) ≅ SL_{2}(q)/Z(SL_{2}(q)) ≅ P SL_{2}(q) ≅ Inn(P SL_{2}(q)).

Next, compute the automorphisms of the generalized quaternion 2-groups,

Q_{2^{s+1}} = ⟨x, y | x^{2^{s−1}} = y^{2}, x^{2^{s}} = 1, yxy^{−1} = x^{−1}⟩ ,

which appear as the Sylow 2-subgroups of SL_{2}(q), q odd. Thus for Q_{2^{s+1}} one has the automorphisms g_{l} defined by g_{l}(x) = x and g_{l}(y) = x^{l}y for l ∈ Z/2^{s} and f_{a} such that f_{a}(x) = x^{a} and f_{a}(y) = y for a ∈ (ℤ/2^{s})^{∗}. This gives Aut(Q_{2^{s+1}}) ≅ ℤ/2^{s} ⋊ (ℤ/2^{s})^{∗} if s ≥ 3.

The center of Q_{2^{s+1}} is cyclic of order 2 generated by x^{2^{s−1}} = y^{2} and the quotient of Q_{2^{s+1}} by its center is isomorphic to D_{2^{s}}. There is an induced homomorphism Aut(Q_{2^{s+1}}) ^{}Aut(D_{2^{s}}), which is an epimorphism with kernel ℤ/2×ℤ/2 generated by f_{2^{s−1}+1}:

f_{2^{s−1}+1}(x) = x^{2^{s−1}+1}, f_{2^{s−1}+1}(y) = y

and g_{2^{s−1}}:

g_{2^{s−1}}(x) = x, g_{2^{s−1}}(y) = x^{2^{s−1}}y .

Notice that g_{2^{s−1}} is inner while f_{2^{s−1}+1} it not inner unless s = 2, and therefore

Out(Q_{8}) ≅ Out(D_{4}) ≅ Σ_{3}

and for s ≥ 3, we obtain an extension

1 ^{}⟨f̄_{2^{s−1}+1}⟩ℤ/2 ^{}Out(Q_{2^{s+1}}) ^{}Out(D_{2^{s}}) ^{}1.

The following Proposition describes the relationships between the outer automorphism groups computed above for q = 3^{2^{s}}, s ≥ 1.

Proposition 5.5. Assume that q = 3^{2^{s−2}}, s ≥ 3. There is an isomorphism

π : Out(SL_{2}(𝔽_{q})) ^{}Out(Q_{2^{s+1}})

and a commutative diagram

ℤ/2

[figure: Out(SL_{2}(q)); ≅; Out(Q_{2^{s+1}})]

[figure: ℤ/2; Out(P SL_{2}(q)); ≅; Out(D_{2^{s}})]

where the bottom row and right column are exact.

Homology decomposition of SL_{2}(q) and P SL_{2}(q). A homology decomposition of P SL_{2}(q) is described in detail in [1]. We refer to [4] for the general theory. Restrict attention to the case where q = 3^{2^{s−2}}, s ≥ 3. Choose a Sylow subgroup S ≅ D_{2^{s}}. Then there are subgroups Z < S of order 2 given by the center of S and two non-conjugate elementary abelian 2-subgroups V, W < S of rank 2, which give an ample collection E_{2} = {Z, V, W } of elementary abelian 2-subgroups of P SL_{2}(𝔽_{q}). The associated conjugacy category A_{E_{2}} can be described by means of the following diagram

Σ_{3} ⟳ W [figure: Σ_{3}/Σ_{2}] Z [figure: Σ_{3}/Σ_{2}] V ⟳ Σ_{3} .

The centralizer diagram α_{E_{2}} : A_{E_{2}}^{op} ^{}Spaces is up to homotopy and 2-completion described by

Σ_{3} ⟳ BW [figure] BD_{2^{s}} [figure] BV ⟳ Σ_{3}.

and the natural map

a_{E_{2}} : hocolim α_{E_{2}} ^{}BP SL_{2}(𝔽_{q}) (9)

induces a mod-2 homology isomorphism.

For SL_{2}(q) one obtains a homology decomposition by pulling back α_{E_{2}} along the projection SL_{2}(q) ^{}P SL_{2}(q) (q = 3^{2^{s−2}}, s ≥ 3 as above). We obtain a new strictly commutative diagram β_{E_{2}} of the form

Σ_{3} ⟳ BQ_{8}  [figure] BQ_{2^{s+1}} [figure]  BQ_{8} ⟳ Σ_{3}.

One can view β_{E_{2}} as a functor from A_{E_{2}} to Spaces and there is a map b_{E_{2}} out of the diagram to BSL_{2}(q) given by the pull-back process described above.

Lemma 5.6. The map

b_{E_{2}} : hocolim_{A_{E_{2}}} β_{E_{2}} ^{}BSL_{2}(q)

induces a mod-2 equivalence.

Proof. This follows from the homology decomposition in equation (9).

The group Out(BG_{2}^{∧}) for G = SL_{2}(q) and P SL_{2}(q). The left column of diagram (8) induces a principal fibration

Bℤ/2 ^{Bi}^{}BSL_{2}(q)_{2}^{∧} ^{Bp}^{}BP SL_{2}(q)_{2}^{∧}. (10)

Lemma 5.7. There is an isomorphism

θ: Out(BSL_{2}(q)_{2}^{∧}) ^{}Out(BP SL_{2}(q)_{2}^{∧}) .

Proof. There is a homotopy equivalence Map(Bℤ/2, BP SL_{2}(q)_{2}^{∧})_{c} ≃ BP SL_{2}(q)_{2}^{∧}, where c denotes the constant map (cf. [1]). Then the Zabrodsky lemma [14, 10] applies to the principal fibration (10) to give a homotopy equivalence

Map(BSL_{2}(q)_{2}^{∧}, BP SL_{2}(q)_{2}^{∧})_{{f|f∘Bi≃∗}} ≃ Map(BP SL_{2}(q)_{2}^{∧}, BP SL_{2}(q)_{2}^{∧}) .

Now, any homotopy equivalence g ∈ Out(BSL_{2}(q)_{2}^{∧}) induces the identity in mod-2 cohomology. Hence g ∘ Bi ≃ Bi and so Bp ∘ g ∘ Bi ≃ ∗. It follows that there exists ḡ ∈ Out(BP SL_{2}(q)_{2}^{∧}) satisfying Bp ∘ g ≃ ḡ ∘ Bp. Define θ(g) = ḡ.

Finally, we observe that [BSL_{2}(q)_{2}^{∧}), Bℤ/2] ≅ [BSL_{2}(q)_{2}^{∧}), B^{2}ℤ/2] ≅ 0, hence θ turns out to be an isomorphism.

Lemma 5.8. Let G be a finite group and let S be its Sylow p-subgroup. Assume that S is self normalizing in G. Then there is a homomorphism res : Out(G) ^{res}^{}Out(S), which factors through Out(BG_{p}^{∧}).

Proof. Let ϕ̄ be an outer automorphism of G represented by some automorphism ϕ. Then ϕ carries S into another Sylow p-subgroup S′. But there is an inner automorphism c_{g} of G which carries S′ back to S. Define res(ϕ̄) to be the class of the composition c_{g} ∘ ϕ in Out(S). If g′ ∈ G is another element conjugating S′ to S then g′g^{−1} ∈ N_{G}(S) = S. Hence c_{g} ∘ ϕ and c_{g′} ∘ ϕ differ only by an inner automorphism of S. Also if ϕ′ is another representative for ϕ̄ then ϕ and ϕ′ differ by an inner automorphism of G and the procedure carries the difference again into an inner automorphism of S. Thus the restriction map is well defined and obviously a group homomorphism.

Similarly, if ψ ∈ Out(BG_{p}^{∧}) then the composition

BS ^{Bι}^{}BG_{p}^{∧} ^{ϕ}^{}BG_{p}^{∧},

where ι denotes the inclusion, is homotopic to a map induced by a homomorphism (see Proposition 2.1). Thus the same argument implies that there is an automorphism α of S such that Bια ≃ ψBι.

Finally if ψ ∈ Out(BG_{p}^{∧}) is induced by an automorphism ϕ of G, then the procedure described above gives that α is conjugate to res(ϕ) and the lemma follows.

Remark 5.9. Lemma 5.8 applies to G = SL_{2}(q) and G = P SL_{2}(q), q ≡ ±1 (mod 8). In fact, the centralizer in P SL_{2}(q) of the center Z ≅ ℤ/2 of its Sylow 2-subgroup is isomorphic to D_{q−1}, the dihedral group of order 2q − 2, generated by the classes of the matrices ( \begin{matrix}ζ & 0 \\ 0 & ζ^{−1} \end{matrix} ) and ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ), where ζ is a generator of 𝔽_{q}^{∗} (Proposition 4.2 of [1]). Then N_{P SL_{2}(q)}(D_{2^{s}}) ⊂ D_{q−1} and a quick calculation shows that N_{P SL_{2}(q)}(D_{2^{s}}) = D_{2^{s}}. From this equality it follows that N_{SL_{2}(q)}(Q_{2^{s+1}}) = Q_{2^{s+1}} as well.

In particular, for q = 3^{2^{s−2}}, s ≥ 3, we have a commutative diagram

Out(SL_{2}(q)) \overline{^{≅}} Out(Q_{2^{s+1}})
[figure: res]
Out(BSL_{2}(q)_{2}^{∧})^{}

(11)

Proposition 5.10. For q = 3^{2^{s−2}}, s ≥ 3, the natural map

B: Out(SL_{2}(q)) ^{}Out(BSL_{2}(q)_{2}^{∧})

is an isomorphism.

Proof. It suffices to show that

res: Out(BSL_{2}(q)_{2}^{∧}) ^{}Out(BQ_{2^{s+1}})

is an isomorphism. It is clear from Diagram (11) that res is an epimorphism. We use the homology decomposition for BSL_{2}(q)_{2}^{∧} of Lemma 5.6 in order to prove injectivity.

The class of a self equivalence f of BSL_{2}(q)_{2}^{∧} is in the kernel the restriction map if and only if the diagram

\begin{matrix}BQ_{2^{s+1}} & = = = = = & BQ_{2^{s+1}} \\ BSL_{2}(q)_{2}^{∧} & ^{Bι} & \overline{^{f}} & BSL_{2}(q)_{2}^{∧} & ^{Bι} \end{matrix}

is homotopy commutative, where ι denotes the inclusion map. Then, for every object in the diagram β_{E_{2}} we have

f|BQ_{2^{s+1}} ≃ Id|BQ_{2^{s+1}}
f|BQ_{8} ≃ Id|BQ_{8} .

The obstructions for f to be homotopic to the identity lie in the groups

\begin{matrix}lim \\ ←------- \end{matrix}^{i}_{A_{E_{2}}} π_{j}( Map(β_{E_{2}}, BSL_{2}(q)_{2}^{∧})_{Bι_{·}}) , i ≥ 1

and according to Proposition 5.12 these groups are trivial. Hence f ≃ Id, that is res is injective.

Proposition 5.11. For q = 3^{2^{s−2}}, s ≥ 3,

1. Map(BQ_{8}, BSL_{2}(q)_{2}^{∧})_{Bι_{ϵ}} ≃ Bℤ/2, where ϵ = 1, 2 denotes the two different inclusions.

2. Map(BQ_{2^{s+1}}, BSL_{2}(q)_{2}^{∧})_{Bι} ≃ Bℤ/2.

Proof. By Proposition 2.1 these spaces are equivalent to the 2-completion of the respective centralizer. The claim now follows by direct calculation using Remark 5.9

Proposition 5.12. For q = 3^{2^{s−2}}, s ≥ 3,

\begin{matrix}lim \\ ←------- \end{matrix}^{i}_{A_{E_{2}}} π_{j} Map(β_{E_{2}}, BSL_{2}(q)_{2}^{∧})_{Bι_{·}} ≅ \begin{cases}ℤ/2 & i = 0, j = 1 \\ 0 & otherwise. \end{cases}

Proof. This follows at once from Proposition 5.11 and [1, §10].

The discussion above implies Theorem 1.6. Specifically, part 1 follows by combining Proposition 5.5, Lemma 5.7 and Proposition 5.10. Part 2 follows from the calculation of Proposition 5.12 and part 3 follows from [1, Lemma 6.4].

6. Mod-p Homotopy Group Extensions

In this final section we return to the motivation for our study and discuss mod-p homotopy group extensions. The following table compares the classification of group extensions to the classification of homotopy group extensions. The last is of course just a special case of the general classification problem for fibrations.

Group Extensions  Mod-p Homotopy Group Extensions
π by G  π by G
1 A homomorphism π ^{α}^{}Out(G)  A map Bπ_{p}^{∧} ^{α}^{}B Out(BG_{p}^{∧})
Obstructions to the existence  Obstructions to the existence of any
2 of any extension of π by G mod-p homotopy extension of π by G in
in H^{3}(π; Z(G)), where Z(G)  H^{3}(Bπ_{p}^{∧}; Z(G/O_{p′}G)), where Z(G/O_{p′}G)
becomes a π-module via α  becomes a π_{1}(Bπ_{p}^{∧})-module via α
If an extension exists then all  If an extension exists
3 extensions are classified by then all the others are classified by
H^{2}(π; Z(G))  H^{2}(Bπ_{p}^{∧}; Z(G/O_{p′}G))

Corollary 6.1. Let G be a finite group such that the natural map

γ_{G} : Out(G/O_{p′}G) ^{}Out(BG_{p}^{∧})

is an isomorphism. Let π be a finite p-group. Then there is a 1-1 correspondence between fibre homotopy classes of mod-p homotopy extensions of π by G and equivalence classes of ordinary group extensions of π by G/O_{p′}G.

Proof. Under our hypotheses, there is a homotopy equivalence

B Aut(B(G/O_{p′}G)) ^{≃}^{}B Aut(BG_{p}^{∧}).

The result follows.

Recall that a p-group P is called a Swan group if for any finite group G containing P as a Sylow p-subgroup, the inclusion of the normalizer N_{G}(P) ^{}G induces a mod-p homology isomorphism.

Corollary 6.2. Let G be a finite group with a Sylow p-subgroup P and assume either 1. P is normal in G or

2. the inclusion N_{G}(P) ^{}G induces a mod-p homology equivalence or

3. P is a Swan group. In either case let H denote the mod-p′ reduction of N_{G}(P) and let π be any finite p-group. Then fibre homotopy classes of mod-p homotopy extensions of π by G are in 1-1 correspondence with ordinary group extensions of π by H.

Proof. Under either one of 1, 2 or 3, we may replace BG_{p}^{∧} by BH_{p}^{∧} up to homotopy. The result follows at once from Corollary 6.1.

Our calculation for the special linear groups implies a similar result. Specifically we have

Proposition 6.3. Assume q = 3^{2^{s−2}}, s ≥ 3, then there is a homotopy equivalence

B Aut(BSL_{2}(q)) ^{≃}^{}B Aut(BSL_{2}(q)_{2}^{∧}) .

Proof. Since p-completion is a continuous functor, there is a diagram of fibrations

[figure: B SAut(BSL_{2}(q)); B Aut(BSL_{2}(q)); B Out(SL_{2}(q))]

[figure: B SAut(BSL_{2}(q)_{2}^{∧}); B Aut(BSL_{2}(q)_{2}^{∧}); B Out(BSL_{2}(𝔽_{q})_{2}^{∧})]

where the left vertical arrow is a homotopy equivalence by Theorem 1.1 and the right vertical arrow is a homotopy equivalence by Proposition 5.10, hence the result follows.

Remark 6.4. Notice that we have actually computed the spaces B Aut(BSL_{2}(q)_{2}^{∧}) and B Aut(BP SL_{2}(q)_{2}^{∧}) for every odd prime power q = p^{k}. In fact, according to [1] the homotopy types of BSL_{2}(q)_{2}^{∧} and BP SL_{2}(q)_{2}^{∧} depends only on the order of the Sylow 2-subgroup rather than on the concrete odd prime power q. Observe that any order of a Sylow 2-subgroup in SL_{2}(q) can be obtained by letting q = 3^{2^{s}} for some s ≥ 1 if q ≡ ±1 mod 8 and that if q ≡ ±3 mod 8 then BSL_{2}(q)_{2}^{∧} ≃ BSL_{2}(3)_{2}^{∧}, in which case the Sylow 2-subgroup Q_{8}, and respectively BP SL_{2}(3)_{2}^{∧} ≃ (BA_{4})_{2}^{∧} with Sylow 2-subgroup elementary abelian of rank 2. In these two cases the Sylow 2-subgroups are Swan groups.

Corollary 6.5. Let π be a finite 2-group and let q be any odd prime power. Then there is a 1-1 correspondence between fibre homotopy classes mod-2 homotopy extensions of π by SL_{2}(q) and equivalence classes of ordinary group extensions of π by SL_{2}(t) where t = 3^{2^{s}} for some s such that SL_{2}(q) and SL_{2}(t) have Sylow 2-subgroups of the same order if q ≡ ±1 mod 8 and t = 3 otherwise. The corresponding result applies to homotopy extensions of a finite 2-group by P SL_{2}(q).

Our interest in this project was motivated by a rather simple minded question, namely homotopy uniqueness of the space BQ_{2^{n}} [2]. Our results here enable us to give an easy solution of this problem.

Corollary 6.6. Let Q_{2^{r}} be a generalized quaternion group of order 2^{r}. Let X be a 2complete space with H^{∗}(X) ≅ H^{∗}(BQ_{2^{r}}) as an algebra over the Steenrod algebra and assume further that there is an isomorphism between the Bockstein spectral sequences of H^{∗}(X) and H^{∗}(BQ_{2^{r}}) in the sense of [2].Then there is a homotopy equivalence X ≃ BQ_{2^{r}}

Proof. It is shown in [2] that under our hypotheses if X is not equivalent to BQ_{2^{r}} then π_{1}(X) ≅ Q_{2^{s}} for some s < r −2 and its universal cover X̃ has the cohomology of BSL_{2}(q) for some appropriate q, again as an algebra over the Steenrod algebra and with the same Bockstein spectral sequence. By [1] it follows that X̃ ≃ BSL_{2}(q)_{2}^{∧}. The previous corollary thus implies that X is the 2-completion of the classifying space of an extension of Q_{2^{s}} by SL_{2}(t) for an appropriate t. But one now easily checks that no such extension has the cohomology assumed for X. The result follows.

To conclude this paper we comment that homotopy group extensions were defined to be fibrations where both base and fibre are p-completed classifying spaces. In some contexts, in particular if whether or not the total space is p-complete has no significance, it makes sense to consider fibrations with base Bπ (rather than Bπ_{p}^{∧}), and fibre BG_{p}^{∧}.

Corollary 6.7. Let G be a finite group and assume that the natural map

γ_{G} : Out(G/O_{p′}G) ^{}Out(BG_{p}^{∧})

is an isomorphism. Then for any discrete group π there is a 1-1 correspondence between fibre homotopy equivalence classes of fibrations with base Bπ and fibre BG_{p}^{∧} and equivalence classes of ordinary group extensions of π by G/O_{p′}G.

Finally, notice that in the case of the foregoing corollary, the correspondence is given via the Bousfield-Kan fibrewise p-completion functor.

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

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

Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, U.K.

E-mail address: ran@maths.aberdeen.ac.uk