DIAGRAMS UP TO COHOMOLOGY

W.G. Dwyer and C.W. Wilkerson

University of Notre Dame Purdue University

1. Introduction

Let Sp denote the category of spaces, Ho the associated homotopy category, and D a small (index) category. A diagram in Ho with the shape of D is by definition a functor F : D ⟶ Ho. Given such a diagram one can ask whether or not it has a realization, i.e., a lift to a functor D ⟶ Sp, and if so, how many realizations there are up to an appropriate kind of equivalence. This question is studied in [3] and [5] for general diagrams and in [6] for the special case of “centric” diagrams (see §6). In this paper we look at a related question. Let p be a fixed prime number and let H^{∗} denote the cohomology functor H^{∗}(–; F_{p}). If F : D ⟶ Ho is a functor, an H^{∗}-clone of F is by definition a collection (G, {s_{d}}) where

(1) G : D ⟶ Ho is a functor,

(2) for each object d of D, s_{d} : F (d) ⟶ G(d) is an isomorphism, and

(3) for each morphism g : d ⟶ e of D, the diagram

H^{∗} F (d) ⟵^{F (g)^{∗}} H^{∗} F (e)
s_{d}^{∗}↑  ^{s_{e}^{∗}}↑
H^{∗} G(d) ⟵^{G(g)^{∗}} H^{∗} G(e)

commutes. In other words, an H^{∗}-clone of F is another diagram in the homotopy category which is built from essentially the same spaces as F and looks exactly the same as F from the point of view of mod p cohomology. An equivalence or isomorphism between two H^{∗}-clones (G, {s_{d}}) and (G′, {s′_{d}}) is a natural equivalence t : G ⟶ G′ with the property that for each object d of D, the composite

F (d) ⟶^{s_{d}} G(d) ⟶^{t_{d}} G′(d) ⟶^{(s′_{d})^{−1}} F (d)

Typeset by AMS-TEX

The authors were supported in part by the National Science Foundation.

gives the identity map of H^{∗} F (d). Our main theorem (2.4) shows how to compute the set of isomorphism classes of H^{∗}-clones of F in the special case in which F satisfies a “centric” condition (2.1) analogous to that of [6]. We also work out some examples (§3, §5) and explain (6.7) how our results relate to those of [6].

The motivation for this comes from the following example. Suppose that X is a pcompact group [10], for instance, X might be the p-completion G_{p̂} of a compact Lie group G such that π_{0}G is a finite p-group. Given X, the decomposition method of [11] (see also [8] and [12]) produces

(1) a category A_{X},

(2) a functor α_{X} : A_{X}^{op} ⟶ Sp, and

(3) a natural homotopy equivalence BX ∼ (hocolim α_{X})_{p̂}. Let ᾱ_{X} denote the diagram in the homotopy category which underlies α_{X}. We make the following conjecture, which is related to the question of whether or not the homotopy type of BX is determined by the cohomology ring H^{∗} BX as an algebra over the mod p Steenrod algebra.

1.1 Conjecture. Let X be a p-compact group (perhaps connected). Then ᾱ_{X} has no nontrivial H^{∗}-clones, i.e., any H^{∗}-clone of ᾱ_{X} is isomorphic to ᾱ_{X}.

We show below (§3) that the machinery of this paper applies to ᾱ_{X}; this gives a way to check 1.1 in particular cases.

Organization of the paper. Section 2.4 enumerates the set of H^{∗}-clones of diagrams of a certain special type, and §3 shows that this enumeration applies to the case of the decomposition diagram of a p-compact group. In §4 there is a detailed computational analysis of the “lim^{1}” which comes up in the examples from §3; we use this in §5 to prove Conjecture 1.1 for the F_{2}-completion of the exceptional Lie group G_{2}. Calculations similar to the ones in §5 also appear in the work of K. Premadasa [18]. Finally, §6 describes a homotopy limit which combines the ideas here with those of [6] to give, for a suitable diagram in the “cohomology category” an efficient computation of the set of equivalence classes of its realizations. At the end of the paper (6.8) there is a conjecture which is in a sense a generalization of 1.1.

Notation and terminology. The category Sp can be taken to be either the category of topological spaces or of simplicial sets. When forming a function space Map(A, B), we assume in the first case that A has been replaced if necessary by a weakly equivalent cofibrant object (i.e. CW-complex) and in the second that B has been replaced if necessary by a weakly equivalent fibrant object (i.e. Kan complex). The homotopy category Ho is the category obtained from Sp by formally inverting all weak equivalences.

If A is a space, A_{p̂} denotes the F_{p}-completion of A in the sense of [1].

§2. The main theorem

If f : A ⟶ B is a map in Ho, let [A, B]_{{f}} denote the set of maps f′ : A ⟶ B in Ho such that H^{∗}(f′) = H^{∗}(f).

2.1 Definition. A map f : A ⟶ B in Ho is said to be H^{∗}-centric up to homotopy if (1) [A, A]_{{id}} is a group under composition, and

(2) composition with f induces a bijection [A, A]_{{id}} ⟶^{≅} [A, B]_{{f}}.

A functor F : D ⟶ Ho is said to be H^{∗}-centric up to homotopy if F (g) satisfies the above conditions for each morphism g of D.

2.2 Remarks. We will define the stronger notion of H^{∗}-centric later on (6.3). Note that if A is a space with is F_{p}-complete [1] then A satisfies condition (1) above; this follows from the fact that a map between F_{p}-complete spaces is a weak equivalence if and only if it induces an isomorphism on mod p cohomology.

Example. If α : G_{1} ⟶ G_{2} is a map of finite groups, denote by α^{∗} the induced cohomology map H^{∗}(Bα). Let i : G ⟶ H be an inclusion of finite p-groups, N(G) the normalizer of G in H, and N^{in}(G) ⊂ N(G) the subgroup of N(G) consisting of elements whose conjugation action on G is via inner automorphisms. Since inner automorphisms of a group act trivially on the cohomology of the classifying space, there is a natural homomorphism N(G)/N^{in}(G) ⟶ Aut(H^{∗} BG). It is not hard to see that the map Bi : BG ⟶ BH is H^{∗}-centric up to homotopy if and only if

(1) for any homomorphism j : G ⟶ H with H^{∗}(Bj) = H^{∗}(Bi) the subgroup j(G) of H is conjugate to G, and

(2) the N(G)/N^{in}(G) ⟶ Aut(H^{∗} BG) is injective, and contains α^{∗} for every automor-

phism α of G such that α^{∗}i^{∗} = i^{∗}. It follows from 3.1 that these conditions are satisfied if G is the centralizer in H of an elementary abelian p-group.

Suppose that F : D ⟶ Ho is a functor which is H^{∗}-centric up to homotopy. Let Grp denote the category of groups, and construct a functor ϕ_{F} : D^{op} ⟶ Grp by setting

ϕ_{F} (d) = [F (d), F (d)]_{{id}}

for each object d of D. For a map g : d ⟶ e of D the induced homomorphism ϕ_{F} (g) : ϕ_{F} (e) ⟶ ϕ_{F} (d) assigns to a map v ∈ [F (e), F (e)]_{{id}} the unique element u of ϕ_{F} (d) such that the following diagram

(2.3)  \begin{matrix}F (d) & ⟶^{u} & F (d) \\ ^{F (g)}↓ & ^{F (g)}↓ \\ F (e) & ⟶^{v} & F (e) \end{matrix}

commutes. The existence and uniqueness of u follow immediately from the fact that F (g) is H^{∗}-centric up to homotopy. Given elements v and v′ in ϕ_{F} (e), let u and u′ denote their respective images in ϕ_{F} (d) and consider the following diagram

\begin{matrix}F (d) & ⟶^{u} & F (d) & ⟶^{u′} & F (d) \\ ^{F (g)}↓ & ^{F (g)}↓ & ^{F (g)}↓ \\ F (e) & ⟶^{v} & F (e) & ⟶^{v′} & F (e) \end{matrix} .

By choice of u and u′ each small square commutes. It follows that the large square commutes, which shows that the image under ϕ_{F} (g) of v′v is u′u and thus that ϕ_{F} (g) is a group homomorphism.

Given a functor F : D ⟶ Ho, let Cℓ(F) denote the set of equivalence classes (§1) of H^{∗}-clones of F. This is a pointed set with the class of (F, {id_{d}}) as basepoint.

2.4 Theorem. If F : D ⟶ Ho is a functor which is H^{∗}-centric up to homotopy, then there is a natural bijection of pointed sets

Cℓ(F) ≅ lim^{1}ϕ_{F}

and a natural isomorphism of groups

Aut(F, {id_{d}}) ≅ lim ϕ_{F} .

2.5 Description of lim^{1}. Before proving 2.4 we will give an explicit description of the functor “lim^{1}” (also see [1, p. 307]). Suppose that D is a small category and that ϕ : D^{op} ⟶ Grp is a functor. For convenience, if g : d_{0} ⟶ d_{1} is a morphism of D we will write g^{♯} : ϕ(d_{1}) ⟶ ϕ(d_{0}) for the map ϕ(g). Let O denote the set of objects of D, M the set of morphisms of D, and S the disjoint union

S = ⋃_{d∈O} ϕ(d) .

The object S is just a set, or, perhaps better, the collection of morphisms in a groupoid with object set O. Let C^{0}(ϕ) denote the set of all functions c : O ⟶ S such that for each d ∈ O, c(d) ∈ ϕ(d). It is clear that C^{0}(ϕ) is a group under pointwise multiplication. Let C^{1}(ϕ) denote the set of all functions c : M ⟶ S such that for each g : d_{0} ⟶ d_{1} in M, c(g) ∈ ϕ(d_{0}). An element c ∈ C^{1}(ϕ) is called a “normalized 1-cocycle” if

(1) for each identity map g : d_{0} ⟶ d_{1}, c(g) is the identity element of ϕ(d_{0}), and

(2) for each pair g_{0} : d_{0} ⟶ d_{1} and g_{1} : d_{1} ⟶ d_{2} of composable elements of M, the

product

g_{0}^{♯}(c(g_{1}))c(g_{1}g_{0})^{−1}c(g_{0})

is the identity element of ϕ(d_{0}). Let Z^{1}(ϕ) ⊂ C^{1}(ϕ) denote the collection of normalized 1-cocycles; this is a pointed set with basepoint given by the element z ∈ Z^{1}(ϕ) which assigns to each g : d_{0} ⟶ d_{1} in M the identity element of ϕ(d_{0}). There is an action of C^{0}(ϕ) on Z^{1}(ϕ) such that if c ∈ C^{0}(ϕ) and z ∈ Z^{1}(ϕ) then c · z = z′, where for each g : d_{0} ⟶ d_{1} in M,

z′(g) = c(d_{0})z(g)g^{♯}(c(d_{1})^{−1}) .

By definition, the orbit set of this action is lim^{1}ϕ. The basepoint of Z^{1}(ϕ) projects to a basepoint for lim^{1}ϕ.

Proof of 2.4. Say that an H^{∗}-clone (G, {s_{d}}) of F is special if for each object d of D, G(d) = F (d) and s_{d} is the identity map of F (d). It is immediate that any H^{∗}-clone (G′, {s′_{d}}) of F is equivalent to a special H^{∗}-clone (G, {s_{d}}); for instance, set G(g) = (s′_{d_{1}})^{−1}G′(g)s′_{d_{0}} for each morphism g : d_{0} ⟶ d_{1} of D. It follows that we can interpret Cℓ(F) as the set of equivalence classes of special H^{∗}-clones of F. If G is a special H^{∗}-clone of F, let z_{G} ∈ C^{1}(ϕ_{F}) be the function which assigns to each morphism g : d_{0} ⟶ d_{1} of D the unique element h ∈ [F (d_{0}), F (d_{0})]_{{id}} which makes the following diagram commute

\begin{matrix}F (d_{0}) & ⟶^{h} & F (d_{0}) \\ ^{F (g)}↓ & ^{G(g)}↓ \\ F (d_{1}) & ⟶^{id} & F (d_{1}) \end{matrix} .

The existence and uniqueness of h follows immediately from the fact that F (g) is H^{∗}-centric up to homotopy. Clearly z_{G}(g) is the appropriate identity element if g is an identity map. The commutative diagram

F (d_{0}) ⟶^{g_{0}^{♯}(z_{G}(g_{1}))} F (d_{0}) ⟶^{z_{G}(g_{0})} F (d_{0})
F (g_{0})↓  ^{F (g_{0})}↓  ^{G(g_{0})}↓
F (d_{1}) ⟶^{z_{G}(g_{1})} F (d_{1}) ⟶^{id} F (d_{1})
F (g_{1})↓  ^{G(g_{1})}↓  ^{G(g_{1})}↓
F (d_{2}) ⟶^{id} F (d_{2}) ⟶^{id} F (d_{2})

shows that z_{G}(g_{1}g_{0}) = z_{G}(g_{0})g_{0}^{♯}(z_{G}(g_{1})) and thus that z_{G} ∈ Z^{1}(ϕ_{F}). It is easy to check directly that the image ⟨z_{G}⟩ of z_{G} in lim^{1}ϕ_{F} depends only on the equivalence class of the special H^{∗}-clone G, and that the assignment G ↦ ⟨z_{G}⟩ gives the desired bijection Cℓ(F) ⟶ lim^{1}ϕ_{F}. The identification of lim ϕ_{F} is straightforward. □

§3. The basic example

In this section we will give the motivating example of a diagram which is H^{∗}-centric up to homotopy. Suppose that X is a p-compact group with classifying space BX. Let A_{X} denote the category whose objects are the pairs (V, Bf), where V is a nontrivial elementary abelian p-group and f : V ⟶ X is a conjugacy class of monomorphisms [11, §8]. By definition, giving f amounts to giving an ordinary homotopy class of maps Bf : BV ⟶ BX such that H^{∗} BV is finitely generated as a module over (Bf)^{∗}(H^{∗} BX). A morphism (V, f) ⟶ (V′, f′) in A_{X} is an injection i : V ⟶ V′ such that f′ · i is conjugate to f (equivalently, (Bf′) · (Bi) is homotopic to Bf). There is a functor α_{X} : A_{X}^{op} ⟶ Sp given by

α_{X} (V, f) = Map(BV, BX)_{Bf}

where the subscript “Bf” denotes the mapping space component corresponding to the homotopy class Bf. This diagram is interesting because there is a natural map hocolim α_{X} ⟶ BX which induces an isomorphism on mod p homology as well as a weak equivalence (hocolim α_{X})_{p̂} ∼ BX [11, 8.1].

Let ᾱ_{X} : A_{X}^{op} ⟶ Ho denote the diagram in the homotopy category which underlies α_{X}.

3.1 Theorem. For any p-compact group X the diagram ᾱ_{X} is H^{∗}-centric up to homotopy.

The proof depends on three lemmas. In these lemmas, Z denotes a p-compact group, V an elementary abelian p-group, and i : V ⟶ Z a monomorphism (see above). Let BY denote the mapping space component Map(BV, BZ)_{Bi}. (The space Y = Ω BY is a p-compact group which is called the centralizer of V in Z.) Evaluation at the basepoint of BV gives a map Bj : BY ⟶ BZ. The map Bi lifts to a map Bi′ : BV ⟶ BY such that (Bj) · (Bi′) = Bi. This lift has two key properties.

• The map Bi′ is central [11, 2.7] in the sense that evaluation at the basepoint of BV gives a weak equivalence

(3.2)  Map(BV, BY)_{Bi′} ⟶^{∼} BY .

• The map Bi′ extends to a principal fibration sequence [11, 2.8]

(3.3)  BV ⟶^{Bi′} BY ⟶ B(Y/V) .

The lift Bi′ is obtained from the usual abelian group structure on BV [10, 8.2], and sends y ∈ BV to the map BV ⟶ BX which takes x ∈ BV to Bj(x + y).

3.4 Lemma. Consider the commutative diagram

(3.5)

Map(BV, BY)_{Bi′}  ⟶^{v=Map(id,Bj)} BY = Map(BV, BZ)_{Bi}
u↓  ↓^{Bj}
BY = Map(BV, BZ)_{Bi}  ⟶^{Bj}  BZ

in which the vertical maps are given by evaluation at the basepoint of BV. Then the maps u and v are homotopy equivalences and, if u^{−1} is the inverse of u in Ho, the composite vu^{−1} is equal in Ho to the identity map of Y.

Proof. Let k be the composite of Bi with the map μ : BV ×BV = B(V ×V) ⟶ BV induced by addition V × V ⟶ V. The space in the upper left hand corner of 3.5 can be identified as Map(BV × BV, BZ)_{k}. Under this identification the map u corresponds to restriction to the factor ∗ × BV and v corresponds to restriction to the factor BV × ∗. Composition with μ thus gives a map

w : Map(BV, BZ)_{Bi} ⟶ Map(BV × BV, BZ)_{k}

such that uw and vw are identity maps. The map u is an equivalence as above (3.2); this implies that w and hence v are also equivalences. The composite v · u^{−1} is then the same in Ho as the composite (vw)(uw)^{−1} of two identity maps. □.

Let Map(BY, BY)_{[Bi′]} denote the space of all maps h : BY ⟶ BY such that h · (Bi′) is homotopic to Bi′. Similarly, let Map(BY, BZ)_{[Bi]} denote the space of all maps h : BY ⟶ BZ such that h · (Bi) is homotopic to Bi.

3.6 Lemma. The map Bj : BY ⟶ BZ induces an equivalence

Map(BY, BY)_{[Bi′]} ⟶^{∼} Map(BY, BZ)_{[Bi]} .

Proof. Since (Bj) · (Bi′) = Bi, it is clear that composition with Bj gives a map of the indicated type. By elementary homotopy theory, the fibration sequence 3.3 gives rise to a fibration p_{1} : E_{1} ⟶ B(Y/V) with fibre Map(BV, BY)_{Bi′} and space of sections Map(BY, BY)_{[Bi′]}. There is a similar fibration p_{2} : E_{2} ⟶ B(Y/V) with fibre Map(BV, BZ)_{Bi} and space of sections Map(BY, BZ)_{[Bi]}. The map Bj induces a map between these two fibrations which by 3.4 is a fibrewise equivalence; it follows that the corresponding map on spaces of sections is an equivalence. □

3.7 Lemma. The map Bj : BY ⟶ BZ is H^{∗}-centric up to homotopy.

Proof. Since homotopy classes of maps from BV to BZ or BY are detected by their effect on mod p cohomology [14, 3.1.4], it is clear that there is a commutative diagram

[BY, BY]_{{id}}  ⟶^{(Bj)·(–)} [BY, BZ]_{{Bj}}
↓  ↓
π_{0} Map(BY, BY)_{[Bi′]} ⟶^{≅} π_{0} Map(BY, BZ)_{[Bi]}

in which the vertical arrows are monomorphisms. By 3.6, the lower horizontal arrow is a bijection. To show that Bj is H^{∗}-centric up to homotopy it is enough to show that the upper horizontal arrow is a surjection.

Let h : BY ⟶ BZ be a map which has the same effect on mod p cohomology as Bj; it is necessary to find a map w : BY ⟶ BY which induces the identity map on mod p cohomology and such that (Bj) · w is homotopic to h. Consider the commutative diagram

(3.8)

Map(BV, BY)_{Bi′}  ⟶^{v′=Map(id,h)} BY = Map(BV, BZ)_{Bi}
u↓  ↓^{Bj}
BY = Map(BV, BZ)_{Bi}  ⟶^{h}  BZ

in which the vertical arrows are given by evaluation at the basepoint of BV. The map u is an equivalence by 3.4. We will be done if we can show both that the map v′ gives an isomorphism on mod p cohomology and that the composite v′u^{−1} induces the identity map on H^{∗} BY; the composite w = v′u^{−1} will then be the required map with (Bj)·w homotopic to h. Let K denote the category of unstable algebras over the mod p Steenrod algebra and T (V, –) : K ⟶ K the functor which is left adjoint to tensor product with H^{∗} BV. For a map γ : H^{∗} BV ⟶ R in K, let T (V, R)_{γ} denote the summand or “component” of T (V, R) corresponding to γ [8, §3]. The inclusion 0 ⟶ V induces a natural map e_{γ} : T (0, R) = R ⟶ T (V, R)_{γ}. If A is a space, g : BV ⟶ A is a map , and γ = g^{∗} : H^{∗} A ⟶ H^{∗} BV, there is a natural commutative diagram

(3.9)

T (V, H^{∗} A)_{γ} ⟶^{λ_{g}} H^{∗} Map(BV, A)
e_{γ} ↑  ↑
H^{∗} A  ⟶^{=}  H^{∗} A

in which the right hand vertical arrow is induced by evaluation at the basepoint of BV. The map λ_{g} is an isomorphism if, for instance, A is the classifying space of a p-compact group [11, proof of 8.1]. It is a consequence of this fact and of the naturality of 3.9 that the diagram obtained by applying the functor H^{∗}(–) to 3.8 is identical to the diagram obtained by applying H^{∗}(–) to 3.5 (note that there are no choices of isomorphisms here; the diagrams are the same). The desired properties of v′ follow from 3.4. □

Proof of 3.1. The values of the functor ᾱ_{X} are F_{p}-complete spaces because they are the classifying spaces of p-compact groups [11, 2.5]. Let ι : (V, f) ⟶ (V′, f′) be a morphism in A_{X} coming from an injection i : V ⟶ V′. Write V′ = V′′ × i(V), and let Bf′′ : BV′′ ⟶ Map(BV, BX)_{Bf} be the map which is adjoint to Bf′ : BV′′ × BV ⟶ BX. The map α_{X} (ι) can be identified as the map

Map(BV′, BX)_{Bf′} = Map(BV′′, Map(BV, BX)_{Bf})_{Bf′′} ⟶ Map(BV, BX)_{Bf}

obtained by evaluating at the basepoint of BV′′. Since Map(BV, BX) is the classifying space of a p-compact group [10, 5.1], it follows from 3.7 that α_{X} (ι) is H^{∗}-centric up to homotopy. □

§4. Vanishing lim^{1}

In this section we will give a simple way to check for vanishing of the lim^{1} sets that arise in applications of 2.4 to the theory of p-compact groups. The formulas in this section can be interpreted as nonabelian generalizations of formulas of Oliver [17].

The first step, which is mostly for notational convenience, is to reduce to certain very explicit categories. For each i ≥ 1 let A_{i} denote the F_{p} vector space (F_{p})^{i} and for n ≥ 1 let A_{n} denote the category whose objects are the vector spaces A_{i}, 1 ≤ i ≤ n, and whose morphisms are F_{p}-module monomorphisms. Suppose that X is a p-compact group and that ϕ : A_{X} ⟶ Grp is a functor (for instance, ϕ might be the functor derived from ᾱ_{X} (3.1) by the procedure of §2). Let n be the largest integer such that there is a monomorphism A_{n} ⟶ X (see §3); for the existence of such an n see [11, 8.3]. Let Φ : A_{n} ⟶ Grp be the functor given by the formula

Φ(A) = ∏_{f} ϕ(A, f) ,

in which the product is taken over the set Mono(A, X) of all monomorphisms f : A ⟶ X. The behavior of Φ on a morphism i : A ⟶ B of A_{n} is as follows: the (algebraic) monomorphism i induces a map

i^{∗} : Mono(B, X) ⟶ Mono(A, X)

and, given x = {x_{f}} ∈ Φ(A), the image of x under Φ(i) is the element y = {y_{g}} ∈ Φ(B) with y_{g} = ϕ(i)(x_{i^{∗}(g)}).

4.1 Lemma. In the above situation, there is a natural isomorphism of pointed sets between lim^{1}Φ and lim^{1}ϕ.

Proof. (cf. [8, 2.3]) Let ϕ′ be the restriction of ϕ to the full subcategory A′_{X} of A_{X} given by the objects (V, f) such that V is an object of A_{n}. It is clear that the inclusion A′_{X} ⟶ A_{X} is an equivalence of categories, and so lim^{1}ϕ′ is naturally isomorphic to lim^{1}ϕ. By inspection, however, the set Z^{1}(ϕ′) corresponds bijectively to Z^{1}(Φ) in a way which respects the equivalence relations giving lim^{1}. □

From now on, then, we will deal only with functors Φ : A_{n} ⟶ Grp. Let Φ_{1} be the restriction of Φ to the subcategory A_{1} of A_{n}. To simplify both the exposition and the formulas we will assume that n ≥ 3 and that lim^{1}Φ_{1} = ∗. This last condition is always satisfied if p = 2, since in this case A_{1} is a trivial category; the condition is also frequently satisfied for p odd, since A_{1} is then the category of a group of order prime to p (i.e. GL(1, F_{p})) and in examples arising from 2.4 the group Φ(A_{1}) is usually p-complete in some sense.

Let e^{i} : A_{i} ⟶ A_{i+1} be the standard inclusion obtained by adding a zero as the last coordinate and, for j > i, let e^{i,j} : A_{i} ⟶ A_{j} be the composite e^{j−1} ⋯ e^{i}. For j > i we will identify A_{i} with its image in A_{j} under e^{i,j}. Let P (i, j) ⊂ GL(i + j, F_{p}) be the subgroup of linear transformations which carry the subspace A_{i} to itself; these are block matrices of the form

( \begin{matrix}M_{1} & M_{2} \\ 0 & M_{3} \end{matrix} )

in which M_{1} is square of size i × i. There is a surjection r_{i} : P (i, j) ⟶ GL(i, F_{p}) which maps the above block matrix to M_{1}.

Recall that the Tits building T (A_{3}) of A_{3} is the graph with a vertex for each 1dimensional subspace V or 2-dimensional subspace W of A_{3} and an edge for each pair (V, W) with V ⊂ W.

4.2 Definition. A 1-cycle in T (A_{3}) is a circular sequence of vertices of T (A_{3}) such that any two adjacent ones are connected by an edge; equivalently, C is a a sequence

(4.3)  C = (V_{0}, W_{1}, V_{1}, W_{2}, V_{2}, …, W_{k}, V_{k})

of subspaces of A_{3} such that each V_{i} has dimension 1, each W_{i} has dimension 2, W_{i} ⊃ V_{i−1}, W_{i} ⊃ V_{i}, and V_{k} = V_{0}. A polarization of C is a collection {α_{i}, β_{i}}_{i=1}^{k} of elements of GL(3, F_{p}) such that α_{i} carries the subspace pair (W_{i}, V_{i−1}) to the standard pair (A_{2}, A_{1}), and β_{i} carries (W_{i}, V_{i}) to (A_{2}, A_{1}). The difference elements {a_{i}, b_{i}} (i = 1, ⋯, k) associated to the polarization are given by b_{i} = β_{i}α_{i}^{−1}, a_{i} = α_{i+1}β_{i}^{−1} (i < k), and a_{k} = α_{1}β_{k}^{−1}. Note that a_{i} ∈ P (1, 2) and b_{i} ∈ P (2, 1).

Suppose that n ≥ 3 and that Φ : A_{n} ⟶ Grp is a functor. If x ∈ Φ(A_{i}) and f : Φ_{i} ⟶ Φ_{j} is a morphism of A_{n}, write f_{#}(x) for the image of x under Φ(f). Observe that GL(i, F_{p}) is the group of self-maps in A_{n} of A_{i} (i ≤ n) so that there is an action of GL(i, F_{p}) on Φ(A_{i}) which sends (g, x) to g_{#}(x).

4.4 Definition. The set Z_{sp}^{1}(Φ) of special 1-cocycles for Φ is the collection of set maps ζ : GL(2, F_{p}) ⟶ Φ(A_{2}) which have the following three properties.

(1) For g, h ∈ GL(2, F_{p}), ζ(gh) = ζ(g)h_{#}(ζ(h)).

(2) If g ∈ P (1, 1), then ζ(g) = 1.

(3) Suppose that C is a 1-cycle of T (A_{3}) (4.3) with polarization {α_{i}, β_{i}} and associated difference elements {a_{i}, b_{i}}. Let ζ̃ denote the composite function

P (2, 1) ⟶^{r_{2}} GL(2, F_{p}) ⟶^{ζ} Φ(A_{2}) ⟶^{e_{#}^{2}} Φ(A_{3})

and for each i let γ_{i} denote the map (α_{1}β_{i}^{−1})_{#}. Then the following equality holds in Φ(A_{3}):

γ_{k}(ζ̃(b_{k}))γ_{k−1}(ζ̃(b_{k−1})) ⋯ γ_{1}(ζ̃(b_{1})) = 1 .

Remark. It can be checked that given conditions (1) and (2) of 4.4, condition (3) holds for all polarizations of a 1-cycle C if and only if it holds for any single polarization.

Let x be an element of the fixed set F of the action of GL(1, F_{p}) on Φ(A_{1}), and let y ∈ Φ(A_{2}) be the image of x under e_{#}^{1}. Given ζ ∈ Z_{sp}^{1}(Φ), there is another element ζ′ ∈ Z_{sp}^{1}(Φ) with ζ′(g) = yζ(g)g_{#}(y)^{−1} and the formula x · ζ = ζ′ gives an action of F on Z_{sp}^{1}(Φ). Let Z_{sp}^{1}(Φ)/∼ denote the orbit set of this action; this has a basepoint given by the orbit of the element ζ ∈ Z_{sp}^{1}(Φ) with ζ(g) = 1 for all g ∈ GL(2, F_{p}).

4.5 Proposition. Suppose that Φ : A_{n} ⟶ Grp is a functor, n ≥ 2. Let Φ_{1} be the restriction of Φ to A_{1}, and assume that lim^{1}Φ_{1} = ∗. Then there is a natural monomorphism of pointed sets

lim^{1}Φ ⟶ Z_{sp}^{1}(Φ)/∼ .

Remark. In fact, the monomorphism in 4.5 is an isomorphism, although we will not write down the proof of this. The argument consists in following what appears below and observing, with the help of the action of GL(3, F_{p}) on T (A_{3}), that GL(3, F_{p}) is the quotient of an amalgamated sum P (1, 2) ∗_{P (1,1,1)} P (2, 1) by a normal subgroup isomorphic to the fundamental group of T (A_{3}). For i > 3, GL(i, F_{p}) is isomorphic to the appropriate analog of this amalgamated sum, essentially because T (A_{i}) is 1-connected.

The proof of 4.5 relies on the following more elementary fact from linear algebra.

4.6 Lemma. If n ≥ 3 then the group GL(n, F_{p}) is generated by the subgroups P (i, j), i + j = n.

Proof of 4.5. Let z be an element of Z^{1}(Φ). According to 2.5 (adjusted for the fact that Φ is a covariant functor) z is a function which assigns to each morphism g : A_{i} ⟶ A_{j} of A_{n} an element z(g) ∈ Φ(A_{j}), such that z(g) = 1 if g is an identity morphism and

(4.7)  z(gh) = z(g)g_{#}z(h) .

If c ∈ C^{0}(Φ), then z represents the same element of lim^{1}Φ as the “cohomologous” cocycle c · z = z′ with

(4.8)  z′(g) = c(A_{j})z(g)g_{#}c(A_{i})^{−1} .

Since lim^{1}Φ_{1} = ∗ the restriction of z to A_{1} is cohomologous to the trivial cocycle; in other words, there is an element x ∈ Φ(A_{1}) such that z(g) = xg_{#}(x)^{−1} for each g ∈ GL(1, F_{p}). Define an element c ∈ C^{0}(Φ) inductively by setting c(A_{1}) = x^{−1} and c(A_{i+1}) equal to the inverse of z(e^{i})e_{#}^{i}(c(A_{i})). A calculation with 4.8 shows that the cocycle c · z = z′ satisfies two conditions:

(1) z′(g) = 1 for each g ∈ GL(1, F_{p}), and

(2) z′(e^{i}) = 1 (i = 1, …, n − 1).

Let Z′ ⊂ Z^{1}(Φ) denote the collection of all cocycles which satisfy the above two conditions. Let F denote the fixed set of the action of GL(1, F_{p}) on Φ(A_{1}), and, given x ∈ F, let c_{x} ∈ C^{0}(Φ) be defined inductively by c(A_{1}) = x, c(A_{i+1}) = e_{#}^{i}c(A_{i}). The construction (x, z) ↦ c_{x} · z gives an action of F on Z′, and by elementary calculation the subgroup {c_{x} : x ∈ F } of C^{0}(Φ) is equal to the set of elements c ∈ C^{0}(Φ) such that Z′ and c · Z′ intersect nontrivially. It follows immediately that lim^{1}Φ is isomorphic to the orbit set of the action of F on Z′. Given z ∈ Z′, let ζ be the restriction of z to GL(2, F_{p}). We will show that ζ ∈ Z_{sp}^{1}(Φ) and that the map z ↦ ζ is injective; the proposition then follows from the obvious fact that this map respects the actions of F on the objects involved.

Condition 4.4(1) for ζ follows from the cocycle condition 4.7. Observe that if g ∈ P (i, j) ⊂ GL(i + j, F_{p}) (i + j ≤ n), then there is an equality

(4.9)  z(g) = e_{#}^{i,i+j}z(r_{i}(g)) ∈ Φ(A_{i+j})

which follows from the chain

\begin{aligned}z(g) = z(g)1 & = z(g)g_{#}(z(e^{i,i+j})) = z(ge^{i,i+j}) = z(e^{i,i+j}r_{i}(g)) \\ & = z(e^{i,i+j})e_{#}^{i,i+j}(r_{i}(g))) = e_{#}^{i,i+j}(r_{i}(g))) \end{aligned} .

Here we have used that z ∈ Z′ and hence z(e^{i,i+j}) = 1. Condition 4.4(2) is derived by applying 4.9 to g ∈ P (1, 1) and noting that z(g) = 1 for g ∈ GL(1, F_{p}). Let C be a 1-cycle in T (A_{3}) (4.3) with polarization {α_{i}, β_{i}} and associated difference elements {a_{i}, b_{i}}. It is clear that there is an identity

(4.10)  a_{k}b_{k}a_{k−1}b_{k−1} ⋯ b_{1}a_{1} = 1

in GL(3, F_{p}). Since a_{i} ∈ P (1, 2) and b_{i} ∈ P (2, 1), it follows from 4.9 that z(a_{i}) = 1 and z(b_{i}) = ζ̃(b_{i}) (where the notation ζ̃ is from 4.4). Applying z to equation 4.10 and using the cocycle property of z (4.7) to expand the resulting expression now gives condition 4.4(3); part of the “expansion” is actually a contraction that uses the identity

a_{k}b_{k} ⋯ a_{i+1}b_{i+1}a_{i} = α_{1}β_{i}^{−1} .

It remains to prove that the assignment z ↦ ζ is injective. Suppose that z and z′ are two elements of Z′ which agree on GL(2, F_{p}). We will first prove by induction on m that for any 2 ≤ m ≤ n the cocycles z and z′ agree on GL(i, F_{p}) for all i ≤ m. In fact, if g ∈ P (i, j), i + j = m, then z(g) = e_{#}^{i,m}z(r_{i}(g)) = e_{#}^{i,m}z′(r_{i}(g)) = z′(g) by 4.9; thus z and z′ agree on a set of elements which generate GL(m, F_{p}) (4.6), and hence agree on the whole group by the cocycle property 4.7. Suppose now that f : A_{i} ⟶ A_{j} is an arbitrary morphism in A_{n} with i < j. It is clear that there is an element g ∈ GL(j, F_{p}) with ge^{i,j} = f, and hence

z(f) = z(ge^{i,j}) = z(g)g_{#}z(e^{i,j}) = z′(g)g_{#}z′(e^{i,j}) = z′(f) .

This shows that z and z′ are identical. □

§5. The exceptional group G_{2}

In this chapter we use 4.5 to sketch a proof of Conjecture 1.1 in the case in which p = 2, and X is the 2-completion of the exceptional compact Lie group G_{2}. In other words, we will prove that ᾱ_{X} has no nontrivial H^{∗}-clones.

Suppose that G is a compact Lie group (eventually G_{2}). Let A_{G} be the category whose objects are the non-trivial elementary abelian subgroups of G; a morphism V ⟶ V′ in A_{G} is a monomorphism f : V ⟶ V′ of abelian groups with the property that there exists an element g ∈ G such that f(x) = gxg^{−1} for all x ∈ V. As in [12], there is a functor α_{G}^{op} : A_{G} ⟶ Sp which sends V to the Borel construction EG×_{G}(G/C_{G}(V)), where C_{G}(V) is the centralizer of V in G. (Note that this Borel construction has the homotopy type of the classifying space BC_{G}(V).)

5.1 Remark. The effect on a morphism f : V ⟶ V′ of α_{G} is as follows. Let g ∈ G be an element such that f(x) = gxg^{−1} for all x ∈ V. The map h ↦ g^{−1}hg gives a homomorphism C_{G}(V′) ⟶ C_{G}(V), and so there is a G-equivariant map G/C_{G}(V′) ⟶ G/C_{G}(V) given by xC_{G}(V′) ↦ xgC_{G}(V). The induced map of Borel constructions is α_{G}(f). It does not depend upon the choice of the element g.

The following proposition is well-known; it is a consequence of [8, 2.2], [11, pf. of 8.1], and the fact that if G is a compact Lie group with π_{0}G a p-group, and V is an elementary abelian p-group, then there is a natural weak equivalence Map(BV, BG)_{p̂} ⟶ Map(BV, BG_{p̂}) [6, 4.5].

5.2 Proposition. Suppose that G is a compact Lie group such that π_{0}G is a p-group, and let X be the p-compact group G_{p̂}. Then there is an equivalence of categories e : A_{G} ⟶ A_{X} such that the composite functor α_{X} · e is weakly equivalent to (α_{G})_{p̂}.

Remark. A weak equivalence between two functors α, α′ : A_{G} ⟶ Sp is a natural transformation which gives an ordinary weak equivalence of spaces for each object of A_{G} (cf.

6.1). The functors α and α′ are weakly equivalent if they are connected by a zigzag of weak equivalences.

We will need to deal with certain elements and subgroups of SO(4). Let τ_{1} denote the central diagonal matrix diag(−1, −1, −1, −1) in SO(4), τ_{2} the matrix diag(−1, −1, 1, 1) and τ_{3} the block matrix

τ_{3} = ( \begin{matrix}N & 0 \\ 0 & N \end{matrix} ) with N = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

These matrices generate a subgroup of SO(4) isomorphic to (Z/2)^{3}. The symbol T^{2} will denote the maximal torus of SO(4) given by block matrices

( \begin{matrix}M_{1} & 0 \\ 0 & M_{2} \end{matrix} ) with M_{i} = ( \begin{matrix}cos θ_{i} & − sin θ_{i} \\ sin θ_{i} & cos θ_{i} \end{matrix} )

and T_{ext}^{2} ⊃ T^{2} the subgroup of SO(4) generated by T^{2} and τ_{3}.

If A is a subgroup of G_{2} let N_{G_{2}}(A) denote its normalizer; note that the quotient N_{G_{2}}(A)/C_{G_{2}}(A) acts faithfully on A by conjugation.

5.3 Proposition. For each integer i with 1 ≤ i ≤ 3, the group G_{2} contains up to conjugacy a unique subgroup V_{i} isomorphic to (Z/2)^{i}. The group C_{G_{2}}(V_{1}) is isomorphic to SO(4), the group C_{G_{2}}(V_{2}) is isomorphic to T_{ext}^{2}, and the group C_{G_{2}}(V_{3}) to V_{3} itself. In each case the conjugation action gives an isomorphism

N_{G_{2}}(V_{i})/C_{G_{2}}(V_{i}) ⟶^{≅} Aut(V_{i}) ≅ GL(i, F_{2}) .

Proof. This is stated for i = 1 and i = 3 in [13, 2.3]. For the case i = 2, note that since SO(4) is the centralizer of a non-trivial involution σ in G_{2} and σ is unique up to conjugacy, the conjugacy classes of homomorphisms Z/2 × Z/2 ⟶ SO(4) correspond bijectively to conjugacy classes in SO(4) of non-central involutions. Up to conjugacy there is only one such non-central involution in SO(4), which can be taken to be the matrix τ_{2}. The centralizer of τ_{2} in SO(4), which is the same as the centralizer of ⟨σ, τ_{2}⟩ in G_{2}, is T_{ext}^{2}. Since there is only one conjugacy class of homomorphism Z/2 × Z/2 ⟶ G_{2}, every automorphism of ⟨σ, τ⟩ must be realized by conjugation with an element of G_{2}; this gives the isomorphism N_{G_{2}}(V_{2})/C_{G_{2}}(V_{2}) ≅ GL(2, F_{2}). □

Choose specific subgroups V_{1} ⊂ V_{2} ⊂ V_{3} of G_{2} as in 5.3, together with an isomorphism C_{G_{2}}(V_{1}) ≅ SO(4) ⊂ G_{2}; the groups V_{i} are then subgroups of SO(4), and the choices can be made in such a way that V_{1} = ⟨τ_{1}⟩, V_{2} = ⟨τ_{1}, τ_{2}⟩ and V_{3} = ⟨τ_{1}, τ_{2}, τ_{3}⟩. Let C be the full subcategory of A_{G_{2}} generated by the V_{i} (i = 1, 2, 3) and A_{3} the category of §4. The unique basis-preserving vector space isomorphisms V_{i} ≅ A_{i} give a functor ι : C ⟶ A_{3} which by 5.3 is an isomorphism of categories. Since C is a skeletal subcategory of A_{G_{2}} (i.e., a full subcategory which contains one object of each isomorphism type) the composite of ι^{−1} : A_{3} ⟶ C with the inclusionC ⟶ A_{G_{2}} is an equivalence of categories. Let α : A^{op} ⟶ Sp be the restriction of α_{G_{2}} to A_{3}, α̂ the 2-completion of α, and F, F̂ the diagrams in Ho underlying α, α̂ respectively.

By 5.2, the following proposition is equivalent to the result stated at the beginning of this section.

5.4 Proposition. The functor F̂ : A^{op} ⟶ Ho has no H^{∗}-clones.

5.5 Notation. If G is a compact Lie group, we will let B̂G denote BG_{ˆ2}. The symbol Z_{2} will stand for the ring of 2-adic integers, and Γ_{i}(n) for the subgroup of GL(n, Z_{2}) consisting of matrices which are congruent to the identity matrix mod 2^{i}.

Proof of 5.4. Since the functor F̂ is H^{∗}-centric up to homotopy (3.1), the proposition is equivalent to the assertion that lim^{1}ϕF̂ is trivial.

The functor α̂ has the following properties:

(1) α̂(A_{1}) ∼ B̂ SO(4), α̂(A_{2}) ∼ B̂T_{ext}^{2}, and α̂(A_{3}) ∼ B̂A_{3}.

(2) The map α̂(e^{1}) : α̂(A_{2}) ⟶ α̂(A_{1}) is up to homotopy the map B̂T_{ext}^{2} ⟶ B̂ SO(4)

induced by the inclusion T_{ext}^{2} ⊂ SO(4). To study the functor ϕF̂ it is necessary to calculate the groups Aut_{i} of homotopy classes of homotopy self-equivalences of α̂(A_{i}), i = 1, 2, 3, and then determine the subgroups ϕF̂(A_{i}) of equivalences which induce the identity on H^{∗}. Picking out the groups ϕF̂(A_{i}) turns out to be relatively easy, because for i = 1, 2, 3 the restriction map H^{∗} α̂(A_{i}) ⟶ H^{∗} BV_{3} is a monomorphism. The case i = 1 of this assertion is classical, the case i = 3 is trivial, and the case i = 2 is proved either by direct calculation or by combining the exactness of Lannes’ functor T with the case i = 1 [14, 2.1, p. 203].

Let L denote the Z_{2} module given by π_{2}B̂T^{2}. The group V_{2} is the subgroup of elements of exponent 2 in T^{2}, and so the long exact homotopy sequence associated to the fibration sequence

BV_{2} ⟶ B̂T^{2} ⟶^{B̂(t↦t^{2})} B̂T^{2}

gives a natural isomorphism

(5.6)  V_{2} ≅ Z/2 ⊗ L .

Let W_{1} be the rank 2 elementary abelian 2-subgroup of GL(2, Z_{2}) generated by the matrices

( \begin{matrix}−1 & −1 \\ 0 & 1 \end{matrix} ) and ( \begin{matrix}1 & 1 \\ 0 & −1 \end{matrix} ) .

(Note that the square of each matrix is the identity, and the product of these matrices in either order is the negative of the identity matrix.) The group SO(4) is isomorphic to the quotient of Spin(4) ≅ SU(2) × SU(2) by a diagonal central Z/2, and from this it follows easily that it is possible to choose a basis {ℓ_{1}, ℓ_{2}} for L (equivalently, an isomorphism L ≅ (Z_{2})^{2}) in such a way that the conjugation image of the Weyl group of SO(4) in Aut(L) is exactly W_{1} (see [9,§3] for a similar calculation). After a possible replacement of ℓ_{2} by ℓ_{1} − ℓ_{2} the basis can be adapted to the basis {τ_{1}, τ_{2}} of V_{2}, in the sense that the reduction mod 2 (5.6) of ℓ_{i} is τ_{i}.

Let N_{1} denote the normalizer of W_{1} in GL(2, Z_{2}). By [9, §5] there is a natural isomorphism Aut_{1} ≅ N_{1}/W_{1}. The group N_{1}/W_{1} is itself isomorphic to the wreath product Σ_{2} ≀ Γ_{2}(1) [9, 5.4, pf. of 5.5]. (Note in checking this reference that the obvious map Γ_{2}(1) ⟶ GL(1, Z_{2})/⟨ ± 1⟩ is an isomorphism.) It follows easily that the subgroup ϕF̂(A_{1}) of Aut_{1} is the kernel Γ_{2}(1) × Γ_{2}(1) of the projection Aut_{1} ⟶ Σ_{2}. The inclusion of this product into N_{1}/W_{1} is induced by the map Γ_{2}(1) × Γ_{2}(1) ⟶ Γ_{1}(2) given by the formula

(5.7 .)  (u, v) ↦ ( \begin{matrix}u & (u − v)/2 \\ 0 & v \end{matrix} )

Let W_{2} denote the central subgroup of GL(2, Z_{2}) generated by the negative of the identity matrix and N_{2} = GL(2, Z_{2}) its normalizer. An analysis along the lines of [9, §5] but substantially more elementary shows that there is a natural isomorphism Aut_{2} ≅ N_{2}/W_{2}. Denote this quotient by ḠL(2, Z_{2}). The group ϕF̂(A_{2}) is the image Γ̄_{1}(2) in ḠL(2, Z_{2}) of Γ_{2}(1).

It is clear that Aut_{3} is isomorphic to GL(3, F_{2}) and that its subgroup ϕF̂(A_{3}) is trivial. The next step is to obtain information about the maps in the diagram ϕF̂. A naturality

argument shows that the map ϕF̂(e^{1}) : Γ_{2}(1) × Γ_{2}(1) ⟶ Γ̄_{1}(2) is induced by formula 5.7. The action of Aut(A_{2}) on F̂(A_{2}) is induced by the conjugation action of N_{G_{2}}(V_{2})/C_{G_{2}}(V_{2}) on BC_{G_{2}}(V_{2}); homotopically this conjugation action is the one provided by the fact that BC_{G_{2}}(V_{2}) is a regular covering space of BN_{G_{2}}(V_{2}) with covering group N_{G_{2}}(V_{2})/C_{G_{2}}(V_{2}). From this it follows that the map f : Aut(A_{2}) ⟶ Aut_{2} ≅ ḠL(2, Z_{2}) given by F̂ induces an isomorphism Aut(A_{2}) ≅ GL(2, F_{2}) upon reducing mod 2. Since the basis {ℓ_{1}, ℓ_{2}} used in making the identification Aut_{2} ≅ ḠL(2, Z_{2}) is adapted to the basis {τ_{1}, τ_{2}} for L, this isomorphism Aut(A_{2}) ≅ GL(2, F_{2}) is actually the identity map. The action of Aut(A_{2}) on Γ̄_{1}(2) given by ϕF̂ is obtained by letting Aut(A_{2}) act on Γ̄_{1}(2) by conjugation via the homomorphism f.

Denote by gl(2, F_{2}) the vector space of 2 × 2 matrices over F_{2}, and by pgl(2, F_{2}) its quotient by the subgroup generated by the identity matrix. Matrix conjugation gives actions of Aut(A_{2}) = GL(2, F_{2}) on both gl(2, F_{2}) and pgl(2, F_{2}). Let Γ̄_{i}(2) be the image in Γ̄_{1}(2) of Γ_{i}(2). By the discussion in the last paragraph there are Aut(A_{2})-equivariant isomorphisms

Γ̄_{i}(2)/Γ̄_{i+1}(2) ≅ \{ \begin{matrix}pgl(2, F_{2}) & i = 1 \\ Γ_{i}(2)/Γ_{i+1}(2) ≅ gl(2, F_{2}) & i > 1 \end{matrix}

Consider now an element ζ ∈ Z_{sp}^{1}(ϕF̂) (§4); ζ is a function GL(2, F_{2}) ⟶ Γ̄_{1}(2) which satisfies conditions (1) and (2) of 4.4. (Note that condition 4.4(3) is automatically satisfied, because ϕF̂(A_{3}) is the trivial group.) Consider the elements of GL(2, F_{2}) given by the matrices

s = ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ) and t = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) .

These generate GL(2, F_{2}) subject to the relations s^{3} = 1, t^{2} = 1, tst = s^{2}. Since ζ(t) = 1, the cocycle ζ is determined by ζ(s). The idea of the argument is now to gradually deform ζ toward the trivial cocycle, and use the completeness of Z_{2} to pass to the limit. More precisely, we will inductively construct elements w_{i} ∈ Γ_{i+1}(1) × Γ_{i+1}(1) such that if x_{i} = w_{i} ⋯ w_{1} and y_{i} = e_{#}^{1}(x_{i}), then y_{i}ζ(s)s_{#}(y_{i})^{−1} ∈ Γ_{i+1}(2). The sequence {x_{i}} then converges in the 2-adic topology on Γ_{2}(1) × Γ_{2}(1), and its limit x gives a trivialization of the cocycle ζ.

The construction of w_{1} is left to the reader since it is very similar to the inductive step we are about to describe. Assume that suitable w_{i}, together with the associated elements x_{i} and y_{i}, have been chosen for i < n (n > 1). Let ξ ∈ Z_{sp}^{1}(ϕF̂) be determined by ξ(s) = y_{n−1}ζ(s)s_{#}(y_{n−1})^{−1}, and let ξ̄ : GL(2, F_{2}) ⟶ gl(2, F_{2}) be the reduction of ξ modulo Γ_{i+1}(2). With the group operation in gl(2, F_{2}) written additively, ξ̄ satisfies the cocycle condition

(5.8)  ξ̄(gh) = ξ̄(g) + g_{#}ξ̄(h)  g, h ∈ GL(2, F_{2}) .

Here, as noted above, the action of g on ξ̄(h) is by conjugation. Expanding the left hand side of the equation ξ̄(s^{3}) = 0 with 5.8 gives ξ̄(s) + s_{#}ξ̄(s) + s_{#}^{2}ξ̄(s) = 0, which by explicit calculation implies that ξ̄(s) has the form

(5.9)  ( \begin{matrix}b + c & b \\ c & b + c \end{matrix} )

Expanding both sides of ξ̄(tst) = ξ̄(s^{2}) by the same technique gives

( \begin{matrix}b & b + c \\ c & b \end{matrix} ) = t_{#}ξ̄(s) = ξ̄(s) + s_{#}ξ̄(s) = ( \begin{matrix}b & c \\ b + c & b \end{matrix} )

which implies b = 0. If c = 0 let w_{n} be the identity element, otherwise choose w_{n} = (u, v) ∈ Γ_{2}(1)n + 1 × Γ_{2}(1)n + 1 such that (u − v)/2 is not congruent to zero mod 2^{n}. If M is the reduction mod Γ_{n}(2) of e_{#}^{1}(w_{n}), then (5.7)

M + ξ̄(s) + s_{#}(M) = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) + ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ) = 0 .

This shows that w_{n} has the required inductive property. □

§6. The cohomology category

For convenience of exposition, in this section “space” means “simplicial set”. Let CoHo denote the category whose objects are spaces and whose morphisms are cohomology classes of maps; more formally, CoHo is the quotient category of Ho in which two maps f, g : X ⟶ Y are considered equivalent if they induce the same map H^{∗} Y ⟶ H^{∗} X. Let π : Sp ⟶ CoHo be the obvious functor.

6.1 Definition. Suppose that X̄ : D ⟶ CoHo is a functor. A realization of X̄ is a pair (X, s), where X : D ⟶ Sp is a functor and s : πX ⟶ X̄ is a natural equivalence. A weak equivalence t : (X, s) ⟶ (X′, s′) between two such realizations is a natural transformation from X to X′ such that

(1) for each object d of D, t_{d} : X(d) ⟶ X′(d) is a weak equivalence of spaces, and (2) the composite natural transformation s′ · π(t) is equal to s.

The realization complex rX̄ of X̄ is defined to be the nerve of the category whose objects are the realizations of X̄ and whose morphisms are the weak equivalences between these realizations.

The aim of this section is to calculate the homotopy type of rX̄ for diagrams X̄ which satisfy a special condition. The main result, which requires some preparation to state, is a combination of 2.4 and the results in [6].

Remark. The realization complex rX̄ of 6.1 is the nerve of a category which is not small. Nevertheless this nerve is homotopically small [3] and it is possible to make standard homotopy theoretic constructions with it. The components of rX̄ are in 1−1 correspondence with weak equivalence classes {(X, s)} of realizations of X̄, and the component corresponding to a particular (X, s) is equivalent to the classifying space of an appropriate complex [3, 3.5] of homotopy automorphisms of X (in our situation this is the complex of homotopy automorphisms of X which for each object d of D induce the identity automorphism of H^{∗} X(d)).

6.2 Remark. Let X̄ : D ⟶ CoHo be a functor, D′ a small category, and F : D′ ⟶ D a functor. Let F^{∗}X̄ : D′ ⟶ CoHo be the composite of X̄ with F. It is easy to see that F induces a natural map rX̄ ⟶ r(F^{∗}X̄).

If f : A ⟶ B is a map in Sp or CoHo, let Map(A, B)_{{f}} denote the subspace of the mapping space Map(A, B) consisting of maps f′ such that H^{∗}(f′) = H^{∗}(f).

6.3 Definition. A map f : A ⟶ B between fibrant simplicial sets is said to be H^{∗}-centric if (1) [A, A]_{{id}} = π_{0} Map(A, A)_{{id}} is a group under composition, and

(2) composition with f induces a weak equivalence Map(A, A)_{{id}} ⟶ Map(A, B)_{{f}}. A map f in CoHo is said to be H^{∗}-centric if any representative of f is H^{∗}-centric (equivalently, if all representatives are H^{∗}-centric). A functor F : D ⟶ CoHo is said to be H^{∗}-centric if F (g) is H^{∗}-centric for each morphism g of D.

Remark. It is clear that a map f : A ⟶ B is H^{∗}-centric if and only if f is both H^{∗}-centric up to homotopy in the sense of §2 and centric in the sense of [6]. In particular, if X is a p-compact group, then the diagram in CoHo underlying α_{X} is H^{∗}-centric (3.1, [6, §4]).

6.4 Remark. If A is a fibrant simplicial set let h(A) denote Map(A, A)_{{id}}. Suppose that f : A ⟶ B is a map in CoHo, let M = Map(A, B)_{{f}}, G = h(A) and H = h(B). Let _{G}M denote the Borel construction of the right action of G on M, M_{H} the Borel construction of the left action of H on M, and _{G}M_{H} the corresponding double Borel construction (which is the Borel construction of the right action of G on M_{H} or equivalently the Borel construction of the left action of H on _{G}M). There are fibration sequences

\begin{matrix}M ⟶ & _{G}M ⟶ BG \\ _{G}M ⟶ & _{G}M_{H} ⟶ BH \\ M_{H} ⟶ & _{G}M_{H} ⟶ BG \end{matrix}

Suppose that f is H^{∗}-centric; this implies that _{G}M is weakly contractible and hence that GM_{H} ⟶ BH is an equivalence, so that the third fibration sequence above determines up to homotopy a map BH ⟶ BG. We will denote this map f^{♭}. The construction of f^{♭} is parallel to what was done in §2 using 2.3 to construct ϕ_{F} (g).

For any object d of D, let D↓d denote the over category [1, XI, §2] [15, p. 46] of the identity functor. The objects of this category are pairs (d′, g) where d′ is an object of D and f : d′ ⟶ d is a map; a morphism (d′, f) ⟶ (d′′, g) is a map h : d′ ⟶ d′′ such that gh = f. There is a forgetful functor O_{d} : D↓d ⟶ D which sends a pair (d, f) to d. Given a diagram X̄ : D ⟶ CoHo, there is an induced diagram

O_{d}^{∗}X̄ = X̄ · O_{d} : D↓d ⟶ CoHo

for each object d of D, as well (6.2) as a functor r_{D^{op}} X̄ : D^{op} ⟶ Sp which maps an object d ∈ D^{op} to the realization complex r(O_{d}^{∗}X̄).

As in [6, p. 577], in order to form homotopy limits [1] which have homotopy invariant meaning we sometimes have to replace a given functor Y : D ⟶ Sp by a fibrant one Y′, i.e., by a weakly equivalent functor Y′ such that Y′(d) is a Kan complex for each object d of D. We will write Y^{f} to denote such a replacement. The following theorem is proved in exactly the same way as [6, 2.1]; it is a derivative form of the basic diagram classification result in [2].

6.5 Theorem. Suppose that D is a small category and that X̄ : D ⟶ CoHo is a functor. Then the natural map

rX̄ = lim r_{D^{op}} X̄ ⟶ holim(r_{D^{op}} X̄)^{f}

is a weak equivalence.

We have now come to the main result of this section. In combination with 6.5, it expresses rX̄, for X̄ a functor which is H^{∗}-centric, as the homotopy limit of a diagram in which the constituents are classifying spaces of self-equivalences of the spaces X̄(d).

6.6 Theorem. Suppose that D is a small category and that X̄ : D ⟶ CoHo is a functor which is H^{∗}-centric. Then for each object d of D the space r_{D^{op}} X̄ is weakly homotopy equivalent in a natural way to Bh(X̄(d)). Under these equivalences, for each g : d ⟶ d′ in D the map r_{D^{op}} X̄(g) is homotopic to the map

X̄(g)^{♭} : Bh(X̄d′) ⟶ Bh(X̄d) .

described in 6.4.

Proof. This is essentially the same as the proof of [6, 2.2]. The main issue (cf. the proof of [6, 2.3]) is to show that if

A_{0} ⟶^{f_{1}} A_{1} ⟶^{f_{2}} ⋯ ⟶^{f_{n}} A_{n}

is a chain of maps in CoHo such that each f_{i} is H^{∗}-centric, then the natural projection

B(h(A_{0}), Map(A_{0}, A_{1})_{{f_{1}}}, h(A_{1}), …, Map(A_{n−1}, A_{n})_{{f_{n}}}, h(A_{n}))
↓
Bh(A_{n})

is an equivalence; the iterated bar construction on the left here is described in [4, 9.1]. This is proved by an induction [4, 9.2(vii)] that depends only on the observation made in 6.4 about the contractibility of certain ordinary Borel constructions. □

Let Grpd denote the category of groupoids, so that Grp is the full subcategory of Grpd consisting of groupoids with a single object.

6.7 Definition. If H, H′ : D ⟶ Grpd are functors, a natural transformation t : H ⟶ H′ is said to be a weak equivalence if t_{d} is an equivalence of categories for each object d ∈ D.

Let D and X̄ be as in 6.6, let Π_{1} be the composite of r_{D^{op}} X̄ with the fundamental groupoid functor, and let BΠ_{1} be the further composite with the classifying space functor. It is possible to check that finding a point in holim(BΠ_{1}) is equivalent to lifting X̄ to a functor F : D ⟶ Ho or, from another point of view, to finding a weak equivalence (6.7) between Π_{1} and a diagram of groups. If holim(BΠ_{1}) is nonempty, i.e., such a lift F exists, then Π_{1} is weakly equivalent to the diagram ϕ_{F} and holim(BΠ_{1}) is weakly equivalent as a space to holim(Bϕ_{F}). In particular π_{0} holim(BΠ_{1}) is isomorphic to lim^{1}ϕ_{F} [1, p. 309] and so corresponds bijectively (2.4) to the set of isomorphism classes of H^{∗}-clones of F. There is a natural map r_{D^{op}} X̄ ⟶ BΠ_{1} and, given a component of holim(BΠ_{1}) corresponding to an H^{∗}-clone G, the problem of lifting this component to a component of holim(r_{D^{op}} X̄)^{f} is equivalent to the problem of realizing G by a diagram of spaces.

In this way computing π_{0} holim(r_{D^{op}} X̄)^{f} can be broken down into two steps: first, compute the set of lifts of X̄ to Ho (and observe that if any such lift exists, the others are exactly its H^{∗}-clones and so are enumerated by 2.4); second, determine which of these lifts of X̄ to Ho lift further to Sp, and analyze in each case how many of these second-level lifts there are (this can be handled by [6]).

We end with a conjecture related to 1.1.

6.8 Conjecture. Let X be a p-compact group (perhaps connected) and ᾱ_{X}^{H} the diagram in CoHo which underlies α_{X}. Then the realization space rᾱ_{X}^{H} is connected.

Conjecture 6.8 asserts that the trivial clone of ᾱ_{X} is the only one which can be realized as a diagram of spaces, and adds that up to weak equivalence there is only one such realization. Technically, Conjecture 6.8 does not include 1.1; it would be possible for 6.8 to hold and 1.1 to fail if there existed nontrivial H^{∗}-clones of ᾱ_{X} which could not be realized as diagrams of spaces. In fact we conjecture that this does not happen, and that more generally the E^{2}-term of the homotopy spectral sequence for π_{∗} holim(r_{D^{op}}ᾱ_{X}^{H})^{f} (see [6, §3] and [1, XI, 6.2]) is trivial away from the y-axis.

Remark. Conjecture 6.8 can be proved for the special case in which X is the 2-completion of G_{2} by combining the result of §5 with an argument along the lines of the one in [9, §7].

References

[1] A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics vol. 304, Springer-Verlag, Berlin, 1972.

[2] , A classification theorem for diagrams of simplicial sets, Topology 23 (1984), 139–155.

[3] W. G. Dwyer and D. M. Kan, Realizing diagrams in the homotopy category by means of diagrams of simplicial sets, Proc. Amer. Math. Soc. 91 (1984), 456–460.

[4] W. G. Dwyer and D. M. Kan, Equivariant homotopy classification, J. Pure and Applied Algebra 35 (1985), 269–285.

[5] W. G. Dwyer, D. M. Kan, and J. H. Smith, Homotopy commutative diagrams and their realizations, J. Pure and Applied Algebra 57 (1989), 5–24.

[6] W. G. Dwyer and D. M. Kan, Centric maps and realization of diagrams in the homotopy category, Proc. Amer. Math. Soc. 114 (1992), 575–584.

[7] W. G. Dwyer, H. R. Miller and C. W. Wilkerson, Homotopical uniqueness of classifying spaces, Topology 31 (1992), 29–45.

[8] W. G. Dwyer and C. W. Wilkerson, A cohomology decomposition theorem, Topology 31 (1992), 433–443.

[9] W. G. Dwyer and C. W. Wilkerson, A new finite loop space at the prime two, J. Amer. Math. Soc. 6 (1993), 37–63.

[10] W. G. Dwyer and C. W. Wilkerson, Homotopy fixed point methods for Lie groups and finite loop spaces, Annals of Math. (to appear).

[11] W. G. Dwyer and C. W. Wilkerson, The center of a p-compact group, preprint (Notre Dame) 1993.

[12] S. Jackowski and J. E. McClure, Homotopy decomposition of classifying spaces via elementary abelian p-groups, Topology 31 (1992), 113–132.

[13] S. Jackowski, J. McClure and R. Oliver, Maps between classifying spaces revisited.

[14] J. Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un p-groupe abélien élémentaire, Publications Mathématiques de l’Institut des Hautes Études Scientifiques 75 (1992), 135–244.

[15] S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics vol. 5, Springer, Berlin, 1971.

[16] D. Notbohm, Homotopy uniqueness of classifying spaces of compact connected Lie groups at primes dividing the order of the Weyl group, Habilitationsschrift, Göttingen, 1992.

[17] R. Oliver, Higher limits of functors on categories of elementary abelian p-groups.

[18] K. Premadasa, Thesis (Purdue University) 1994.

University of Notre Dame, Notre Dame, Indiana 46556

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