THE CENTER OF A p–COMPACT GROUP

W.G. Dwyer and C.W. Wilkerson

University of Notre Dame Purdue University

§1. Introduction

Compact Lie groups appear frequently in algebraic topology, but they are relatively rigid analytic objects, and partially for that reason are a challenge to understand homotopically. Since H. Hopf (and H-spaces) topologists have aspired to escape the analytic straitjacket by finding some homotopy theoretic concept which would capture enough of the idea of “compact Lie group” to lead to rich and interesting structure theorems. In [12] we came up with our own candidate, the notion of p-compact group, and studied these objects to the extent of constructing maximal tori, Weyl groups, etc. In this paper we continue the study by looking at the idea of the “center” of a p-compact group and showing that two very different ways of defining the center are equivalent. This leads for instance to a reproof and generalization of a theorem from [15] about the identity component of the space of self homotopy equivalences of BG (G compact Lie). Along the way we find various familiar-looking elements of internal structure in a p-compact group X, enumerate the X ’s which are abelian in the appropriate sense, and construct what might be called the “adjoint form” of X.

Before describing in more detail the main results we are aiming at, we have to introduce some ideas from [12]. A loop space X is by definition a triple (X , BX , e) in which X is a space, BX is a connected pointed space (called the classifying space of X), and e : X → ΩBX is a homotopy equivalence from X to the space ΩBX of based loops in BX. A p-compact group is a loop space X such that

(1) X is F_{p}-finite (in the sense that H^{∗}(X; F_{p}) is finite dimensional),

(2) π_{0}X is a finite p-group, and

(3) π_{i}X (i ≥ 1) is a finitely generated module over the ring Z_{p} of p-adic integers. A homomorphism f : X → Y between loop spaces is a pointed map Bf : BX → BY; if X and Y are p-compact groups the homomorphism f is said to be a monomorphism if the homotopy fibre Y/f(X) of Bf is F_{p}-finite (in this case, if the homomorphism f is understood, we will refer to X as a subgroup of Y). If f : X → Y is a homomorphism of loop spaces, the centralizer of f(X) in Y, denoted C_{Y}(f(X)), is the loop space Ω Map(BX , BY)_{Bf}. Here Map(BX , BY)_{Bf} is the component containing Bf of the space of (unpointed) maps

Typeset by AMS-TEX

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

from BX to BY. There is a natural loop space homomorphism C_{Y}(f(X)) ⟶ Y, induced by evaluation of maps at the basepoint of BX.

A p-compact group X is said to be abelian if C_{X} (id(X)) ⟶ X is an equivalence, where id : X ⟶ X is the identity homomorphism. One of the first things we prove in the paper is

1.1 Theorem. Any abelian p-compact group is equivalent to the product of a finite abelian p-group and a p-compact torus.

Note that a p-compact torus is just the F_{p}-completion of an ordinary torus.

A homomorphism f : X → Y of p-compact groups is said to be central if the homomorphism C_{Y}(f(X)) ⟶ Y is an equivalence.

1.2 Theorem. Let X be a p-compact group. Then any central subgroup of X is abelian. Moreover, X has (in an appropriate sense) a unique maximal central subgroup, denoted C^{c}(X). If X is connected, then C^{c}(X) is a subgroup of the maximal torus of X and can be described explicitly in terms of the normalizer of the torus.

The above maximal central subgroup C^{c}(X) is called the p-compact center of X. It is possible to form a quotient p-compact group X /C^{c}(X), and it turns out that if X is connected this quotient has a trivial p-compact center. The quotient X /C^{c}(X) is what we call the adjoint form of X.

For a p-compact group X, C^{c}(X) is one candidate for the “center” of X. There is another candidate, already mentioned above. This is the loop space C_{X} (id(X)), which is denoted C^{h}(X) of X and called the homotopy center of X. We will prove the following theorem (see [11, §5] for what is in retrospect a special case).

1.3 Theorem. Let X be a p-compact group. Then the evident (§11) loop space homomorphism C^{c}(X) ⟶ C^{h}(X) is an equivalence.

This gives a generalization to non-simple groups of a fact which was proved for G = SO(3) in [9] and for general simple groups G in [15, Theorem 3].

1.4 Theorem. Let G be a compact connected Lie group. Then the natural map

B Center(G) ⟶ Map(BG, BG)_{id}

induces an isomorphism on homology with any finite coefficients.

CENTER OF A p–COMPACT GROUP 3

1.5 Organization of the paper. Section 2 gives some general properties of p-compact groups, including a recognition principle (2.15) for maximal tori, and §3 describes the sense in which certain p-compact groups have discrete approximations. Section 4 defines the notion of maximal rank subgroup and gives an “infinitesimal” criterion (4.7) for the inclusion of such a subgroup to be an isomorphism. Section 5 contains the proof of 1.1, while §6 contains a construction of the p-compact center and a study of the adjoint form. In §7 there is an explicit calculation of the center of a connected p-compact group in terms of the normalizer of a maximal torus; most of the interest here is at the prime 2, where the same complications arise for p-compact groups as for compact Lie groups. Section 7 also gives the Weyl group structure of the centralizer of a subgroup of the torus (7.6). Section 8 shows that the homology decomposition theorem of Jackowski and McClure [14] can be extended to p-compact groups, and §9 illustrates how to use this decomposition theorem to construct inductive proofs; the actual statement proved in §9 is a generalization of Miller’s Theorem [17]. There are a few technical results in §10, including a key splitting theorem (10.7). Finally, §11 has the proof of 1.3 and §12 the proof of 1.4.

1.6 Notation and terminology. In this paper p denotes a fixed prime number, F_{p} the field with p elements, Z_{p} the ring of p-adic integers, and Q_{p} the field Q ⊗ Z_{p}.

For a space X the symbol H^{∗}X denotes the mod p cohomology ring H^{∗}(X , F_{p}) and H_{Q_{p}}(X) the ring Q ⊗ H^{∗}(X , Z_{p}). The space X is F_{p}-finite if H^{∗}X is finite dimensional, and f : X ⟶ Y is an F_{p}-equivalence if it induces an isomorphism H^{∗}Y ≅ H^{∗}X. If X is F_{p}-finite then the Euler characteristic of X is the sum ∑_{i}(−1)^{i} rk_{F_{p}} H^{i}X; it turns out that this is the same as ∑_{i}(−1)^{i} rk_{Q_{p}} H_{Q_{p}}^{i}(X). The ring H_{Q_{p}}^{∗}(X) serves as a replacement for rational cohomology in cases in which the actual rational cohomology ring may be too large to be useful [12, §4].

We assume that any space in this paper has the homotopy type of a CW-complex; if not, the space can be replaced by the geometric realization of its singular complex. The word “equivalence” stands for “homotopy equivalence”. We will use Comp_{F_{p}}(–) to denote the homotopical F_{p}-completion functor constructed in [3] (see [12, §11] for a list of some of the properties of this functor); a space X is F_{p}-complete if the natural map X ⟶ Comp_{F_{p}}(X) is an equivalence.

1.7 Remark. Some of the results in this paper, including Theorem 1.1 and the first part of Theorem 1.2, have been obtained independently by Moller and Notbohm in [18].

§2. Loop spaces and p-compact groups

In this section we will summarize a few basic facts about loop spaces and p-compact groups. Note that any topological group G is a loop space in a natural way: the space BG is the usual classifying space of G. For example, any discrete group is a loop space. One way to motivate some of the definitions below is to look at what they specialize to in the case of discrete groups [12, §3].

2.1 Loop spaces. If X and Y are loop spaces, a homomorphism f : X → Y is by definition a pointed map Bf : BX ⟶ BY; the space of homomorphisms X → Y is the space Map_{∗}(BX , BY) of pointed maps BX → BY. An outer homomorphism X → Y is a map BX → BY which does not necessarily preserve the basepoints. Two homomorphisms f and g are homotopic if Bf and Bg are homotopic as pointed maps; the homomorphisms are conjugate if they are homotopic as outer homomorphisms, i.e., if Bf and Bg are freely homotopic. A homomorphism f is trivial if Bf is the constant map, and an equivalence if Bf is an equivalence. A short exact sequence

(2.2)  {1} ⟶ X ⟶^{f} Y ⟶^{g} Z ⟶ {1}

of loop spaces is by definition a fibration sequence BX ⟶^{Bf} BY ⟶^{Bg} BZ.

If f : X → Y is a homomorphism of loop spaces, the homogeneous space Y/f(X) is defined to be the homotopy fibre of Bf over the basepoint of BY; this space is denoted Y/X if f is understood. The centralizer of f(X) in Y, denoted C_{Y}(f(X)) (or C_{Y}(X) if f is understood) is the loop space of the mapping space component Map(BX , BY)_{Bf}. (The map Bf is used as a basepoint in forming this loop space.) Evaluation at the basepoint of BY gives a loop space homomorphism C_{Y}(X) ⟶ Y, and the homomorphism f : X → Y is said to be central if the map C_{Y}(X) ⟶ Y is an equivalence.

2.3 p-compact groups. A p-compact group is a loop space X such that X is F_{p}-finite and BX is F_{p}-complete, or equivalently [12, §2] a loop space X which satisfies the three conditions listed in §1. If X is a p-compact group and H ⊂ π_{0}X a subgroup, the inverse image X_{H} in X of H is a p-compact group with B(X_{H}) an appropriate covering space of BX; for instance, the identity component X_{1} = X_{{1}} of X is a p-compact group. If f : X → Y is a homomorphism of p-compact groups, then f is said to be a monomorphism if Y/X is F_{p}-finite and an epimorphism if Y/X = BZ for a p-compact group Z. In the first case X is said to be a subgroup of Y.

2.4 Proposition. [12, 2.3, 9.11] For any p-compact group X the cohomology ring H^{∗}BX is noetherian. If f : X ⟶ Y is a homomorphism of p-compact groups, then f is a monomorphism if and only if under (Bf)^{∗} the ring H^{∗}BX is finitely generated as a module over H^{∗}BY.

Note that a composite of monomorphisms is a monomorphism. In a short exact sequence as in 2.2 of p-compact groups, the homomorphism f is a monomorphism and g is an epimorphism.

If G is a compact Lie group such that π_{0}G is a p-group, we will let Ĝ denote the pcompact group with B(Ĝ) = Comp_{F_{p}} (BG) [12, §11]; as a space Ĝ is just Comp_{F_{p}} G. A p-compact torus is a p-compact group of the form T̂, where T = SO(2)^{r} is an ordinary torus; more generally, a p-compact toral group G is a p-compact group such that the identity component G_{1} is a p-compact torus. We will work extensively with p-compact toral groups because they have good technical properties, some of which are listed in the next two propositions.

2.5 Proposition. [12, §6] Suppose that f : G → X is a homomorphism of p-compact groups, where G is a p-compact toral group. Then

(1) C_{X} (G) is a p-compact group, and

(2) the map C_{X} (G) ⟶ X is a monomorphism.

2.6 Proposition. [12, 8.6] Suppose that f : G → X is a homomorphism of p-compact groups, where G is a p-compact toral group. Then f is central if and only if there exists a homomorphism G × X ⟶ X which is homotopic to f on G × {1} and to id_{X} on {1} × X.

Remark. In saying that a homomorphism h : G × X ⟶ X is “homotopic to f on G × {1}”, for instance, we mean that the restriction of Bh to BG ≅ BG × ∗ ⊂ BG × BX is homotopic to Bf. Proposition 2.6 is true without the assumption that G is a p-compact toral group (10.2), but we do not know how to give a similar generalization of 2.5.

A loop space X is said to be abelian if the identity map of X is central. Any p-compact torus is abelian, as is more generally the product of a p-compact torus and a finite abelian p-group. (In 5.2 we will show that any abelian p-compact group is of this second form.)

2.7 Proposition. [12, 8.2] Let f : G → X be a homomorphism of p-compact groups, where G is an abelian p-compact toral group. Then f lifts naturally up to homotopy to a central homomorphism G ⟶ C_{X} (G).

2.8 Proposition. [12, 8.3] Suppose that f : G ⟶ X is a central monomorphism of pcompact groups, where G is an abelian p-compact toral group. Then there is a p-compact group structure on the homogeneous space X /G and a natural short exact sequence

{1} ⟶ G ⟶ X ⟶ X /G ⟶ {1} .

Maximal tori and Weyl groups. If X is a p-compact group, a homomorphism f : T ⟶ X with T a p-compact torus is said to be a maximal torus for X if the map T ⟶ C_{X} (T) of 2.7 gives an equivalence between T and the identity component C_{X} (T)_{1} of its centralizer. Sometimes for brevity we will simply say that T is a maximal torus for X. For a maximal torus f : T ⟶ X the Weyl space W_{X ,f} of T is defined to be the space of self-equivalences of BT over BX (constructed after replacing Bf : BT ⟶ BX by an equivalent fibration).

2.9 Theorem. [12, 9.5] Any p-compact group X has a maximal torus f : T ⟶ X, unique up to conjugacy. If T is such a maximal torus, then the space W_{X ,f} is homotopically discrete and π_{0}W_{X ,f} is a finite group (under composition). The order of π_{0}W_{X ,f} is equal to the Euler characteristic χ(X /T).

If f : T ⟶ X is a maximal torus for X, the Weyl group W_{X ,f} is defined to be the group π_{0}W_{X ,f}; this is sometimes denoted W_{X ,T} or W_{X} and called the Weyl group of X. The normalizer N(T) of T is the loop space whose classifying space is the Borel construction of the action of W_{X ,f} on BT. It is clear that BN(T) is up to homotopy a fibration over BW_{X} with fibre BT, so that N(T) is not a p-compact group in general unless W_{X} is a p-group. However, the loop space N_{p}(T) obtained as the union of components of N(T) corresponding to a Sylow p-subgroup of W_{X} is always a p-compact group. The p-compact group N_{p}(T) is called a “p-normalizer” of T. By construction, the homomorphism T ⟶ X factors as a composite

(2.9)  T ⟶ N_{p}(T) ⟶ N(T) ⟶ X .

2.10 Proposition. [12, 9.9] If X is a p-compact group with maximal torus T, then the above homomorphism N_{p}(T) ⟶ X is a monomorphism, and the Euler characteristic χ(X /N_{p}(T)) is relatively prime to p.

2.11 Remark. The loop space homomorphism N(T) ⟶ X in 2.9 induces a group homomorphism f : W_{X} = π_{0}N(T) ⟶ π_{0}X. By definition the kernel of f is the Weyl group of the identity component X_{1} of X. The map f is onto; this comes down to showing that if U is any component of X /T then the homotopy fixed point set U^{hT} is nonempty [12, proof of 9.5], which is proved as in [12, proof of 8.11] by observing that χ(U) ≠ 0. Since π_{0}X is a finite p-group, it follows that the map π_{0}N_{p}(T) ⟶ π_{0}X is also onto.

Recall that an element of finite order in GL(r, Q_{p}) or GL(r, Z_{p}) is said to be a reflection (or sometimes a pseudoreflection) if it pointwise fixes a codimension 1 subspace of (Q_{p})^{r}. The rank of a p-compact torus T is the number r such that T ≅ Ĝ with G = SO(2)^{r}; this equals the rank over Z_{p} of the free module H^{2}(BT, Z_{p}).

2.12 Proposition. [12, 9.7] Let X be a connected p-compact group with maximal torus T, and let r be the rank of T. Then the natural action of the Weyl group W_{X} on H^{2}(BT, Z_{p}) is faithful and and represents W_{X} as a finite subgroup of GL(r, Z_{p}) generated by reflections.

2.13 Remark. The number r is 2.12 is called the rank of X; it is also the number of generators in the polynomial algebra H_{Q_{p}}^{∗}BX [12, 9.7]. Since H^{2}(BT, Z_{p}) is the Z_{p}-dual of π_{2}BT, it follows that the natural action of W_{X} on π_{2}BT is also faithful and represents W_{X} as a group generated by reflections.

Euler characteristics. Here we extract two useful properties of Euler characteristics from [12].

2.14 Proposition. Let f : X ⟶ Y be a monomorphism of p-compact groups, G a pcompact toral group, and g : G → Y a homomorphism of loop spaces. Suppose that one of the following conditions holds.

(1) G is a p-compact toral group and χ(Y/X) is not divisible by p.

(2) G is a p-compact torus and χ(Y/X) is not zero. Then g lifts up to conjugacy to a homomorphism g̃ : G ⟶ X.

Sketch of proof. According to [12, 3.3] this is a question of proving that in the given cases the homotopy fixed point set (Y/X)^{hG} is nonempty. Case (2) then follows from combining [12, 4.7, 5.7, 6.7] (see also the proof of [12, 8.11]). Case (1) is proved in essentially the same way, but using [12, 4.6] and the inductive argument in the proof of [12, 4.7] instead of using [12, 4.7] itself. □

2.15 Proposition. Suppose X is a p-compact group, T is a p-compact torus, and i : T ⟶ X is a monomorphism. Then i is a maximal torus for X if and only if χ(X /T) ≠ 0.

Proof. If i is a maximal torus, then χ(X /T) ≠ 0 by 2.9. Suppose then that χ(X /T) ≠ 0. Let j : T′ ⟶ X be a maximal torus for X; note that j is also a monomorphism. According to 2.14, there are homomorphisms f : T ⟶ T′ and g : T′ ⟶ T such that j · f is conjugate to i and i · f′ is conjugate to j, and a simple argument using 2.4 shows that the composites f · g and g · f are also monomorphisms. It is elementary to prove using 2.4 that any monomorphism T ⟶ T or T′ ⟶ T′ is an equivalence; for instance, this applies to f · g and g · f. It follows that f and g are equivalences, and hence that T is conjugate in X to the maximal torus T′. □

2.16 Remark. Let G be a compact Lie group with π_{0}G a p-group, and T ⟶ G a Lietheoretic maximal torus; it is a classical result that the Euler characteristic χ(G/T) is nonzero. An easy argument with the fibre lemma (cf. [12, proof of 5.7]) shows that G/T is F_{p}-good and that Ĝ/T̂ is equivalent to the F_{p}-completion of G/T. It follows that χ(Ĝ/T̂) ≠ 0 and thus by 2.15 that T̂ is a maximal torus for Ĝ.

§3. Discrete approximations

Surprisingly, some questions about p-compact groups can be reduced to questions about discrete groups. This stems mostly from the fact that there is an effective way of constructing “discrete approximations” for p-compact toral groups.

Let Z/p^{∞} denote the group Z[1/p]/Z = ∪_{n}Z/p^{n}. By definition, a p-discrete torus is a discrete group A which is isomorphic to (Z/p^{∞})^{r} for some r. A p-discrete toral group is a discrete group G with a normal subgroup A such that A is a p-discrete torus and G/A is a finite p-group. The p-discrete torus A is then the maximal divisible subgroup of G. Any p-discrete toral group G can be expressed as an increasing union of finite p-groups [12, 6.19].

The closure Ḡ of a p-discrete toral group G is defined to be the loop space Ω Comp_{F_{p}}(BG). The completion map BG ⟶ Comp_{F_{p}}(BG) is an isomorphism on mod p cohomology [12, 6.9] and gives a homomorphism G ⟶ Ḡ.

3.1 Proposition. If G is a p-discrete toral group, then Ḡ is a p-compact toral group and the homogeneous space Ḡ/G is an Eilenberg-MacLane space of type K(V, 1) for some rational vector space V.

Proof. All except the last statement is proven in [12, proof of 6.9]. By the arguments there, and the fact that the completion functor Comp_{F_{p}}(–) preserves products [3, I, §7], it is enough to show that if G = Z/p^{∞} then the homotopy fibre of the map ϵ : BG ⟶ BḠ is of the indicated type. There is a short exact sequence

{1} ⟶ Z_{p} ⟶ Q_{p} ⟶ Z/p^{∞} ⟶ {1}

of abelian groups, which gives rise to a fibration f : K(Z/p^{∞}, 1) ⟶ K(Z_{p}, 2) with fibre K(Q_{p}, 1). The target of the map f is F_{p}-complete, and f induces an isomorphism on mod p homology because the fibre of f has the mod p homology of a point. This implies that map f is in fact equivalent to the completion map ϵ and gives the required calculation of the homotopy fibre of ϵ. □

It is also possible to reverse the closure construction. A discrete approximation for a p-compact toral group G is defined to be a p-discrete toral group Ğ together with a homomorphism α : Ğ ⟶ G which induces an isomorphism H^{∗}BG ⟶ H^{∗}BĞ.

3.2 Proposition. Any p-compact toral group G has a discrete approximation Ğ ⟶ G. If G and H are p-compact toral groups with discrete approximations Ğ and H̆, then any homomorphism f : G ⟶ H lifts uniquely to a homomorphism f̆ : Ğ ⟶ H̆. The homomorphism f is central if and only if its lift f̆ is central, which is the case if and only if f̆(Ğ) is contained in the center of H̆.

Sketch of proof. The fact that discrete approximations exist is proven in [12, 6.8]. Suppose that a : Ğ ⟶ G is a discrete approximation. The map Ba : BĞ ⟶ BG induces an isomorphism on mod p homology and the target of Ba is F_{p}-complete; this implies that Ba is equivalent to the F_{p}-completion map BĞ ⟶ Comp_{F_{p}}(BĞ) and thus (3.1) that the homotopy fibre of Ba is a space of type K(V, 1) for a rational vector space V. The required lifting result now comes from obstruction theory. The final statement comes out of the observation that f is central if and only if there is a suitable homomorphism G × H ⟶ H (2.6), whereas the map f̆of K(π, 1)’s is visibly central if and only if there is a corresponding homomorphism Ğ × H̆ ⟶ H̆; one type of homomorphism can be obtained from the other by an appropriate closure/discrete approximation construction. □

3.3 Remark. Note that by 3.2 any two discrete approximations of a p-compact toral group G are canonically isomorphic (as groups). If Ğ ⟶ G is a discrete approximation, then by [12, 6.8] there is a short exact sequence

{1} ⟶ (π_{1}G) ⊗ Z/p^{∞} ⟶ Ğ ⟶ π_{0}G ⟶ {1} .

The previous two propositions allow us to shift back and forth easily between p-compact toral groups and their discrete approximations. Because of the following proposition, which we think of as expressing the density in G of a discrete approximation, this shifting process is usually innocuous.

3.4 Proposition. (Density) [12, 6.7] Let G be a p-compact toral group, α : Ğ ⟶ G a discrete approximation, and U an F_{p}-complete space (e.g., U = BX for a p-compact group X). Then Bα induces equivalences Map(BG, U) ⟶ Map(BĞ, U) and Map_{∗}(BG, U) ⟶ Map_{∗}(BĞ, U).

Remark. For example, if G is a p-discrete toral group and f : G ⟶ X is a homomorphism with X a p-compact group, then “passing to the closure” (i.e. applying the functor Comp_{F_{p}}(–) to Bf : BG ⟶ BX) gives a homomorphism f̄: Ḡ ⟶ X.

Kernels and exact sequences. One of the most useful features of a p-discrete toral group is that it has concrete elements, so that it is possible to describe kernels (or later on in §7 centers) as explicit subgroups.

Let G be a p-discrete toral group , x ∈ G an element and f : G → X a homomorphism of loop spaces. We will say that f(x) is trivial if the restriction of f to the cyclic subgroup of G generated by x is trivial (2.1). The kernel of f, denoted ker(f) is the subset of G given by {x | f(x) is trivial}. The kernel of f is said to be trivial if it contains only the identity element of G.

3.5 Proposition. [12, §7]. Suppose that G is a p-discrete toral group, X a p-compact group, and f : G → X a homomorphism. Then

(1) ker(f) is a normal subgroup of G,

(2) up to homotopy there is a unique homomorphism f′ : G/ ker(f) ⟶ X which extends f,

(3) ker(f′) is trivial, and

(4) the induced homomorphism C_{X} (G/ ker(f)) ⟶ C_{X} (G) is an equivalence.

The next proposition provides a simple way to recognize monomorphisms.

3.6 Proposition. [12, 7.3] Suppose that G is a p-compact toral group, and f : G → X is a homomorphism of p-compact groups. Let α : Ğ ⟶ G be a discrete approximation. Then f is a monomorphism if and only if ker(f · α) is trivial.

It is also useful to be able to recognize exact sequences.

3.7 Proposition. Consider a commutative diagram of loop spaces and homomorphisms

Ğ ⟶ H̆ ⟶ K̆
↓  ↓  ↓
G ⟶ H ⟶ K

in which the lower row is a sequence of p-compact toral groups and the upper row is the corresponding (3.2) sequence of their discrete approximations. Then the lower sequence is exact if and only if the upper one is.

Proof. Suppose that the upper sequence is exact; in this case the problem is to show that the induced fibre sequence of classifying spaces remains a fibre sequence after F_{p}- completion. By the fibre lemma [12, 11.7] this will be the case if the fundamental group of the base, namely K̆, acts nilpotently on the mod p homology of the fibre. Since the mod p homology groups of the fibre are finite dimensional in each dimension [12, §12], the action of K on these homology groups must factor through the finite p-group obtained by dividing K by its maximal divisible subgroup. This implies that the action is nilpotent [12, 11.6].

Suppose on the other hand that the lower sequence is exact. The map H̆ ⟶ K̆ is onto by the description of discrete approximations given in 3.3: clearly π_{0}H ⟶ π_{0}K is onto, and, since the cokernel of π_{1}H ⟶ π_{1}K is a finite abelian p-group, (π_{1}H) ⊗ Z/p^{∞} ⟶ (π_{1}K) ⊗ Z/p^{∞} is onto. Let Ğ′ be the kernel of H̆ ⟶ K̆. Elementary algebra shows that Ğ′ is a p-discrete toral group, and by the uniqueness provision of 3.2 the map Ğ ⟶ H̆ lifts to a map i : Ğ ⟶ Ğ′. Let G′ be the closure of Ğ. The argument in the paragraph above shows that G′ ⟶ H ⟶ K is an exact sequence of p-compact toral groups, so by uniqueness of homotopy fibres the map G ⟶ G′ induced by the closure of i is an equivalence, and hence (3.3) the map i is an isomorphism. □

3.8 Remark. One consequence of 3.7 is the following: if G ⟶ T ⟶ K is a short exact sequence of p-compact toral groups in which T is a p-compact torus, then G is abelian and K is a p-compact torus. The proof of this amounts to inspecting the corresponding exact sequence of discrete approximations and noticing that any subgroup of (Z/p^{∞})^{r} is abelian, and any quotient group of (Z/p^{∞})^{r} is (Z/p^{∞})^{s} for some s.

3.9 One parameter subgroups. The p-discrete torus Z/p^{∞} and its closure function like “one-parameter subgroups” of p-compact groups.

If f : X ⟶ Y is a homomorphism of loop spaces, we will write f(X) ⊆ Y_{1} (and say that f(X) lies in the identity component of Y) if f lifts to a homomorphism X ⟶ Y_{1}.

3.10 Proposition. Suppose that f : Z/p^{n} ⟶ X is a homomorphism, where X is a p-compact group . Then f(Z/p^{n}) ⊆ X_{1} if and only if f extends to a homomorphism f′ : Z/p^{∞} ⟶ X.

Proof. First suppose that the extension f′ exists, and let A be the closure of Z/p^{∞}. Then A is connected (proof of 3.1) and the fact that f(Z/p^{n}) ⊆ X_{1} follows from the fact that passing to closure gives an extension of f to a homomorphism A ⟶ X. If f(Z/p^{n}) ⊆ X_{1} then the required extension f′ exists by an inductive argument using [12, 5.5]. □

3.11 Proposition. Suppose that X is a p-compact group. Then X is connected if and only if every homomorphism f : Z/p^{n} ⟶ X (n ≥ 1) extends to a homomorphism f′ : Z/p^{∞} ⟶ X.

Proof. If X is connected, the desired extensions exist by 3.10. Suppose conversely that X is not connected, and let N̆_{p}(T) be a discrete approximation to a p-normalizer (2.10) of a maximal torus T for X. By 2.11 the map N̆_{p}(T) = π_{0}N̆_{p}(T) ⟶ π_{0}X is surjective. Since N̆_{p}(T) is a union of finite p-groups, this implies that there exists a composite homomorphism f : Z/p^{n} ⟶ N̆_{p}(T) ⟶ X such that f(Z/p^{n}) ⊈ X_{1}. □

3.12 A slight generalization. At some points in this paper it will be convenient for us to have a notion of discrete approximation somewhat more general than the one above.

Definition. A loop space G is an extended p-compact torus if π_{0}G is finite and the identity component G_{1} is a p-compact torus. A discrete group G is an extended p-discrete torus if G has a normal subgroup T of finite index such that T is a p-discrete torus.

Note that if G is an extended p-discrete torus there is a unique normal p-discrete torus T in G such that G/T is finite; in fact T is the maximal divisible subgroup of G. We will denote this unique p-discrete torus by G_{(1)}. If f : G ⟶ X is a homomorphism of loop spaces and π_{0}X is a finite group, then f induces a homomorphism G/G_{(1)} ⟶ π_{0}X and the restriction of f to G_{(1)} lifts uniquely to a homomorphism G_{(1)} ⟶ X_{1}.

Definition. A discrete approximation for an extended p-compact torus G is a homomorphism f : Ğ ⟶ G, where Ğ is an extended p-discrete torus and f induces

(1) an isomorphism Ğ/Ğ_{(1)} ⟶ π_{0}G, and

(2) an isomorphism H^{∗}BG_{1} ⟶ H^{∗}BĞ_{(1)} .

It is not hard to see that the above notion specializes to the previous notion of discrete approximation (3.2) if G is a p-compact toral group (i.e., if π_{0}G is a p-group). The following proposition is proved in exactly the same way as 3.2 (cf. [12, 6.8]).

3.13 Proposition. Any extended p-compact torus G has a discrete approximation Ğ ⟶ G. If G and H are extended p-compact tori with discrete approximations Ğ and H̆, then any homomorphism f : G ⟶ H lifts uniquely to a homomorphism f̆: Ğ ⟶ H̆.

In particular, any two discrete approximations for an extended p-compact torus are canonically isomorphic (as ordinary discrete groups).

§4. Monomorphisms of maximal rank

In this section we will work out some points connected with maximal rank subgroups of a p-compact group. The conclusion is a theorem which can be interpreted as giving an “infinitesimal” criterion for a homomorphism of connected p-compact groups to be an equivalence (4.7).

4.1 Definition. A homomorphism f : X → Y of p-compact groups is said to be of maximal rank if f is a monomorphism and there is a maximal torus i : T → X for X such that f · i is a maximal torus for Y.

4.2 Remark. It follows from the uniqueness of maximal tori (2.9) that if f : X → Y is of maximal rank then any maximal torus for X gives a maximal torus for Y. Similarly, if f is a monomorphism then f is of maximal rank if and only if some maximal torus T ⟶ Y for Y lifts to a torus T ⟶ X; such a lift is necessarily a maximal torus for X (to see this, combine 2.15 with the multiplicativity of the Euler characteristic in the fibration X /T ⟶ Y/T ⟶ X /Y, e.g. 10.6).

There is one simple way to construct homomorphisms of maximal rank.

4.3 Propositions. Let X be a p-compact group with maximal torus T ⟶ X, suppose that A is a p-compact toral group, and let j : A → T be a monomorphism. Then the natural monomorphism C_{X} (A) ⟶ X is of maximal rank.

Proof. There is a commutative diagram

C_{T} (T) ⟶ C_{T} (A) ⟶ C_{X} (A)
≅↓  ↓  ↓
T ⟶^{=} T ⟶ X

in which the left vertical arrow is an equivalence because T is abelian. This provides a lift of T up to conjugacy to C_{X} (A). □

4.4 Lemma. Suppose that f : X → Y is of maximal rank, i : T ⟶ X is a maximal torus for X, and f · i : T → Y the corresponding maximal torus for Y. Then there is a natural monomorphism of Weyl groups W_{X ,i} ⟶ W_{Y,fi}.

Remark. The above monomorphism W_{X ,i} ⟶ W_{Y,fi} depends only up to inner automorphism on the choice of a torus T ⟶ X for X. Sometimes for brevity we will say that f induces a monomorphism W_{X} ⟶ W_{Y}.

Proof of 4.4. By definition, the Weyl group W_{X ,i} is π_{0}W_{X ,i}, where W_{X ,i} is the space of self-maps over BX of a fibration u : U → BX equivalent to i : BT ⟶ BX. Similarly, W_{Y,fi} = π_{0}W_{Y,fi} where W_{Y,fi} is the space of self-maps over BY of a fibration v : V → BY equivalent to fi : BT ⟶ BY. We can assume without loss of generality that f : BX ⟶ BY is a fibration and choose v = f · u; in this case there is essentially an inclusion W_{X ,i} ⟶ W_{Y,fi} of topological monoids and it is not hard to check that the induced component homomorphism W_{X ,i} ⟶ W_{Y,fi} does not depend on the fibration choices involved. This gives the required map W_{X ,i} ⟶ W_{Y,fi}; it remains to show that the map is a monomorphism.

As in [12, 9.5] the group W_{X ,i} is isomorphic to the set of components of the homotopically discrete space (X /T)^{hT}, and W_{Y,fi} to the set of components of (Y/T)^{hT} (see [12, §10] for a discussion of these homotopy fixed point sets). The map between these groups can be produced on the level of sets by applying the functor (–)^{hT} to the map X /T ⟶ Y/T induced by f. Let C be the component of (Y/X)^{hT} corresponding to i : T → X [12, 3.3]. Applying (–)^{hT} to the fibration X /T ⟶ Y/T ⟶ Y/X [12, 10.6] shows that the map (X /T)^{hT} ⟶ (Y/T)^{hT} is an injection on π_{0} if and only if the natural action of π_{1}C on π_{0}(X /T)^{hT} is trivial. To prove this action is trivial, we will prove that C is simply connected. Applying (–)^{hT} to the fibration Y/X ⟶ BX ⟶ BY (with T “acting” [12, 10.8] trivially on base and total space) shows that C is the basepoint component of C_{Y}(T)/C_{X} (T). This last homogeneous space is in fact homotopically discrete because C_{X} (T) ⟶ C_{Y}(T) is a homomorphism of p-compact groups which is an equivalence on identity components, both identity components being equivalent to T (2.9). □

If X is a space let cd_{F_{p}} X denote the largest integer n such that H^{n}X ≠ 0. Similarly, let cd_{Q_{p}} X denote the largest integer n such that H_{Q_{p}}^{n}X ≠ 0.

4.5 Lemma. If X is a p-compact group then cd_{F_{p}} X = cd_{Q_{p}} X.

Proof. This follows from the Bockstein spectral sequence arguments in [4]. The arguments depend only on the fact that X is an F_{p}-finite H-space. □

4.6 Lemma. Suppose that f : X ⟶ Y is a monomorphism of p-compact groups. Then (1) cd_{F_{p}} Y = cd_{F_{p}} X + cd_{F_{p}} Y/X,

(2) cd_{F_{p}} Y = cd_{F_{p}} X if and only if the map f gives an equivalence between X and a union of components of Y, and

(3) If X is connected then cd_{Q_{p}} Y = cd_{Q_{p}} X + cd_{Q_{p}} Y/X.

Proof. The statements involving cd_{F_{p}} Y are from [12, 6.14]. The equality in (3) follows along the lines of [12, proof of 6.14] from an examination of the Serre spectral sequence of the fibration X ⟶ Y ⟶ Y/X. Note that in this fibration the fundamental group of the base acts trivially on the (co)-homology of the fibre, since the fibration is pulled back from the path fibration over the simply connected space BX. □

4.7 Theorem. A monomorphism f : X → Y between connected p-compact groups is an equivalence if and only if f is of maximal rank and induces (4.4) a Weyl group isomorphism W_{X} ⟶ W_{Y}.

Proof. It is only necessary to show that if f is of maximal rank and induces a Weyl group isomorphism then f is an equivalence. Let T be a maximal torus for X, let W denote W_{X ,T} ≅ W_{Y,T}, and let R denote the ring of invariants (H_{Q_{p}}^{∗}BT)^{W}. Write S for the ring Q_{p} ⊗_{R} H_{Q_{p}}^{∗}BT. According to [12, 9.7], H_{Q_{p}}^{∗}(X /T) and H_{Q_{p}}^{∗}(Y/T) are both isomorphic to S. By 4.6(3), cd_{Q_{p}} X = cd_{Q_{p}} Y and hence cd_{F_{p}} X = cd_{F_{p}} Y by 4.5. The result now follows from 4.6(2). □

§5. Abelian p-compact groups and central monomorphisms

The main goal of this section is to prove the following theorem.

5.1 Theorem. If f : X → Y is a central monomorphism of p-compact groups, the X is equivalent to the product of a p-compact torus and a finite abelian p-group.

This is analogous to the theorem that if f : G → H is a monomorphism of compact Lie groups with f(G) contained in the center of H, then G is isomorphic to the product of a torus with a finite abelian group. Applying 5.1 to the identity map of a p-compact group X gives the following corollary (the “if” part of which is essentially obvious).

5.2 Corollary. A p-compact group X is abelian if and only if X is equivalent to the product of a p-compact torus and a finite abelian p-group.

Remark. A beautiful result of Bousfield [2, 6.9] implies that if X is a loop space which is abelian in the sense of §2 then BX is equivalent to a product of Eilenberg-MacLane spaces. Given this result, it is possible to prove 5.2 by just enumerating the products of Eilenberg-MacLane spaces which can be classifying spaces of p-compact groups. We will use a different approach.

The proof of 5.1 depends on a few lemmas.

5.3 Lemma. Suppose that f : X → Z and g : Y ⟶ Z are homomorphisms of loop spaces and that g is central. Then there is up to conjugacy a unique homomorphism μ(f, g) : X × Y ⟶ Z which is conjugate to f on X × {1} and to g on on {1} × Y. The homomorphism μ(f, g) is central if f is.

Proof. This follows from the equivalence

Map(BX × BY, BZ) ≅ Map(BX , Map(BY, BZ)) .

The assumption that g is central gives Map(BY, BZ)_{Bg} ≅ BZ. □

5.4 Lemma. Let G be a p-discrete toral group, X a p-compact group, and f : G ⟶ X a homomorphism. Suppose that A is a cyclic p-group with generator a and i, j : A ⟶ G two homomorphisms such i(a) commutes with j(a), f · i is conjugate to f · j, and f · i is central. Then i(a)^{−1}j(a) ∈ ker(f).

Proof. Let s : A × A ⟶ G be given by s(x) = i(x)j(x). By the uniqueness provision of 5.3, the composite f · s is equal to the composite of f · i with the sum map A × A ⟶ A. This implies directly that s(a^{−1}, a) ∈ ker(f). □

5.5 Lemma. Let g : G ⟶ H be a homomorphism of p-discrete toral groups and h : H ⟶ X a homomorphism with ker(h) = {1}, where X is a p-compact group. Suppose that h · g is central. Then g is central.

Proof. Since h·g is central there is by 5.3 a homomorphism G×X ⟶ X which is conjugate to h · g on G × {1} and to id_{X} on {1} × X. This can be restricted to give a homomorphism f : G × H ⟶ X which is conjugate to h · g on G × {1} and to h on {1} × H. By 5.4, for each x ∈ G the element (x, g(x)^{−1}) in G × H lies in ker(f). Since ker(f) is a normal subgroup of G × H which intersects {1} × H only in the identity element, it follows from elementary algebra that g(G) must lie in the center of H. □

5.6 Lemma. Suppose that P is a p-compact toral group, X is a p-compact group, and f : P → X is a central monomorphism. Then P is equivalent to the product of a p-compact torus and a finite abelian p-group.

Proof. Let P̆ be a discrete approximation (3.2) for P, and f̆ the composite of f with the map P̆ → P. Since ker(f̆) is trivial (3.6), it follows from 5.5 that P̆ is abelian and hence for algebraic reasons isomorphic to the product of a finite abelian p-group and a group of the form (Z/p^{∞})^{r}. This implies that P, which is equivalent to the closure (3.1) of P̆, has the desired form. □

Proof of 5.1. Let i : T → X be a maximal torus for X with discrete approximation T̆ → T. Let ĭ be the composite of i with T̆ → T. The composites f · i and f · ĭ are central by 2.6. Suppose that the action of the Weyl group W of the identity component X_{1} of X on H^{2}(BT; Z_{p}) is nontrivial. In this case there must be a self homotopy equivalence Bw : BT → BT, not homotopic to the identity, such that Bi · Bw is homotopic to Bi. Such a w lifts uniquely (3.2) to an automorphism w̆ : T̆ → T̆, not equal to the identity, such that ĭ· w̆ is conjugate to ĭ. Since w̆ is not the identity, there must be some element x ∈ T̆ such that w̆(x) ≠ x. By 5.4, then, (x^{−1}w̆(x)) ≠ 1 belongs to ker(f̆· i), which is impossible (3.6) because f · i is a monomorphism. This shows that W acts trivially on H^{2}(BT; Z_{p}) and hence, since this action is faithful [12, 9.7], that W itself is trivial. By 4.7 (applied to T ⟶ X_{1}), the p-compact group X_{1} is a torus. Since X is thus a p-compact toral group, the desired result is given by 5.6. □

§6. p-compact centers and adjoint forms.

Let f : C → X be a central monomorphism of p-compact groups. The map f is said to be a p-compact center for X (or more informally C is said to be a p-compact center for X) if for any central monomorphism g : A → X of p-compact groups there exists up to homotopy a unique homomorphism h : A → C such that f · h is homotopic to g. This signifies that up to homotopy f is a terminal object in the category of central monomorphisms A → X.

6.1 Theorem. Any p-compact group X has a p-compact center f : C → X. Suppose that f is such a p-compact center and that g : A → X is a central homomorphism in which A is a p-compact toral group; let L(g, f) be the space of all homomorphisms h : A ⟶ C such that f · h = g. Then L(g, f) is contractible.

Remark. Note above that if A ⟶ X is a central monomorphism, then A is necessarily a p-compact toral group (5.1).

6.2 Remark. Let (Bf)^{fib} : (BC)^{fib} ⟶ BX be a pointed fibration equivalent to Bf. By the above phrase the space of all homomorphisms h : A ⟶ C such that f · h = g we mean the space of all basepoint preserving maps Bh : BA ⟶ (BC)^{fib} such that (Bf)^{fib} · Bh = Bg.

Theorem 6.1 implies that p-compact centers are homotopically unique in a very strong sense. For this reason we will sometimes speak of the p-compact center of X and denote it by C^{c}(X). According to 2.8 there is a natural p-compact group structure on X /C^{c}(X).

6.3 Theorem. If X is a connected p-compact group then the p-compact center of X /C^{c}(X) is trivial.

Remark. If X is a connected p-compact group, the quotient X /C^{c}(X) is called the adjoint form X_{ad} of X. Note that if X is not connected then X /C^{c}(X) may have a nontrivial p-compact center; consider, for instance, a nonabelian finite p-group or the Lie group O(2).

Let X be a p-compact group, C a p-discrete toral group, and f : C ⟶ X a central homomorphism such that ker(f) = {1}. We will say that f is a p-discrete center for X if the induced map f̄: C̄ ⟶ X is a p-compact center for X. Such a p-discrete center is just a discrete approximation to C^{c}(X) and has very similar uniqueness properties. In the course of proving 6.1 we will give the following description of the p-discrete center of a p-compact group X; for connected X a more explicit description appears in §7.

6.4 Theorem. Let X be a p-compact group, T a maximal torus for X, N_{p}(T) the inverse image in the normalizer of T of a p-Sylow subgroup of the Weyl group of X, and N̆_{p}(T) a discrete approximation for N_{p}(T). Let C̆ ⊂ N̆_{p}(T) be the set of elements x ∈ N̆_{p}(T) such that ⟨x⟩ ⟶ X is central. Then C̆ is a subgroup of N̆_{p}(T) and C̆ ⟶ X is a p-discrete center for X.

6.5 Lemma. Let G be a p-compact toral group and g : G ⟶ X a monomorphism of p-compact groups. Suppose that that H is a p-compact toral group, and that h : H ⟶ X is a central homomorphism. Then if h lifts to any homomorphism h̃ : H ⟶ G it lifts to exactly one, in the sense that the space (cf. 6.2) of all such lifts h̃ is contractible.

Proof. Suppose that h lifts to a homomorphism h̃ : H ⟶ G. Let i : H̆ ⟶ H be a discrete approximation, and let K̆ ⊂ H̆ be the kernel (3.5) of h · i. It is clear from 3.6 that K̆ is also the kernel of h̃ · i, since otherwise, after (3.2) lifting h̃ · i to a map H̆ ⟶ Ğ for some discrete approximation Ğ of G, we would find a nontrivial element in the kernel of the composite Ğ ⟶ G ⟶ X. Let H′ be the closure (3.1) of H̆/K̆. By naturality of the closure construction there is a homomorphism H ⟶ H′. The map h extends (3.5, 3.4) to a central monomorphism h′ : H′ ⟶ X. Similarly, h̃ extends to a monomorphism h̃′ : H′ ⟶ G which is a lift of h′. An argument using 5.5 and discrete approximations shows that h̃′ is central.

Consider the diagram (2.8)

BG ⟶ B(G/h̃′(H′))
Bg↓  ↓
BH ⟶^{Bh} BX ⟶ B(X /h′(H′))

in which the right hand square is a homotopy fibre square and the composite of the two lower arrows is null homotopic. Let E be the homotopy pullback of BG ⟶ BX over Bh. It follows that there is a homotopy equivalence over BH between E and BH × (X /G), and so the space of lifts up to conjugacy of h into G (i.e. the homotopy fibre over Bh of Map(BH, BG) ⟶ Map(BH, BX)) is equivalent to Map(BH, X /G). This last space is equivalent by evaluation at the basepoint of BH to X /G itself [12, 6.1 and 5.3]. There is now a 3 × 3 square

F  ⟶  X /G  ⟶^{=} X /G
↓  ↓  ↓
Map_{∗}(BH, BG)_{{Bh}} ⟶ Map(BH, BG)_{{Bh}} ⟶ BG
↓  ↓  ↓
Map_{∗}(BH, BX)_{Bh} ⟶ Map(BH, BX)_{Bh} ⟶ BX

in which the rows and columns are fibration sequences; a subscript “{Bh}” denotes restricting to the components of the indicated mapping space that cover Bh up to homotopy. The space F is the desired space of lifts h̃, and is clearly contractible. □

6.6 Lemma. Let X be a p-compact group, P a p-discrete toral group, and f : P → X a monomorphism. Let C ⊂ P be the set of all elements x ∈ P such that f(⟨x⟩) is central. Then C is a subgroup of P and f(C) is central.

Proof. By 5.5, C is contained in the center of P. It is enough to show that if {C_{σ}}_{σ∈S} is a countable collection of subgroups of the center of P such that f(C_{σ}) is central for each σ ∈ S, then the subgroup of the center of P generated by the C_{σ} is central. For this, observe first that if C_{1} and C_{2} are two subgroups of the center of P such that f(C_{1}) and f(C_{2}) are central, then C_{1} + C_{2} is central; this is a consequence of 5.3 and 3.5. Secondly, apply a homotopy limit argument [12, 6.20] to show that if C_{0} ⊂ C_{1} ⊂ C_{2} ⋯ is an increasing chain of subgroups of the center of P such that f(C_{i}) is central for each i, then f(∪_{i}C_{i}) is central. □

Proof of 6.1 and 6.4. We will use the notation of 6.4. By 6.6 the group C̆ is a subgroup of the center of N̆_{p}(T) and f(C̆) is central. Let C ⟶ X be the closure of C̆; by density the map C ⟶ X is central and by 3.6 it is a monomorphism (cf. 2.10). Suppose that f : A ⟶ X is a central homomorphism in which A is a p-compact toral group; for instance, f might be a central monomorphism (5.1). Since χ(X /N_{p}(T)) is relatively prime to p (2.10), the map f lifts by 2.14 to a central (5.5) homomorphism g : A ⟶ N_{p}(T). Let Ă ⟶ A be a discrete approximation. By 3.2 the composite Ă ⟶ A ⟶ N_{p}(T) lifts to a homomorphism ğ : Ă ⟶ N̆_{p}(T) such that, by 2.6 and the definition of C̆, f(Ă) ⊂ C̆. Passing to closures gives a lift A ⟶ C of the homomorphism f. This proves in particular that C is a p-compact center for X. The statement about L(g, f) follows from 6.5. □

6.7 Remark. The subgroup C̆ of N̆_{p}(T) considered in the above proof is contained in the center of N̆_{p}(T) (5.5). Let T̆ be the maximal divisible subgroup of N̆_{p}(T), so that T̆ is a discrete approximation to the maximal torus T. If X is connected the quotient group N̆_{p}(T)/T̆ ⊂ W_{X} acts faithfully on π_{2}T (2.13) and hence faithfully by conjugation on T̆ = π_{2}T ⊗ Z/p^{∞} (3.3). It follows that if X is connected the group C̆ is contained in T̆ and so, after passing to closures, that the map C^{c}(X) ⟶ X lifts to a unique (6.5) homomorphism C^{c}(X) ⟶ T.

6.8 Lemma. Suppose that

{1} ⟶ A ⟶^{i} X ⟶ Y ⟶ {1}

is a short exact sequence of p-compact groups in which A is an abelian p-compact toral group and Y is connected. Then the monomorphism i is central.

Proof. Since A is abelian the identity component Map(BA, BA)_{id} of the space of self homotopy equivalences of BA is equivalent to BA. This implies that any fibration over a simply connected base with fibre BA is principal. In particular the fibration of classifying spaces corresponding to the above short exact sequence is principal, and the corresponding action map BA×BX ⟶ BX has the properties required to show that i is central (2.6). □

Proof of 6.3. Let T be a maximal torus for X. As remarked above (6.7), the homomorphism C^{c}(X) ⟶ X lifts to a homomorphism C^{c}(X) ⟶ T; this lift is a central (5.5) monomorphism (3.6). Let X′ denote X /C^{c}(X) and let T′ denote T/C^{c}(X) (see 2.8). By 3.8 the p-compact group T′ is a p-compact torus. Consider the map of short exact sequences

\begin{aligned}\{1\} & ⟶ & C^{c}(X) & ⟶ & T & ⟶ & T′ & ⟶ & \{1\} \\ & & ^{=}↓ & & ↓ & & ↓ \\ \{1\} & ⟶ & C^{c}(X) & ⟶ & X & ⟶ & X′ & ⟶ & \{1\} \end{aligned} .

Taking vertical fibres in the induced diagram of classifying spaces shows that X /T ≅ X′/T′, so that T′ ⟶ X′ is a maximal torus for X′ (2.15). Suppose that C′ ⟶ X′ is a p-compact center and let C ⟶ X be the “inverse image” of C′ in X, i.e., let BC be given by the homotopy fibre square

(6.9)  \begin{matrix}BC & ⟶ & BX \\ ↓ & & ↓ \\ BC′ & ⟶ & BX′ \end{matrix}

It is clear from a homotopy group calculation that C is a p-compact toral group. Since X′ is connected, the map C′ ⟶ X′ lifts to a monomorphism C′ ⟶ T′ as above, and so the map C ⟶ X lifts to a monomorphism C ⟶ T (the fact C ⟶ T is a monomorphism follows from the fact that T/C is equivalent to T′/C′). By 3.8, then, C is abelian. There is an evident short exact sequence C ⟶ X ⟶ X′/C′, and applying 6.8 to it shows that the map C ⟶ X is central. As a consequence, the homomorphism C ⟶ X lifts to a homomorphism C ⟶ C^{c}(X) and so the composite C ⟶ X ⟶ X′ is trivial. An argument with discrete approximations, for instance, now shows that C′ ⟶ X′ is trivial and thus that C is trivial (3.7, 3.6). □

§7. Computing p-compact centers

In this section we will give an explicit way to compute the p-compact center of a pcompact group X in terms of the normalizer of a maximal torus for X. Suppose that X is a p-compact group of rank r (2.13), T is a maximal torus for X, and W = W_{X} is the Weyl group of X. Let N(T) be the normalizer of T, N̆(T) ⟶ N(T) a discrete approximation in the sense of §3.12, and T̆ the maximal divisible subgroup N̆(T)_{(1)} of N̆(T), so that T̆ is a discrete approximation in the usual sense for N(T)_{1} = T.

The Weyl group W acts on T̆ by conjugation in the group extension

(7.1)  {1} ⟶ T̆ ⟶ N̆(T) ⟶ W ⟶ {1}

and on BT by monodromy in the fibration

(7.2)  BT ⟶ BN(T) ⟶ BW .

The fibration 7.2 is obtained from the exact sequence 7.1 by passing to classifying spaces and taking fibrewise F_{p}-completion; under this construction the two actions correspond.

Let s ∈ W be a reflection (2.12). The order ord(s) of s must divide the order of the multiplicative group of roots of unity in Q_{p}^{∗}, so that ord(s) = 2 if p = 2 and ord(s) divides (p − 1) if p is odd. Since H^{2}(BT, Z_{p}) is the dual over Z_{p} of π_{2}BT = (Z_{p})^{r}, a nonidentity element s ∈ W is a reflection if and only if the fixed point set of the action of s on π_{2}BT = π_{1}T contains a subgroup isomorphic to (Z_{p})^{r−1}, or equivalently if and only if the action of s on T̆ = π_{1}T ⊗ Z/p^{∞} (see 3.3) contains a subgroup isomorphic to (Z/p^{∞})^{r−1}.

7.3 Definition. If s ∈ W is a reflection,

(1) the fixed point set F (s) of s is the fixed point set of the action of x on T̆ by conjugation,

(2) the singular hyperplane H(s) of s is the maximal divisible subgroup of F (s) (so that H(s) ≅ (Z/p^{∞})^{r−1}),

(3) the singular coset K(s) of s is the subset of T̆ given by elements of the form x^{ord(s)}, as x runs through elements of N̆(T) which project to s in W, and

(4) the singular set σ(s) of s is the union σ(s) = H(s) ∪ K(s).

7.4 Remark. The singular coset K(s) is genuinely a coset of H(s) in T̆. On the one hand, suppose that x and x′ are two elements of N̆(T) which project to s. Then x = x′a for some a ∈ T̆, and it is easy to calculate directly that x^{ord(s)} = (x′)^{ord(s)}ν(a), where ν is the endomorphism of T̆ which sends a to the product of the elements s^{i}a, 1 ≤ i ≤ ord(s). The image of ν is a divisible subgroup of T̆ which is pointwise fixed by s and thus contained in H(s). It follows that x^{ord(s)} and (x′)^{ord(s)} are in the same coset of H(s). Conversely, if y = x^{ord(s)} ∈ K(s) and a ∈ H(s), it is possible (because H(s) is divisible) to find b ∈ H(s) with b^{ord(s)} = a. Then ay = (bx)^{ord(s)}.

Our main theorem in this section is the following one.

7.5 Theorem. Let X be a connected p-compact group, T a maximal torus for X, and T̆ ⟶ T a discrete approximation for T. Define C ⊂ T̆ by

C = ∩_{s}σ(s)

where the intersection is indexed by reflections s ∈ W. Then C ⟶ X is a p-discrete center (6.4) for X.

This is actually a corollary of a more general calculation. Suppose that A ⊂ T̆ is a subgroup. Let W_{X} (A) denote the Weyl group of C_{X} (A), and W_{X} (A)_{1} the Weyl group of the identity component C_{X} (A)_{1} of C_{X} (A). There are inclusions W_{X} (A)_{1} ⊂ W_{X} (A) ⊂ W, where the last follows from §4.

7.6 Theorem. Let X be a connected p-compact group with maximal torus T and Weyl group W. Suppose that A ⊂ T̆ is a subgroup. Then

(1) W_{X} (A) is the subgroup of W consisting of the elements which, under the conjugation action of W on T̆, pointwise fix the subgroup A, and

(2) W_{X} (A)_{1} is the subgroup of W_{X} (A) generated by those elements s ∈ W_{X} (A) such that s ∈ W is a reflection and A ⊂ σ(s).

7.7 Remark. The above theorems raise some natural questions about the structure of σ(s). By definition there are inclusions

H(s) ⊂ σ(s) = H(s) ∪ K(s) ⊂ F (s) ,

and K(s) is a coset of H(s). If p is odd, then s acts on T̆/H(s) ≅ Z/p^{∞} with only the identity as a fixed point (this follows from the fact that at an odd prime a p-adic root of unity has a nontrivial reduction mod p), and so the above chain of inclusions collapses to H(s) = σ(s) = F (s). If p = 2 the situation is more complex. In this case the reflection s is of order 2 and acts on T̆/H(s) ≅ Z/2^{∞} by inversion; the fixed subgroup here is of order 2, and this potentially allows H(s) to be of index 2 in F (s). There are three possibilities (we list with each one an example of a compact Lie group G such that the possibility is realized in Ĝ):

(1) H(s) ⊊ σ(s) = F (s): SU(2).

(2) H(s) = σ(s) ⊊ F (s): SO(3).
(3) H(s) = σ(s) = F (s): U(2)

Note that in all cases σ(s) is a subgroup of T̆.

Remark. The above description of σ(s) shows that if p is odd, then

(1) the p-discrete center C of X ( 7.5) is the center of N̆(T) (i.e. the subgroup of T̆ which is fixed by the conjugation action of W), and

(2) the group W_{X} (A)_{1} of 7.6 is the subgroup of W_{X} (A) generated by the reflections contained in W_{X} (A).

If p = 2 then the p-discrete center of X is contained in the center of N̆(T) but is not necessarily equal to it, unless of course the center of N̆(T) is trivial. This last possibility actually comes up if X is the 2-compact group DI(4) from [11]; in this case X is of rank 3 and it is possible to verify that the action of the Weyl group W on the elements of exponent 2 in T̆ gives an epimorphism W ⟶ GL(3, F_{2}) (see [11, §4] for instance). This implies that the center of N̆(T) is trivial, hence that the p-discrete center C of X is trivial and that the p-compact center C^{c}(X) is trivial.

Proof of 7.5 (given 7.6). Let A be a subgroup of T̆. By 7.6 and 4.7 the map A ⟶ X is central if and only if A ⊂ σ(s) for each reflection s in the Weyl group W of X (recall that W is generated by such reflections). The result follows from the description of the p-discrete center of X provided by 6.4, and the fact (6.7) that since X is connected the p-discrete center is contained in T̆. □

7.8 Lemma. Suppose that X is a connected p-compact group, A is a p-discrete torus and i : A → X is a homomorphism. Then C_{X} (i(A)) is connected.

Proof. It is enough (3.11) to show that any homomorphism Z/p^{n} ⟶ C_{X} (A) (n ≥ 0) extends to a homomorphism Z/p^{∞} ⟶ C_{X} (A) or equivalently that any homomorphism Z/p^{n} ×A ⟶ X extends to a homomorphism Z/p^{∞} ×A ⟶ X. This is easily seen to be true if X is a p-compact torus. For the general case, then, it is enough to choose a maximal torus T ⟶ X and show that any homomorphism f : Z/p^{n} × A ⟶ X lifts up to conjugacy to a homomorphism f̃: Z/p^{n} × A ⟶ T. To simplify notation, let C denote Z/p^{n}. Such a lift f̃ exists if and only if the homotopy fixed point set

(X /T)^{h(C×A)} ≅ ((X /T)^{hC})^{hA}

is nonempty (see [12, 3.3, 10.5]). The “left translation action” of C via f on X /T (see [12, 3.3]) extends to an action of the connected loop space X and so is homotopically trivial, i.e., each element of C acts by a self-map of X /T which is homotopic to the identity. This implies that the Lefschetz number of the action of C on X /T is equal to the Euler characteristic of X /T and so in particular is nonzero (2.9). This Lefschetz number is equal to the Euler characteristic (X /T)^{hC} [12, 4.5, 5.7], and so χ((X /T)^{hC}) is nonzero. By [12, 4.7, 5.7], the space ((X /T)^{hC})^{hA} is not empty. □

7.9 Proof of 7.6(1). Let W′ ⊂ W be the group of elements which, under the conjugation action of W on T̆, pointwise fix the subgroup A, and let C denote the centralizer of A in N̆(T). By calculation there is an exact sequence (of discrete groups)

{1} ⟶ T̆ ⟶ C ⟶ W′ ⟶ {1} .

The homomorphism N̆(T) ⟶ X induces a homomorphism C ⟶ C_{X} (A) which immediately gives the inclusion W′ ⊂ W_{X} (A). Conversely, if W_{X} (A) is not contained in W′, then there are elements a ∈ A and w ∈ W_{X} (A) such that w · a ≠ a. Let C ⊂ A be the cyclic subgroup generated by a, let i : C ⟶ T̆ be the inclusion homomorphism, and j : C ⟶ T̆ the composite of i with the conjugation action of w. Clearly i and j become conjugate when composed with the homomorphism T̆ ⟶ C_{X} (A), and so it follows from 5.4 that a(w · a)^{−1} belongs to the kernel of T̆ ⟶ C_{X} (A). This is impossible, since the composite T̆ ⟶ C_{X} (A) ⟶ X has a trivial kernel. □

7.10 Lemma. Let X be a p-compact group, A an abelian p-discrete toral group, and f : A ⟶ X a homomorphism. Let U and V be two subgroups of A and U · V ⊂ A the subgroup generated by both. Then there is a natural map

(7.11)  C_{X} (U · V) ⟶ C_{C_{X} (V)}(U)

which is an equivalence.

Proof. Multiplication gives a surjective homomorphism U × V ⟶ U · V, an induced map BU × BV ⟶ B(U · V), and hence a map

Map(B(U · V), BX) ⟶ Map(BU × BV, BX) = Map(BU, Map(BV, BX)) .

Restricting to appropriate components provides the required natural map 7.11. Since all of the elements in the kernel of the map U × V ⟶ U · V are also in the kernel of the composite U × V ⟶ U · V ⟶ X, the fact that the map 7.11 is an equivalence follows from 3.5. □

7.12 Lemma. In the situation of 7.6, let s ∈ W be a reflection. Then s ∈ W_{X} (A)_{1} if and only if s ∈ W_{C_{X} (H(s))}(A)_{1}.

Proof. The maximal rank map C_{C_{X} (H(s))}(A) ⟶ C_{X} (A) clearly carries W_{C_{X} (H(s))}(A)_{1} into W_{X} (A)_{1}; this gives one of the implications. Suppose then that s ∈ W_{X} (A)_{1}. Let Z be the centralizer of H(s) in C_{X} (A)_{1}. Then s ∈ W_{Z_{1}} because s ∈ W_{Z} by 7.9 and W_{Z_{1}} = W_{Z} because Z is connected (7.8). However, by 7.10 the p-compact group Z_{1} equals the identity component of C_{C_{X} (H(s))}(A), and so it follows that s ∈ W_{C_{X} (H(s))}(A)_{1}. □

Proof of 7.6(2). By [12, 9.7] the subgroup W_{X} (A)_{1} of W is generated by the reflections it contains, so proving (2) comes down to showing that a reflection s ∈ W belongs to W_{X} (A)_{1} if and only if A ⊂ σ(s).

Suppose first that A ⊂ σ(s). To show that s ∈ W_{X} (A)_{1} is enough to show that s ∈ W_{X} (σ(s))_{1}, and it is this second inclusion that we will prove. By 7.12, we can assume that H(s) ⟶ X is central. If H(s) = σ(s), then W_{X} (σ(s))_{1} = W_{X} (H(s))_{1} = W_{X_{1}} = W_{X}, so s ∈ W_{X} (σ(s))_{1} for trivial reasons and we are done. If H(s) ≠ σ(s), then p = 2 and σ(s) = H(s) × Z/2 (see 7.7). Choose t ∈ K(s) ⊂ σ(s) to generate the second factor of σ(s) = H(s)×Z/2, and let x ∈ N̆(T) be an element, with image s ∈ W_{X}, such that x^{2} = t. There is a commutative diagram of loop spaces and homomorphisms

Z/2 ⟶ Z/4 ⟶ Z/2^{∞}
t↓  ^{x}↓  ^{f}↓
T̆ ⟶ N̆(T) ⟶ X

in which the map f is obtained from 3.10 by using the connectivity of X. Taking centralizers of ⟨t⟩ everywhere in this diagram and using 7.10 for the equivalence C_{X} (⟨t⟩) ≅ C_{X} (σ(s)) gives another commutative diagram

Z/2 ⟶  Z/4  ⟶ Z/2^{∞}
t↓  ^{x}↓  ^{f}↓
T̆ ⟶ C_{N̆(T)}(⟨t⟩) ⟶ C_{X} (σ(s))

This shows that x, which represents s ∈ W_{X} (σ(s)), lies in the identity component of C_{X} (σ(s)) (3.10).

Conversely, suppose that A is not contained in σ(s). If A is not contained in F (s) then s ∉ W_{X} (A) by 7.6(1), and the result we want, that s ∉ W_{X} (A)_{1}, certainly follows. If A ⊆ F̆(s) but A ⊈ σ(s), then p = 2, H(s) = σ(s), and F (s) ≅ H(s) × Z/2 (see 7.7). By 7.12, in showing s ∉ W_{X} (A)_{1}, we can replace X by C_{X} (H(s)) or, equivalently, assume that H(s) ⟶ X is central. By 7.10, then, C_{X} (A) is equivalent to C_{X} (F (s)). Choose x ∈ N̆(T) so that the image of x in W is s and such that x^{2} = 1; this is possible because K(s) = H(s) in the current situation and so 1 ∈ K(s). If s ∈ W_{X} (A)_{1} then x lies in the identity component of C_{X} (F (s)) and so the homomorphism f : Z/2 ⟶ C_{X} (F (s)) representing x lifts to a homomorphism Z/2 ⟶ C_{X} (F (s))_{1}. Since the Euler characteristic χ(C_{X} (F (s))_{1}/T) is non-zero (2.9), f lifts further (2.14) to a homomorphism Z/2 ⟶ T and even (3.2) to an actual group homomorphism Z/2 ⟶ T̆. This implies that f(Z/2) is central in C_{X} (F (s)) , since every element of order 2 in T̆ is contained in F (s) and so is central in C_{X} (F (s)). However it is impossible for f to be central in this way, since x acts nontrivially by conjugation on T̆ and so x is not in the center of the evident subgroup of N̆(T) which is a discrete approximation to the 2-normalizer of the torus in C_{X} (F (s)) (see 5.5). This contradiction establishes that A is contained in σ(s). □

§8. A homology decomposition theorem

Suppose that X is a p-compact group. Let A_{X} denote the category whose objects are the pairs (V, f), where V is a nontrivial elementary abelian p-group and f : V ⟶ X is a conjugacy class of monomorphisms. A morphism (V, f) ⟶ (V′, f′) in this category is a injection i : V ⟶ V′ such that f′ · i is conjugate to f. There is a functor α_{X} : A_{X}^{op} ⟶ (Spaces) given by

α_{X} (V, f) = Map(BV, BX)_{Bf} .

By definition, α_{X} (V, f) is equivalent to BC_{X} (g(V)) for any g : V ⟶ X contained in the conjugacy class f.

Evaluation at the basepoint of BV gives a map

α_{X} (V, f) ⟶ BX

which is natural in (V, f) [10, §6]; together these induce a map

a_{X} : hocolim α_{X} ⟶ BX .

In this section we will prove the following theorem.

8.1 Theorem. For any p-compact group X the map a_{X} : hocolim α_{X} ⟶ BX induces an isomorphism on mod p cohomology.

Remark. This is called a homology decomposition theorem because it shows how to express BX up to mod p homology in terms of classifying spaces of other p-compact groups; if X has trivial p-compact center, these other p-compact groups will be strictly smaller than X (cf. proof of 9.10). A decomposition theorem like this was first proved for classifying spaces of compact Lie groups in [14]. We will follow the more algebraic treatment from Proof of 8.1. Let R denote the object H^{∗}BX of the category K of unstable algebras over the mod p Steenrod algebra, and let A_{R} be the category whose objects are the pairs (V, g), where V is a nontrivial elementary abelian p-group and g : R ⟶ H^{∗}BV is a map in K which makes H^{∗}BV into a finitely generated R-module. A morphism (V, g) ⟶ (V′, g′) in this category is an injection i : V ⟶ V′ such that (Bi)^{∗} ·g′ = g. Let T^{V} : K ⟶ K be the functor which is left adjoint to tensor product with H^{∗}BV; this functor is studied by Lannes in [16] . There is a functor α_{R} : A_{R} ⟶ K which assigns to (V, g) the object T (V, R)_{g}, also denoted T_{g}^{V} (R), which is the summand or “component” of T^{V} (R) corresponding to the map g [10, §3]. Let T ⟶ X be a maximal torus, and let S = H^{∗}BN_{p}(T). The monomorphism (2.10) N_{p}(T) ⟶ X induces a map i : R ⟶ S. Since X /N_{p}(T) is F_{p}-finite and has Euler characteristic prime to p, the Becker-Gottlieb transfer construction (as generalized in [5]) provides a left inverse t : S ⟶ R for i, such that t is both a map of R-modules and a map of modules over the mod p Steenrod algebra. Moreover, the object S ∈ K has a nontrivial “center” in the sense of [10, §4] because any discrete approximation N̆_{p}(T) to N_{p}(T) has a nontrivial group theoretic center [10, 1.4]. By [10, 1.2], then, the natural map R ⟶ lim α_{R} is an isomorphism, and the higher limits lim^{i} α_{R} vanish for i > 0. (Note that these higher limits are calculated in the category of graded mod p vector spaces.)

For any elementary abelian p-group V there is a map Φ from the set of conjugacy classes of homomorphisms f : V ⟶ X to the set of K-maps g : H^{∗}BV ⟶ R; the map Φ assigns to f the cohomology map (Bf)^{∗} induced by Bf. According to [16, 3.1.4] the map Φ is a bijection. Since a homomorphism f : V ⟶ X is a monomorphism if and only if Φ(f) makes H^{∗}BV into a finitely generated module over R (2.4), it follows that Φ gives an isomorphism of categories Φ : A_{X} ⟶ A_{R}.

If V is an elementary abelian p-group and f : V ⟶ X is a homomorphism, then the mapping space component Map(BV, BX)_{Bf} is the classifying space of a p-compact group and so in particular it is F_{p}-complete and has finite dimensional mod p cohomology groups. By [16] then, the natural map λ_{f} : T (V, R)_{Φ(f)} ⟶ H^{∗} Map(BV, BX)_{Bf} is an isomorphism; here λ_{f} is adjoint to the map on cohomology induced by the evaluation map BV × Map(BV, BX)_{Bf} ⟶ BX. These isomorphisms combine to give a natural equivalence from the functor α_{R} to the composite H^{∗}(α_{X}) · Φ. It follows then from the remarks above about α_{R} that lim^{i} H^{∗}(α_{X}) = 0 for i > 0 and that the natural map lim H^{∗}(α_{X}) ⟶ H^{∗}(BX) is an isomorphism. The theorem can now be derived by a straightforward application of the spectral sequence of [3, XII, 5.8]; recall that this is a first quadrant spectral converging to H^{∗}(hocolim α_{X}) which at E^{2} contains the groups lim^{i} H^{∗}(α_{X}). □

The above proof of 8.1 reveals a few properties of A_{X} which we will use later on.

8.2 Proposition. If X is a p-compact group, then the nerve (or underlying space [3, XI, §2]) Nerve(A_{X}) of the category A_{X} has the mod p (co-)homology of a point.

Proof. An inspection of [3, XI, §6] shows that H^{i} Nerve(A_{X}) is isomorphic to lim^{i} β, where β is the constant functor on A_{X}^{op} with value F_{p}. Since β is naturally equivalent to the functor H^{0}(α_{X}), this proposition follows from the vanishing result referred to in the proof of 8.1. □

8.3 Proposition. If X is a p-compact group, then A_{X} is equivalent to a category with a finite number of objects and a finite number of morphisms between any two objects.

Proof. By definition the set of morphisms between any two objects of A_{X} is finite, so it is enough to show that up to isomorphism A_{X} has only a finite number of objects. As in the proof of 8.1, such objects correspond to pairs (V, g), where V is a nontrivial elementary abelian p-group and g : H^{∗}BX ⟶ H^{∗}BV makes H^{∗}BV a finitely generated module over H^{∗}BX. Since H^{∗}BX is a finitely generated, i.e. noetherian, algebra over F_{p}, an elementary Poincaré series argument shows that any such pair (V, g) must have the Krull dimension of H^{∗}BV (which equals rk_{F_{p}} V) less than or equal to the Krull dimension of H^{∗}BX; in particular, up to isomorphism there are only a finite number of possible V. For any given V, a K-map g : H^{∗}BX ⟶ H^{∗}BV is determined by the images under g of some finite set of algebra generators for H^{∗}BX; since H^{∗}BV is finite in each dimension, there are only a finite number of choices for these images. □

§9. An inductive principle and the “Sullivan Conjecture”

In this section we will describe an inductive principle which can sometimes be used in conjunction with 8.1 to prove statements about p-compact groups. We then show that the principle leads to a proof for p-compact groups of an analogue of Miller’s Theorem (the “Sullivan Conjecture”) [17].

9.1 Definition. A class Cl of p-compact groups is said to be saturated if it satisfies the following five conditions.

(1) Cl is closed under equivalences, in the sense that if f : X ⟶ Y is a homomorphism of p-compact groups which is an equivalence, then X ∈ Cl if and only if Y ∈ Cl.

(2) The trivial p-compact group belongs to Cl.

(3) If the identity component X_{1} of X is in Cl, then X ∈ Cl.

(4) If X is connected and X /C^{c}(X) ∈ Cl, then X ∈ Cl.

(5) If X is connected, C^{c}(X) is trivial, and Y ∈ Cl for all p-compact groups Y such that cd_{F_{p}} Y < cd_{F_{p}} X, then X ∈ Cl.

9.2 Theorem. Any saturated class Cl contains all p-compact groups.

Proof. We will work by induction on cd_{F_{p}} X to show that X ∈ Cl. If cd_{F_{p}} X = 0 then X_{1} is contractible and X ∈ Cl by the first three conditions. Suppose that cd_{F_{p}} X > 0 and that Y ∈ Cl for all Y with cd_{F_{p}} Y < cd_{F_{p}} X. Let C be the p-compact center of X_{1}. Then X_{1}/C ∈ Cl by (5), X_{1} ∈ Cl by (4), and hence X ∈ Cl by (3). □

9.3 Theorem. (Sullivan conjecture for p-compact groups) Let X be a p-compact group and U a space which is F_{p}-complete and F_{p}-finite. Then inclusion of constant maps gives an equivalence U ⟶ Map(BX , U).

Before proving 9.3 we will list a few lemmas. If X and U are spaces, write “X ⊥ U” if inclusion of constant maps gives an equivalence U ⟶ Map(X , U). The first three lemmas are relatively elementary.

9.4 Lemma. [2, §2-3] Let F ⟶ E ⟶ B be a fibration. If F ⊥ U and B ⊥ U then E ⊥ U.

9.5 Lemma. Let C be a category and F : C ⟶ (Spaces) a functor. If Nerve(C) ⊥ U and F (c) ⊥ U for each object c of C, then hocolim F ⊥ U.

Proof. See [3, XII, §4] and observe that the homotopy limit of a constant functor on C with value U is the space Map(Nerve(C), U). □

9.6 Lemma. [12, 11.13] If U is an F_{p}-complete space and X ⟶ Y is an F_{p}-equivalence, then X ⊥ U if and only if Y ⊥ U.

9.7 Lemma. If X is a p-compact toral group and U is a space which is F_{p}-complete and F_{p}-finite, then BX ⊥ U.

Proof. By density (3.4) we can assume instead that X is a p-discrete toral group. In this case X is an increasing union of finite p-groups [12, 6.19], and BX is equivalent to the corresponding sequential homotopy colimit of classifying spaces of finite p-groups [3, XII, 3.6]. Since BG ⊥ U for any finite p-group G by Miller’s Theorem, the lemma follows from 9.5. □

Let Cl be the class of all p-compact groups X such that BX ⊥ U for each space U which is F_{p}-complete and F_{p}-finite. It is clear that Cl satisfies (1) and (2) of 9.1. In order to complete the proof of 9.3, i.e., to show that Cl is the class of all p-compact groups, it is enough by 9.2 to prove the following three propositions. In each of them X is a p-compact group and U is a space which is F_{p}-complete and F_{p}-finite.

9.8 Proposition. If B(X_{1}) ⊥ U, then BX ⊥ U.

Proof. There is a fibration sequence BX_{1} ⟶ BX ⟶ Bπ_{0}X. Since Bπ_{0}X ⊥ U by [17], the proposition follows from 9.4. □

9.9 Proposition. Suppose that X is connected and that Z = X /C^{c}(X). If BZ ⊥ Y, then BX ⊥ Y.

Proof. There is a fibration sequence BC^{c}(X) ⟶ BX ⟶ BY. Since BC^{c}(X) ⊥ Y by 9.7, the proposition follows from 9.4. □

9.10 Proposition. If X is connected, C^{c}(X) is trivial, and BY ⊥ U for all Y such that cd_{F_{p}} Y < cd_{F_{p}} X, then BX ⊥ U.

Proof. Let f : V ⟶ X be a monomorphism with V a nontrivial elementary abelian pgroup. Since C^{c}(X) is trivial the homomorphism f cannot be central, and so (by 4.6) cd_{F_{p}} C_{X} (f(V)) < cd_{F_{p}} X. It follows that α_{X} (V, f) ⊥ U for each object (V, f) of A_{X}, and so hocolim α_{X} ⊥ U by 8.2 and 9.5. By 9.6, BX ⊥ U. □

§10. A few technical results

In this section we begin by proving a collection of results which are more or less immediate corollaries of 9.3. Some of these are interesting because they are generalizations for arbitrary p-compact groups of results that were previously known only for p-compact toral groups. We finish by giving a splitting result (10.7) for extensions of a p-compact torus by a connected p-compact group in adjoint form; this will be used in the proof of 11.7.

10.1 Proposition. (cf. [12, 6.1 and 5.3]) If f : X → Y is a trivial homomorphism between p-compact groups, then f is central.

Proof. Let f : BX ⟶ BY be the constant map. The proposition follows from the fact that the loop space Ω Map(BX , BY)_{f} is equivalent to Map(BX , Y), which, by 9.3, is in turn equivalent to Y. □

10.2 Proposition. (cf. 2.6) A homomorphism f : X → Y between p-compact groups is central if and only if there is a homomorphism μ : X × Y → Y which restricts to f on X × {1} and to the identity on {1} × Y.

Proof. This is the same as the proof of [12, 8.6]. If f is central then an evaluation map gives the required homomorphism μ. If μ exists, then it induces an equivalence between C_{Y}(f(X)) and C_{Y}(g(X)), where g : X ⟶ Y is the trivial homomorphism. This second centralizer is equivalent to Y by 10.1. □

10.3 Lemma. (cf. 3.5) Let

{1} ⟶ X ⟶^{f} Y ⟶ Z ⟶ {1}

be a short exact sequence of p-compact groups and g : Y ⟶ U a homomorphism such that g·f is a trivial homomorphism. Then up to homotopy there is a unique homomorphism g′ : Z ⟶ U which extends g, and the natural map C_{U}(g′(Z)) ⟶ C_{U}(g(Y)) is an equivalence.

Proof. This is the same as the proof of [12, 7.5]. Let h : X ⟶ U be the trivial homomorphism. The idea is to express the space of maps BY ⟶ BU which up to homotopy are trivial on BX as the space of sections of a fibration over BZ with fibre Map(BX , BU)_{Bh}, and then to use 10.1 to identify this fibration as the product BZ × BU ⟶ BZ. □

10.4 Proposition. Suppose that

{1} ⟶ X ⟶ Y ⟶^{f} Z ⟶ {1}

is a short exact sequence of p-compact groups, and that g : U ⟶ Y is a central homomorphism of p-compact groups. Then the composite f · g : U ⟶ Z is also central.

Proof. By 10.2 it is enough to construct an appropriate homomorphism μ : U × Z ⟶ Z. Consider the corresponding homomorphism U × Y ⟶ Y which expresses the fact that g is central. The adjoint of the associated map of classifying spaces is a map BU ⟶ Map(BY, BY)_{id} which induces (by composing points in the target with Bf) a map BU ⟶ Map(BY, BZ)_{Bf}. By 10.3, however, Map(BY, BZ)_{Bf} is equivalent via precomposition with Bf to Map(BZ, BZ)_{id}, and it is easy to check that the adjoint of the resulting map BU ⟶ Map(BZ, BZ)_{id} gives the required homomorphism μ. □

10.5 Proposition. Let

{1} ⟶ X ⟶^{f} Y ⟶ Z ⟶ {1}

be a short exact sequence of p-compact groups, U a p-compact group, and g : U ⟶ X a homomorphism. Then

(1) if f · g is central then g is central, and

(2) if g is central and Z is connected then f · g is central Proof. Let h : U ⟶ Z denote the trivial homomorphism. Evaluation at the basepoint of BU gives a map of fibration sequences

F ⟶ Map(BU, BY)_{B(f·g)} ⟶ Map(BU, BZ)_{Bh}
a↓  ^{b}↓  ^{c}↓
BX ⟶  BY  ⟶  BZ

where F denotes the space of maps from BU to BX which after composition with Bf are homotopic to B(f · g). Note that if F is connected then F = Map(BU, BX)_{Bg}. By 10.1 the map c is an equivalence. It follows that if b is an equivalence if and only if a is an equivalence. If f · g is central then b is an equivalence, so that a is an equivalence, F is connected, and g is central. This gives (1). If Z is connected then F is connected by the long exact homotopy sequence of the upper fibration; thus if g is central, the map a is an equivalence, so b is an equivalence too and f · g is central. This gives (2). □

10.6 Lemma. Suppose that X ⟶ Y ⟶ Z is a fibration sequence of F_{p}-finite spaces, such that Z is connected and the action of π_{1}Z on H_{∗}X is nilpotent. Then χ(Y) = χ(X)χ(Z).

Proof. It follows from [12, 4.13] that the desired multiplicativity holds in E^{2} of the fibration Serre spectral sequence, and a standard inductive argument shows that this persists to E^{∞}. □

Remark. Lemma 10.6 does not necessarily hold for a general fibration sequence of F_{p}-finite spaces. For example, let p be odd and consider the fibration

S^{2} ⟶ RP^{2} ⟶ RP^{∞} .

The mod p cohomology Euler characteristics of base and total space are 1, while that of the fibre is 2.

10.7 Proposition. Suppose that

{1} ⟶ X ⟶ Y ⟶ A ⟶ {1}

is a short exact sequence of connected p-compact groups, where X has a trivial p-compact center and A is a p-compact torus. Then the sequence is equivalent to the product sequence

{1} ⟶ X ⟶ X × A ⟶ A ⟶ {1} .

Proof. Let T be a maximal torus for Y and T̆ ⟶ T a discrete approximation. Let f be the composite T̆ ⟶ Y ⟶ A, and denote ker(f) by K̆ and T̆/K̆ by Q̆. Let K and Q denote the closures of K̆ and Q̆ respectively. Observe that Q̆, as a quotient of T̆, is a p-discrete torus and hence that Q is a p-compact torus. There is a commutative diagram of loop spaces and homomorphisms (3.5)

K̆ ⟶ T̆ ⟶ Q̆
↓  ↓  ↓
X ⟶ Y ⟶ A

which on passing to closures gives a map of short exact sequences

(10.8)  \begin{aligned}\{1\} & ⟶ & K & ⟶ & T & ⟶ & Q & ⟶ & \{1\} \\ & & ↓ & & ↓ & & ↓ \\ \{1\} & ⟶ & X & ⟶ & Y & ⟶ & A & ⟶ & \{1\} \end{aligned}

By construction the map Q̆ ⟶ A has trivial kernel, and so Q ⟶ A is a monomorphism (3.6). The composite K̆ ⟶ X ⟶ Y has trivial kernel (since T ⟶ Y is a monomorphism) and so K̆ ⟶ X has trivial kernel and K ⟶ X is a monomorphism. Taking vertical homotopy fibres in the diagram of classifying spaces induced by 10.8 gives a fibre sequence

X /K ⟶ Y/T ⟶ A/Q

in which all three spaces are F_{p}-finite and in which it is straightforward to verify that the fundamental group of the base, which is a quotient of π_{1}A, acts nilpotently on the homology of the fibre. By multiplicativity of Euler characteristic (10.6), χ(Y/T) = χ(X /K)χ(A/Q). Since χ(Y/T) ≠ 0, it follows that χ(X /K) ≠ 0 ≠ χ(A/Q) and hence (2.15) that Q is a maximal torus for A and (use in addition 10.6) that the identity component K_{1} is a maximal torus for X. Since A is itself a p-compact torus, the uniqueness property of maximal tori implies that the map Q ⟶ A is an equivalence. As in 3.8, K is a product of a p-compact torus and a finite abelian p-group; since X is connected and a maximal torus in a connected p-compact group is self-centralizing [12, 9.1], we conclude that the finite abelian p-group factor is trivial, K is connected, and K itself is a maximal torus for X.

Let C ⟶ T̆ be the p-discrete center of Y (6.7). By 7.5, C = ∩_{s}σ(s), where s runs through all reflections in the Weyl group W of Y. By 10.5 the intersection C ∩ K̆ is trivial, since X has a trivial p-discrete center. Let T̆^{W} be the fixed point set of the conjugation action of W on T̆, and T̆_{nrm} the image of the norm map T̆ ⟶ T̆ which sends an element of T̆ to the sum of its W -conjugates. Then T̆_{nrm} ⊂ T̆^{W}, C ⊂ T̆^{W}, the quotient T̆^{W} /T̆_{nrm} is finite because it is annihilated by |W |, and T̆^{W} /C is finite by 7.7. The quotient T̆ ⟶ Q̆ is clearly equivariant with respect to the conjugation action of W on T̆ and the trivial action of W on Q̆. Under this trivial action of W on Q̆, the norm map Q̆ ⟶ Q̆ is just multiplication by |W | and so is surjective, because Q̆ is divisible. As a consequence, the composite T̆_{nrm} ⟶ T̆ ⟶ Q̆ is surjective. Since Q̆ has no proper subgroups of finite index, it follows that the maps T̆^{W} ⟶ Q̆ and C ⟶ Q̆ are surjective too, this last map in fact being an isomorphism because as mentioned above the intersection C ∩ K̆ is trivial. Passing to closures shows that the homomorphism Y ⟶ A induces an equivalence C^{c}(Y) ⟶ A. The composite homomorphism C^{c}(Y) × X ⟶ C^{c}(Y) × Y ⟶ Y (where the second map comes from 2.6) is the required equivalence of Y with C^{c}(Y) × X ≅ A × X. It is easy to check that this equivalence is compatible with the original short exact sequence. □

§11. Homotopy centers

In this section we will prove 1.3. Let X be a p-compact group. The multiplication homomorphism (2.6):

μ_{c} : C^{c}(X) × X ⟶ X

gives a classifying space map Bμ_{c} : BC^{c}(X) × BX ⟶ BX. The adjoint of Bμ_{c} is a map BC^{c}(X) ⟶ Map(BX , BX)_{id}, or equivalently a loop space homomorphism j_{X} : C^{c}(X) ⟶ C^{h}(X). This is the homomorphism which will turn out to be an equivalence.

11.1 Remark. Let X be a p-compact group. The construction in [12, proof of 8.3] shows that the fibration sequence BC^{c}(X) ⟶ BX ⟶ BX_{ad} extends one step further to the right to give a fibration sequence

(11.2)  BX ⟶ BX_{ad} ⟶ B^{2}C^{c}(X) .

It follows directly from Theorem 1.3 that 11.2 is equivalent to the universal fibration

BX ⟶ E ⟶ B(Aut(BX)_{id})

with fibre BX over a simply connected space; in particular, the total space of this universal bundle is BX_{ad}.

The next proposition shows that in order to prove that j_{X} is an equivalence it is enough to prove that C^{h}(X) is a p-compact group.

11.3 Proposition. Suppose that X is a p-compact group and that C^{h}(X) is a p-compact group. Then the map j_{X} : C^{c}(X) ⟶ C^{h}(X) is an equivalence.

Proof. The composition operation on Map(BX , BX)_{id} gives a homomorphism C^{h}(X) × C^{h}(X) ⟶ C^{h}(X) which shows (10.2) that C^{h}(X) is abelian and hence (5.2) that C^{h}(X) is a p-compact toral group. Consider the evaluation map

Bμ_{h} : BC^{h}(X) × BX = Map(BX , BX)_{id} × BX ⟶ BX .

This induces a homomorphism μ_{h} : C^{h}(X) × X ⟶ X whose restriction to C^{h}(X) × {1} is a central homomorphism C^{h}(X) ⟶ X (2.6). By 6.1 this central homomorphism lifts uniquely to a homomorphism k_{X} : C^{h}(X) ⟶ C^{c}(X). The homomorphism diagrams

C^{h}(X) × X ⟶^{μ_{h}} X
k_{X} ×id↓  ^{id}↓
C^{c}(X) × X ⟶^{μ_{c}} X

C^{c}(X) × X ⟶^{μ_{c}} X
j_{X} ×id↓  ^{id}↓
C^{h}(X) × X ⟶^{μ_{h}} X

commute up to homotopy, the first by the uniqueness provision of 5.3 and the second by construction. It follows directly that j_{X} · k_{X} : C^{h}(X) ⟶ C^{h}(X) is homotopic to the identity, and from the uniqueness provision of 6.1 that k_{X} · j_{X} is also homotopic to the identity. □

The above proof also gives a slightly more specialized result which we will use later on.

11.4 Proposition. Suppose that X is a p-compact group, and that the identity component C^{h}(X)_{1} is a p-compact torus. Then the restriction of j_{X} : C^{c}(X) ⟶ C^{h}(X) to identity components gives an equivalence C^{c}(X)_{1} ⟶ C^{h}(X)_{1}.

Let Cl be the class consisting of all p-compact groups X such that C^{h}(X) is a p-compact group. By 11.3, Theorem 1.3 is equivalent to the statement that Cl contains all p-compact groups. It is clear that Cl satisfies (1) and (2) of 9.1. By 9.2, then, in order to prove 1.3 it is enough to prove the following three propositions. In each one, X is a p-compact group.

11.5 Proposition. If C^{h}(X_{1}) is a p-compact group, then so is C^{h}(X).

11.6 Proposition. Suppose that X is connected, and let C = C^{c}(X). If C^{h}(X /C) is a p-compact group, then so is C^{h}(X).

11.7 Proposition. Suppose that X is connected, and that C^{c}(X) is trivial. If C^{h}(Y) is a p-compact group for each p-compact group Y with cd_{F_{p}} Y < cd_{F_{p}} X, then C^{h}(X) is a p-compact group.

The proofs of 11.5 and 11.6 amount to fairly standard homotopy theoretic calculations; in order to organize these calculations we set up some notation.

If U is a space, let Aut(U) denote the space of (unpointed) self homotopy equivalences of U; this is a loop space, and B Aut(U) is the classifying space for fibrations with fibre equivalent to U. If f : E ⟶ B is a fibration, let Aut(f) denote the space of commutative diagrams

(11.8)  \begin{matrix}E & ⟶ & E \\ ^{f}↓ & & ^{f}↓ \\ B & ⟶ & B \end{matrix}

in which both of the horizontal arrows are equivalences. Let Aut_{B}(f) denote the subspace of Aut(f) consisting of diagrams as above in which the map B ⟶ B is the identity. The spaces Aut(f) and Aut_{B}(f) are loop spaces such that B Aut(f) and B Aut_{B}(f) classify certain types of iterated fibrations [7].

If X and Y are spaces let [X , Y] denote π_{0} Map(X , Y). Suppose that B is a connected space and that f : E ⟶ B is a fibration with fibre F. We will denote by c(f) the element of [B, B Aut(F)] consisting of classifying maps for f, and by C(f) the subset of [B, B Aut(F)] obtained by taking the orbit of c(f) under the precomposition action of π_{0} Aut(B) on [B, B Aut(F)].

11.9 Proposition. [7] Let f : E ⟶ B be a fibration with fibre F over the connected space B. Then there is a fibration sequence

Map(B, B Aut(F))_{C(f)} ⟶ B Aut(f) ⟶ B Aut(B) .

Remark. The above fibration sequence is associated to the composition action of Aut(B) on Map(B, B Aut(F)_{C(f)}.

If X and U are spaces with U connected, let Map(X , U)_{0} denote the subspace of Map(X , U) consisting of maps which are homotopic to constant maps. Write X ⊥_{0} U if inclusion of constant maps gives an equivalence U ⟶ Map(X , U)_{0} (cf. §9).

11.10 Proposition. Suppose that B is a connected space, that f : E ⟶ B is a fibration with fibre F, and that F ⊥_{0} B. Then the forgetful map u : Aut(f)_{id} ⟶ Aut(E)_{id} (cf. 11.8) is an equivalence.

Proof. Let Aut_{B}(f)_{{id_{E}}} denote the subspace of Aut_{B}(f) consisting of maps of E over B which as maps of E are homotopic to the identity, and let Aut(f)_{{id_{E},id_{B}}} denote the subspace of Aut(f) consisting of commutative diagrams 11.8 in which the upper arrow is homotopic to the identity map of E and the lower one to the identity map of B. Consider the map of fibration sequences

Aut_{B}(f)_{{id_{E}}} ⟶ Aut(f)_{{id_{E},id_{B}}} ⟶^{u′} Aut(B)_{id}
=↓  ^{u}↓  ^{v}↓
Aut_{B}(f)_{{id_{E}}} ⟶ Aut(E)_{id}  ⟶^{w} Map(E, B)_{f}

in which u′ like u is a forgetful map and the maps v and w are given by composition with f (on the left and right, respectively). Since F ⊥_{0} B, the argument used in the proof of 10.3 shows that v is an equivalence. The fibration sequence then shows that u is an equivalence and also that the domain of u, being connected, is the identity component Aut(f)_{id} of Aut(f). □

The next two lemmas are proved by direct homotopy group calculation, and depend on the following observation.

11.11 Remark. A connected space U is the classifying space BG of a p-compact toral group G if and only if

(1) π_{1}U is a finite p-group,

(2) π_{2}U is a finitely generated free module over Z_{p} (since Z_{p} is a principal ideal domain, this is equivalent to the condition that π_{2}U be a finitely generated torsion free module over Z_{p}) and

(3) π_{i}(U) = {0} for i > 2. If U is both a loop space and the classifying space of a p-compact toral group then U is the classifying space of an abelian p-compact toral group (2.6).

11.12 Lemma. Suppose that Y is a connected space with H^{i}Y finite for i = 1, 2 and H^{1}(Y, Z_{p}) = 0. Let U be connected space such that ΩU is the classifying space of a pcompact toral group and let f : Y ⟶ U be a map. Then each component of Ω Map(Y, U)_{f} is the classifying space of a p-compact toral group.

11.13 Remark. The above lemma applies if Y = BX for a p-compact group X.

11.14 Lemma. If G is a p-compact toral group, then Aut(BG)_{id} = BC^{h}(G) is the classifying space of a p-compact toral group.

Proof of 11.5. By 11.3, C^{h}(X_{1}) is a p-compact toral group. The fibration

B(X_{1}) ⟶ BX ⟶^{f} Bπ_{0}X

satisfies the conditions of 11.10, so the map

Aut(f)_{id} ⟶ Aut(BX)_{id} = BC^{h}(X)

is an equivalence. Looping down the fibration sequence from 11.9 then gives a fibration sequence for BC^{h}(X) in which the fibre is a disjoint union of classifying spaces of p-compact toral groups (11.12) and the base is Aut(Bπ_{0}X)_{id} ≅ BC, where C is the center of π_{0}X. It follows from the long exact homotopy sequence of a fibration (cf. 11.11) that BC^{h}(X) is the classifying space of a p-compact toral group. □

Proof of 11.6. The fibration

BC^{c}(X) ⟶ BX ⟶^{f} B(X /C^{c}(X))

satisfies the conditions of 11.10 (see 10.1), so the map

Aut(f)_{id} ⟶ Aut(BX)_{id} = BC^{h}(X)

is an equivalence. Looping down the fibration sequence from 11.9 then gives a fibration sequence for BC^{h}(X) in which the base is contractible (6.3, 11.3) and the fibre is the classifying space of a p-compact toral group (11.12, 11.13). This gives the desired result. □

To prove the final proposition we will need a concept from [6]. A map f : X ⟶ Y of spaces is said to be centric if composition with f induces an equivalence Aut(X)_{id} ⟶ Map(X , Y)_{f}.

11.15 Lemma. Let X be a p-compact group, V an elementary abelian group, f : V ⟶ X a monomorphism, and i : C_{X} (V) ⟶ X the induced monomorphism. Then the map Bi : BC_{X} (V) ⟶ BX is centric.

Proof. This is the same as the proof of [6, 4.2]. The rough idea is that, since V is central in C_{X} (V), the centralizer of C_{X} (V) in X is the same as the centralizer of C_{X} (V) in the centralizer of V in X. Encoding this in mapping space terms gives the stated equivalence. □

The following is clear from the complex given in [3, XI, §6] for computing higher derived functors of the inverse limit.

11.16 Lemma. Let A be a category, R a noetherian ring and α : A ⟶ (R–modules) a functor. Suppose that A is equivalent to a category with a finite number of objects and a finite number of morphisms between any two objects, and that α(x) is a finitely generated R-module for any object x of A. Then lim^{i} α is a finitely generated R-module for each i ≥ 0.

Proof of 11.7. The F_{p}-equivalence a_{X} of 8.1 gives an equivalence [12, 11.3]

a_{X}^{#} : Map(BX , BX) ⟶^{≅} Map(hocolim α_{X} , BX) .

As explained in [3, XII, §4] the target of this equivalence is the same as holim β_{X}, where β_{X} : A_{X} ⟶ (Spaces) sends (V, f) to Map(α_{X} (V, f), BX). For each object (V, f) of A_{X} let

e_{V,f} : α_{X} (V, f) = Map(BV, BX)_{Bf} ⟶ BX

be given by evaluation at the basepoint of BV. The definition of a_{X} implies that a_{X}^{#} carries the component Aut(BX)_{id} of Map(BX , BX) to the basepoint component of holim β′_{X}, where for each object (V, f) of A_{X}, β′_{X} (V, F) is the pointed space given by

β′_{X} (V, F) = Map(α_{X} (V, f), BX)_{e_{V,f}} .

Since C^{c}(X) is trivial, each space α_{X} (V, f) is the classifying space of some p-compact group Y with cd_{F_{p}} Y < cd_{F_{p}} X (cf. proof of 9.10). Consequently, by 11.3 and the given hypotheses, Aut(α_{X} (V, f))_{id} is the classifying space of a p-compact toral group, in fact, of an abelian p-compact toral group (11.11). By 11.15, each space β′_{X} (V, f) is also the classifying space of an abelian p-compact toral group. There is a second quadrant homotopy spectral sequence [3, XI, §7] [1]

E_{i,j}^{2} = lim^{−i} π_{j}β′_{X} ⇒ π_{j−i} holim β′_{X} = π_{j−i} Aut(BX)_{id} = π_{j−i}BC^{h}(X)

The E^{2}-term of the spectral sequence has these properties:

(1) E_{i,j}^{2} = 0 for j ≥ 3, because π_{j}β′_{X} (V, f) = 0 for all (V, f) and all j ≥ 3.

(2) E_{0,2}^{2} = lim π_{2}β′_{X} ⊂ ∏_{(V,f)} π_{2}β′_{X} (V, f) is a torsion-free module over Z_{p}, because π_{2}β′_{X} (V, f) is a torsion-free module over Z_{p} for each (V, f).

(3) E_{i,j}^{2} is a finitely generated module over Z_{p} for j = 1, 2 and all i ≤ 0, by 8.3 and

11.16. It follows from these observations and the structure of the spectral sequence that π_{i} Aut(BX)_{id} = 0 for i ≥ 3, that π_{2} Aut(BX)_{id} is a finitely generated torsion-free (equivalently, free) module over Z_{p}, and that π_{1} Aut(BX)_{id} is a finitely generated module over Z_{p}. (Note that this fundamental group is necessarily abelian, because Aut(BX) is a loop space.) We will be done if we can show that π_{1} Aut(BX)_{id} is finite, since then Aut(BX)_{id} = BC^{h}(X) will be the classifying space of a p-compact toral group. It is clear in any case that the identity component C^{h}(X)_{1} is a p-compact torus, since the nonzero homotopy of this component consists at most of a single finitely generated free module over Z_{p} in dimension 1. Since C^{c}(X) is trivial by assumption, it follows from 11.4 that C^{h}(X)_{1} is contractible, and hence that Aut(BX)_{id} is of type K(M, 1) for some finitely generated module M over Z_{p}. Therefore B(Aut(BX)_{id}) is of type K(M, 2). We will prove directly that M = 0, which is certainly enough to show that M is finite. Pick a homomorphism Z_{p} ⟶ M and consider the fibration sequence

BX ⟶ E ⟶ K(Z_{p}, 2)

obtained from the induced map K(Z_{p}, 2) ⟶ K(M, 2) by pulling back the universal fibration over K(M, 2) = B(Aut(BX)_{id}). This gives a short exact sequence of p-compact groups which by 10.7 is a product sequence. This implies that the above fibration is a product fibration and, by the meaning of universality, that the homomorphism Z_{p} ⟶ M is trivial. Since this is true for any homomorphism Z_{p} ⟶ M, M itself is trivial. □

§12. Relationship to Lie groups

In this section we will prove 1.4; this gives one explicit connection between the theory described in this paper and the homotopy theory of compact Lie groups.

Let G be a compact Lie group and C^{a}(G) the algebraic center of G, so that C^{a}(G) is a compact abelian Lie group. The multiplication homomorphism C^{a}(G) × G ⟶ G induces a map on classifying spaces whose adjoint is a map e_{1} : BC^{a}(G) ⟶ Map(BG, BG)_{id} (cf. [15, §4]). This is the natural map referred to in the statement of 1.4.

Recall from §2 that if G is compact Lie group with π_{0}G a p-group, Ĝ denotes the pcompact group obtained from G by F_{p}-completion. There is a natural homomorphism δ : G ⟶ Ĝ.

12.1 Lemma. Let G be a connected compact Lie group, A a p-discrete toral group, and f : A ⟶ G a homomorphism. Then the maps

Map(BG, BG)_{id} ⟶^{h} Map(BG, BĜ)_{Bδ}
Map(BA, BG)_{Bf} ⟶^{k} Map(BA, BĜ)_{B(δ·f)}

induced by composition with δ are F_{p}-equivalences.

Proof. We will only treat the map h; the case of the map k is similar but simpler. Let Comp′_{F_{p}}(BG) denote the product ∏_{q≠p} Comp_{F_{q}}(BG). By [15, 3.1], the completion map ϵ : BG ⟶ Comp′_{F_{p}}(BG) combines with Bδ to give an F_{p}-equivalence

Map(BG, BG)_{id} ⟶ Map(BG, BĜ)_{Bδ} × Map(BG, Comp′_{F_{p}}(BG))_{ϵ}

To finish the proof it is enough to show that the space

(12.2)  Map(BG, Comp′_{F_{p}}(BG))_{ϵ} ≅ ∏_{q≠p} Map(BG, Comp_{F_{q}}(BG))_{ϵ_{q}}

has the mod p homology of a point, where ϵ_{q} : BG ⟶ Comp_{F_{q}}(BG) is the F_{q}-completion map. However, as described in [12, 11.10] each completion map ϵ_{q} induces an equivalence

Map(Comp_{F_{q}}(BG), Comp_{F_{q}}(BG))_{id} ⟶ Map(BG, Comp_{F_{q}}(BG))_{ϵ_{q}}

whose domain, by 1.3, is the classifying space of an abelian q-compact group. It follows that the space in 12.2 is a nilpotent (even simple) space with uniquely p-divisible homotopy groups, and thus has the desired acyclicity property. □

12.3 Lemma. Suppose that G is a connected compact Lie group and K a subgroup of G such that π_{0}K is a p-group and the inclusion K ⟶ G induces an isomorphism H^{∗}BG ⟶ H^{∗}BK. Then K = G.

Proof. It follows from application of the Eilenberg-Moore spectral sequence or the fibre lemma [12, 11.7] to appropriate path fibrations that the map K ⟶ G is an F_{p}-equivalence. Since K and G are both orientable manifolds, this implies that K and G have the same dimension; the fact that G is connected now easily gives K = G. □

12.4 Lemma. [15, A.4] Suppose that G is a connected compact Lie group, K a subgroup of G which is a finite p-group, and C the centralizer of K in G. Then π_{0}C is a finite p-group.

12.5 Theorem. Suppose that G is a connected compact Lie group and i : K ⊂ G a subgroup of G which is a finite p-group. Then K lies in the center of G if and only if the composite K ⟶ G ⟶ Ĝ is central.

Proof. If K lies in the center of G, then applying the F_{p}-completion functor to the map of classifying spaces induced by the multiplication map K ×G ⟶ G gives the homomorphism K×Ĝ ⟶ Ĝ required for showing that K ⟶ Ĝ is central (2.6). If K ⟶ Ĝ is central, consider the square

Map(BK, BG)_{Bi} ⟶ Map(BK, BĜ)_{B(δ·i)}
↓  ↓
BG  ⟶  BĜ

where the vertical maps are induced by evaluation at the basepoint of BK. Here the two horizontal maps are F_{p}-equivalences (the upper one by 12.3) and the right hand map is an equivalence by assumption. It follows that the map Map(BK, BG)_{Bi} ⟶ BG is an isomorphism on homology. Let H be the centralizer of K in G. The main result of [13] implies that the natural map BH ⟶ Map(BK, BG)_{Bi} is an F_{p}-equivalence, and it follows that the inclusion H ⟶ G induces an F_{p}-equivalence BH ⟶ BG. By 12.4 and 12.5, then, H = G. □

12.6 Lemma. Let G be a connected compact Lie group and C the subgroup of C^{a}(G) given by elements whose order is a power of p. (We are treating C as a discrete group). Then the composite homomorphism C ⟶ G ⟶ Ĝ is a p-discrete center for Ĝ.

Proof. Let T ⟶ G be a Lie-theoretic maximal torus for G; as in 2.16, the induced homomorphism T̂ ⟶ Ĝ is a maximal torus for Ĝ. Let T̆ be the subgroup of T given by elements with order a power of p. The composite of the inclusion T̆ ⟶ T with the completion map δ : T ⟶ T̂ is a discrete approximation for T̂. By ordinary Lie group theory, C^{a}(G) is contained in T and so C is the set of elements x ∈ T̆ such that x lies in the center of G. By 12.5 this is the same as the set of elements x ∈ T̆ such that the composite ⟨x⟩ ⟶ T̆ ⟶ Ĝ is central; by 6.4 and 6.7, this set is the p-discrete center of Ĝ. □

Proof of 1.4. It is enough to show that for each prime p the map e_{1} : BC^{a}(G) ⟶ Map(BG, BG)_{id} is an F_{p}-equivalence. Fix p as usual, and let C ⊂ C^{a}(G) be the group of elements of order a power of p. There are maps

BC ⟶^{u} BC^{a}(G) ⟶^{e_{1}} Map(BG, BG)_{id} ⟶^{h} Map(BG, BĜ)_{Bδ} ⟵^{≅} Map(BĜ, BĜ)_{id}

where u is induced by the inclusion C ⟶ C^{a}(G), h is given by composition with Bδ, and the last equivalence is also induced by composition (on the other side) with Bδ. A straightforward argument using 1.3 and 12.6 shows that the composite BC ⟶ Map(BG, BĜ)_{Bδ} is an F_{p}-equivalence. By inspection the map u is an F_{p}-equivalence (recall that C^{a}(G) is the product of a torus and a finite abelian group), and by 12.1 the map h is an F_{p}-equivalence too. It follows that e_{1} is an F_{p}-equivalence. □

References

[1] A. K. Bousfield, Homotopy spectral sequences and obstructions, Israel J. Math 66 (1989), 54–104.

[2] A. K. Bousfield, Localization and periodicity in unstable homotopy theory, preprint (University of Illinois, Chicago) 1992.

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

[4] W. Browder, On Differential Hopf Algebras, Transactions of the A.M.S. 107 (1963), 153-178.

[5] W. G. Dwyer, Transfer maps for fibrations, preprint (Notre Dame) 1993.

[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, D. M. Kan, and J. H. Smith, Towers of fibrations and homotopical wreath products, J. Pure and Applied Algebra 56 (1989), 9–28.

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

[9] W. G. Dwyer and G. Mislin, On the homotopy type of the components of Map_{∗}(BS^{3}, BS^{3}), Algebraic Topology, Barcelona 1986, Lecture Notes in Math. 1298, Springer, Berlin, 1987, pp. 82–89.

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

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

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

[13] W. G. Dwyer and A. Zabrodsky, Maps between classifying spaces, Algebraic Topology, Barcelona 1986, Lecture Notes in Math. 1298, Springer, Berlin, 1987, pp. 106–119.

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

[15] S. Jackowski, J. E. McClure and R. Oliver, Homotopy classification of self-maps of BG via G-actions, Annals of Math. 135 (1992), 227–270.

[16] 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.

[17] H. R. Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. 120 (1984), 39–87.

[18] J. M. Moller and D. Notbohm, Centers and finite coverings of finite loop spaces, preprint Göttingen) 1993.

[19] D. Notbohm, Abbildungen zwischen klassifizierenden Räume, Dissertation, Göttingen (1988).

University of Notre Dame, Notre Dame, Indiana 46556

Purdue University, West Lafayette, Indiana 47907

Processed February 28, 1994