PRODUCT SPLITTINGS FOR p-COMPACT GROUPS

W.G. Dwyer and C.W. Wilkerson

University of Notre Dame Purdue University

§1. Introduction

One of the fundamental results from the theory of compact Lie groups is the product splitting theorem:

1.1 Theorem. Any connected compact Lie group with a trivial center is isomorphic to a product of simple compact Lie groups.

The purpose of this paper is to prove a theorem similar to 1.1 for p-compact groups, which are homotopy theoretic analogues of compact Lie groups. Suppose that X is a p-compact group (see §2) with maximal torus T_{X} and Weyl group W_{X} (2.4). The finitely generated free Z_{p}-module L_{X} = π_{1}T_{X} has a natural action of W_{X} and is called the dual weight lattice of X.

1.2 Definition. A p-compact group X is said to be simple if the center of X is trivial (2.3) and the action of W_{X} on Q ⊗ L_{X} gives an irreducible representation of W_{X} over Q_{p}.

The main result of this paper is the following one.

1.3 Theorem. Any connected p-compact group X with a trivial center is equivalent to a product of simple p-compact groups.

Remark. For p odd, this has been proved independently by Notbohm (see 1.7). The word “equivalent” here means that the classifying space of X is weakly equivalent to a product of classifying spaces of simple p-compact groups.

Remark. If G is a compact Lie group with maximal torus T_{G} and Weyl Group W_{G}, the finitely generated free Z-module L_{G} = π_{1}T_{G} has a natural action of W_{G} and is called the dual weight lattice of G. The group G is simple in the usual sense if and only if the center of G is trivial and the action of W_{G} on Q ⊗ L_{G} gives an irreducible representation of W_{G} over Q. From this point of view 1.3 is a direct analogue of 1.1.

In order to prove Theorem 1.3 we will first give a general splitting criterion for a p-compact group and then show that the criterion applies in a strong way to a p-compact group with a trivial center. Suppose that X is the product ∏_{i} X_{i},

Typeset by AMS-TEX

The authors were supported in part by the National Science Foundation

where each X_{i} (i = 1, …, n) is a p-compact group with Weyl group W_{i} and dual weight lattice L_{i}. It is easy to see from the definitions that W_{X} is isomorphic to the product group ∏_{i} W_{i} and that L_{X} is isomorphic to the product module ∏_{i} L_{i}. In particular, the product decomposition for X is reflected by a product decomposition for L_{X} as a module over W_{X}. A product decomposition of L_{X} which arises in this way is said to be realized by a splitting of X. The general splitting criterion is this.

1.4 Theorem. If X is a connected p-compact group, then any product decomposition of L_{X} as a module over W_{X} can be realized up to equivalence by a splitting of X.

If X is a p-compact group then the action of W_{X} on Q ⊗ L_{X} is generated by reflections (2.4). This implies that no irreducible representation of W_{X} over Q_{p}, except possibly the trivial one, can appear with multiplicity greater than one in Q ⊗ L_{X}. If X has trivial center then the trivial representation of W_{X} does not appear in Q ⊗ L_{X} (7.2), and so there is a unique expression of Q ⊗ L_{X} as a direct sum of simple Q_{p}[W_{X}]-submodules.

1.5 Theorem. Suppose that X is a connected p-compact group with trivial center, and that

Q ⊗ L_{X} = M_{1} + ⋯ + M_{n}

is the unique expression of Q ⊗ L_{X} as a direct sum of simple Q_{p}[W_{X}]-submodules. Let L_{i} = L_{X} ∩ M_{i}. Then the addition map

α : L_{1} × ⋯ × L_{n} ⟶ L_{X}

is an isomorphism of W_{X} -modules.

If X is a p-compact group with trivial center, Theorem 1.5 gives a product splitting of L_{X} as a module over W_{X}, and Theorem 1.4 provides a corresponding product splitting of X. It is clear that the factors in this product splitting of X are simple p-compact groups. This proves 1.3.

1.6 Remark. Call a connected p-compact group X almost simple if the center of X is finite and the quotient of X by its center is simple. Theorem 1.3 implies that if X is a p-compact group which is connected and simply connected, then X splits as a product of almost simple factors. One way to prove this is to deduce from 6.11 and 7.2 that the center C of X is finite, use 1.3 to split X /C as a product ∏_{i} Y_{i} of simple factors (cf. [6, 6.3]), and then lift the product decomposition B(X /C) ≃ ∏_{i} BY_{i} to a product decomposition for the 2-connected cover BX of B(X /C). See also [10].

1.7 Relationship to [11]. Our results were originally obtained independently of [11]. While we were preparing this manuscript, however, we learned of Notbohm’s work and discovered that our arguments could be drastically simplified by using a beautiful idea which he described to us (5.1, 5.7). We are very grateful to him for his communication.

Guide to the proof. The proof of 1.4 given in this paper is completely parallel to a (slightly unconventional) proof of the corresponding fact for compact Lie groups. We will sketch the group theoretic argument and indicate how it fits in with the argument below. Recall the following classical result:

1.8 Proposition. Suppose that G is a connected compact Lie group. Then

(1) any element of G is conjugate to some element in the maximal torus T_{G} [6, pf. of 2.14], and

(2) if x and y are two elements of T_{G} which are conjugate in G, then there is an element of the Weyl group W_{G} which carries x to y (3.4).

Suppose now that G is a connected compact Lie group with dual weight lattice L_{G} = π_{1}T_{G}, and that L_{G} ≅ L_{1} × L_{2} is a product decomposition of modules over W_{G}. The problem is to produce a group splitting G ≅ G_{1} × G_{2}. The isomorphism T_{G} ≅ (R⊗L_{G})/L_{G} (2.5) gives a splitting T_{G} ≅ T_{1} ×T_{2} of modules over W_{G}, where T_{i} ≅ (R ⊗ L_{i})/L_{i}. Let H′ be the centralizer of T_{2} in G and K′ the centralizer of T_{1}. Denote the quotients H′/T_{2} and K′/T_{1} by H and K respectively. All of these groups are connected, and they can be arranged in a large commutative diagram similar to 6.2. The composite T_{1} ⟶ T_{G} ⟶ H′ ⟶ H is a maximal torus inclusion, as is the corresponding composite T_{2} ⟶ K; for simplicity, then, denote T_{1} by T_{H} and T_{2} by T_{K}.

One checks by elementary methods (6.5) that the central quotient map H′ ⟶ H has a unique section which on T_{H} agrees with the existing homomorphism T_{H} ⟶ H′; there is a similar section for the quotient map K′ ⟶ K. These sections give direct product decompositions H′ ≅ H × T_{K} and K′ ≅ T_{H} × K. Composing the sections with the inclusions H′ ⟶ G and K′ ⟶ G gives homomorphisms H ⟶ G and K ⟶ G, which allow H and K to be considered as subgroups of G. The Weyl group W_{H} is isomorphic to the subgroup of W_{G} which pointwise fixes T_{K}; similarly, W_{K} is isomorphic to the subgroup of W_{G} which pointwise fixes T_{H}. These two subgroups of W_{G} commute, and the product map W_{H} × W_{K} ⟶ W_{G} is an isomorphism (6.3).

By 1.8(2), no element of T_{H} is conjugate in G to any element of T_{K}, and so by 1.8(1) (see 3.3) no element of H is conjugate in G to any element of K. Another way of expressing this last fact is to say that the left action of H on G/K is free, so that the double coset space H\G/K is a manifold and there is a principal fibre bundle sequence

H × K ⟶ G ⟶ H\G/K .

It follows from dimension counting (3.8, 3.9) that H\G/K is a point (6.7), and thus that the multiplication map H × K ⟶ G is a diffeomorphism. From this it is easy to see that the natural map

H ≅ (H × T_{K})/T_{K} = H′/T_{K} ⟶ G/K ≅ H

is also a diffeomorphism (6.9). It is clear that this diffeomorphism respects the left actions of T_{K} on the spaces involved. Since the action of T_{K} on (H × T_{K})/T_{K} is trivial, it follows that the action of T_{K} on G/K is trivial, and thus by 1.8(1) that the action of K on G/K is trivial (5.1). This is one way of saying that K is a normal subgroup of G (4.8). Elementary considerations show that the composite map

H ⟶ G ⟶ G/K

is an isomorphism. By symmetry, H is normal in G, and it follows easily that the product map

G ⟶ (G/K) × (G/H) ≅ H × K

is an isomorphism.

Organization of the paper. Section 2 sketches some background material about pcompact groups. Sections 3 and 4 study respectively the notions of double coset space and normalizer. In §5 there is a method (due to Notbohm) for recognizing the triviality of certain bundles over classifying spaces. Finally, §6 has the proof of 1.4 and §7 the proof of 1.3.

Notation and terminology. The prime p is fixed for the paper; F_{p} is the field with p elements, Z_{p} the ring of p-adic integers, and Q_{p} = Q ⊗ Z_{p} the field of p-adic numbers. All unspecified homology and cohomology is with coefficients in F_{p}. A map f of spaces is an F_{p}-equivalence if H^{∗}(f) is an isomorphism. A space X is said to be F_{p}-finite if H^{∗}X is finite, and F_{p}-complete if it is F_{p}-complete in the sense of [2]. The symbol H_{Q_{p}}^{i}(X) stands for the Q_{p}-module Q ⊗ H^{i}(X , Z_{p}).

We assume that all spaces which arise have been replaced, if necessary, by weakly equivalent CW-complexes. The word equivalence applied to a map between spaces means homotopy equivalence.

§2. p-compact groups

The purpose of this section is to recall some material connected with p-compact groups and establish notation. The basic references are [5], [6] and [10].

2.1 Loop spaces and p-compact groups.. A loop space is a triple (X , BX , e) consisting of a space X, a pointed space BX, and an equivalence e : X ≃ Ω BX. Usually BX and e are omitted from the notation. A loop space X is said to be a p-compact group if X is F_{p}-finite, X is F_{p}-complete, and π_{0}X is a finite p-group, or equivalently if X is F_{p}-finite and BX is F_{p}-complete. The p-compact group X is a p-compact torus if it is the F_{p}-completion of an ordinary torus, and a p-compact toral group if its identity component is a p-compact torus. We will use script letters (e.g., X, Y) for spaces and p-compact groups, and roman letters (G, T) for p-compact toral groups.

2.2 Homomorphisms and centralizers. A homomorphism f : X ⟶ Y of loop spaces is by definition a pointed map Bf : BX ⟶ BY; two homomorphisms f and g are homotopic (resp. conjugate) if Bf and Bg are homotopic as pointed (resp. unpointed) maps. Note that if Y is connected there is no distinction between homotopy and conjugacy. A homomorphism f is trivial if Bf is homotopic to a constant map. Suppose that f : X ⟶ Y is a homomorphism of loop spaces. The centralizer of f(X) in Y, denoted C_{Y}(f(X)) or C_{X} (X) is the loop space Ω Map(BX , BY)_{Bf} and the homogeneous space Y/f(X) (or Y/X) is the homotopy fibre of Bf over the basepoint of BY. There is a natural loop space homomorphism C_{Y}(f(X)) ⟶ Y given by a construction which involves evaluating maps at the basepoint of BX. The homomorphism f is said to be central if C_{Y}(X) ⟶ Y is an equivalence. If f is a homomorphism of p-compact groups, the f is said to be a monomorphism if Y/f(X) is F_{p}-finite. If X is a p-compact toral group and Y is a p-compact group then C_{Y}(f(X)) is a p-compact group and the map C_{Y}(f(X)) ⟶ Y is a monomorphism.

2.3 Abelian p-compact groups and central quotients. A loop space X is said to be abelian if the identity homomorphism of X is central. A p-compact group is abelian if and only if it is equivalent to the product of a p-compact torus and a finite abelian p-group. If A is abelian and f : A ⟶ X is a homomorphism of p-compact groups, then f lifts to a central homomorphism A ⟶ C_{X} (A) which is a monomorphism if f is. In general, if f : A ⟶ X is a central monomorphism of p-compact groups (sometimes expressed by saying that A is a central subgroup of X) then it is possible to form a “quotient” p-compact group X /A. For any p-compact group X there is a universal central monomorphism C ⟶ X called the center of X.

2.4 Maximal tori and Weyl groups. A maximal torus for a p-compact group X is a p-compact torus T together with a monomorphism T ⟶ X with the property that the mod p cohomology Euler characteristic χ(X /T) is nonzero. Any p-compact group X has a maximal torus, unique up to conjugacy; a chosen representative is denoted i_{X} : T_{X} ⟶ X or just T_{X}. The Weyl space W_{X} (T_{X}) of self-maps of BT_{X} over BX (computed after replacing the map BT_{X} ⟶ BX by an equivalent fibration) is homotopically discrete, and π_{0}W_{X} (T_{X}) is a finite group called the Weyl group W_{X} of X. The dual weight lattice L_{X} of X is defined to be the finitely generated free Z_{p}-module π_{1}T_{X} = π_{2} BT_{X}; this is a module over W_{X} in an evident way. If X is connected then W_{X} acts faithfully on L_{X} and the image of W_{X} in Aut_{Q_{p}}(Q⊗L_{X}) is generated by reflections, that is, by automorphisms which are diagonalizable and pointwise fix a codimension 1 subspace. (Sometimes such automorphisms are called pseudoreflections.) An element s ∈ W_{X} is itself called a reflection if its image in Aut_{Q_{p}}(Q ⊗ L_{X}) is a reflection.

The Borel construction of the action of W_{X} (T_{X}) on BT_{X} is denoted BN(T_{X}) and maps naturally to BX; the loop space N(T_{X}) = Ω BN(T_{X}) is called the normalizer of T_{X} (in X). The p-normalizer of T_{X}, denoted N_{p}(T_{X}), is the p-compact group which is the inverse image in N(T_{X}) of a p-Sylow subgroup of π_{0}N(T_{X}) = W_{X}.

2.5 p-discrete approximations. A p-discrete torus is by definition a (discrete) group isomorphic to (Z/p^{∞})^{r} for some r; more generally, a p-discrete toral group is a group which lies in a short exact sequence in which the kernel is a p-discrete torus and the quotient is a finite p-group. If G is a p-compact toral group then there exists an essentially unique homomorphism f : Ğ ⟶ G such that Ğ is a p-discrete toral group and Bf is an F_{p}-equivalence; the F_{p}-completion of BĞ is equivalent to BG. One says that Ğ is a p-discrete approximation to G or that G is the p-closure of Ğ. For example, if T is a p-compact torus then T̆ is the group Z/p^{∞} ⊗ π_{1}T. If X is a p-compact group, G is a p-compact toral group, and i : Ğ ⟶ G is a p-discrete approximation, then the assignment f ↦ f · i gives a bijection between homotopy (resp. conjugacy) classes of homomorphisms G ⟶ X and homotopy (resp. conjugacy) classes of homomorphisms Ğ ⟶ X; moreover, if f : G ⟶ X is a homomorphism, then the natural map C_{X} (f(G)) ⟶ C_{X} (fi(Ğ)) is an equivalence.

If L is a finitely generated free Z_{p}-module we denote Z/p^{∞}⊗_{Z}L = Z/p^{∞}⊗_{Z_{p}}L by L̆, so that for instance if X is a p-compact group there is an isomorphism T̆_{X} ≅ L̆_{X}. The formula L = Hom(Z/p^{∞}, L̆) allows L be recovered from L̆.

§3. Double coset spaces

Suppose that f : Y ⟶ X and g : Z ⟶ X are homomorphisms of loop spaces.

3.1 Definition. The double coset space f(Y)\X /g(Z) (or Y\X /Z for short) is the homotopy pullback of the diagram

BY ⟶^{Bf} BX ⟵^{Bg} BZ

For example, if Y = ∗ is trivial, then Y\X /Z is the coset space X /Z. The following proposition is interesting even in this case: it implies that a homomorphism g : Z ⟶ X of p-compact groups is a monomorphism (2.2) if and only if every nontrivial homomorphism Z/p ⟶ Z remains nontrivial when composed with g.

3.2 Proposition. Suppose that f : Y ⟶ X and g : Z ⟶ X are homomorphisms of p-compact groups. Then the following conditions are equivalent

(1) the space Y\X /Z is F_{p}-finite, and

(2) whenever i : Z/p ⟶ Y and j : Z/p ⟶ Z are homomorphisms with fi conjugate to gj, then i and j are trivial.

Proof. Suppose first that i and j are homomorphisms as described with fi conjugate to gj and i, say, nontrivial. It follows from the basic property of homotopy pullbacks that there exists a map h : BZ/p ⟶ Y\X /Z which lifts the map Bi : BZ/p ⟶ BY. Clearly h is not homotopic to a constant map. By Lannes [7, 3.1.1] (and the fact that H^{∗} BZ/p is an exterior algebra on a one-dimensional class tensored with a polynomial algebra on its Bockstein) the map h induces a nontrivial map on mod p cohomology in infinitely many dimensions, and so Y\X /Z is not F_{p}-finite.

To prove the implication in the other direction, we first work in the case Y = ∗. The hypothesis is now that if j : Z/p ⟶ Z is a nontrivial homomorphism then gj is also nontrivial. As in [5, 3.4], this hypothesis is equivalent to the statement that the homotopy fixed point set of the action of gj(Z/p) on X /∗ = X is empty for each such homomorphism j. There is a fibration sequence

Z/N_{p}(T_{Z}) ⟶ X /N_{p}(T_{Z}) ⟶ X /Z

in which the fibre has mod p Euler characteristic prime to p [5, pf. of 2.4], so that in order to show that ∗\X /Z = X /Z is F_{p}-finite it is enough by a transfer argument [5, 9.13] to show that X /N_{p}(T_{Z}) is F_{p}-finite. For this it is enough, as in the proof of [5, 7.3] to show that X /N̆_{p}(T_{Z}) is F_{p}-finite (2.5) or even that X /G is F_{p}-finite for each finite subgroup G of N̆_{p}(T_{Z}). However, X /G can be interpreted as the homotopy orbit space X_{hG} for a suitable “action” of G on X [5, §10] and so the fact that X /G is F_{p}-finite follows by [5, 7.4] from the fact that X^{hK} is empty for each order p cyclic subgroup K of G.

Now consider the general case in which Y is not necessarily contractible. The assumption is that whenever i : Z/p ⟶ Y and j : Z/p ⟶ Z are homomorphisms with fi is conjugate to gj, then i and j are trivial. By the above considerations this implies first of all that X /Z is F_{p}-finite, and secondly that the homotopy fixed point set of the action of fi(Z/p) on (X /Z) is empty for each nontrivial homomorphism i : Z/p ⟶ Y. Now copy the above argument. There is a fibration sequence

Y/N_{p}(T_{Y}) ⟶ N_{p}(T_{Y})\X /Z ⟶ Y\X /Z

in which the fibre has Euler characteristic prime to p, so that in order to show that Y\X /Z is F_{p}-finite, it is enough by a transfer argument to show that N_{p}(T_{Y})\X /Z is F_{p}-finite. For this it is enough to show that N̆_{p}(T_{Y})\X /Z is F_{p}-finite or even that G\X /Z is F_{p}-finite for each finite subgroup G of N̆_{p}(T_{Y}). This however follows by [5, 7.4] from the fact that (X /Z)^{hK} is empty for each order p cyclic subgroup K of G. □

3.3 Corollary. Suppose that f : Y ⟶ X and g : Z ⟶ X are homomorphisms of connected p-compact groups. Then Y\X /Z is F_{p}-finite if and only if T_{Y}\X /T_{Z} is F_{p}-finite.

Proof. This follows immediately from 3.2 and the fact that if U is a connected p-compact group, then any homomorphism Z/p ⟶ U lifts up to homotopy to a homomorphism Z/p ⟶ T_{U} [5, pf. of 9.1]. □

3.4 Proposition. Let X be a p-compact group with maximal torus i : T_{X} ⟶ X, and let j : T_{X} ⟶ N(T_{X}) be the usual homomorphism. Suppose that A is an abelian p-compact toral group and f, g : A ⟶ T_{X} a pair of monomorphisms such that if is conjugate to ig. Then f and g differ by the action of an element of W_{X}, and so jf is conjugate to jg.

Proof. We can assume after adjusting the homomorphisms up to conjugacy that B(if) = B(ig) = Bk. Let Y be the identity component of C_{X} (k(A)). Since A and T = T_{X} are abelian (2.3) and T is connected there are homomorphisms

\begin{matrix}A ⟶^{f′} & C_{T} (f(A)) ≃ T ⟶^{u} Y ⟶^{e} X \\ A ⟶^{g′} & C_{T} (g(A)) ≃ T ⟶^{v} Y ⟶^{e} X \end{matrix}

such that both horizontal composites are conjugate to k, f′ is conjugate to f, g′ is conjugate to g, uf′ is conjugate to vg′, and the composites eu and ev are conjugate to i. The homomorphisms u and v are both maximal tori for Y [6, 4.2]. Since maximal tori in Y are unique up to conjugacy [5, 9.4] there is a homomorphism h : T ⟶ T such that vh is conjugate to u. This implies that evh ∼ ih is conjugate to eu ∼ i, so that h represents an element in W_{X} (2.4). As above, the homomorphism vhf′ ∼ uf′ is conjugate to vg′, and so by [6, 6.5], hf′ is conjugate to g′ (this is a homotopical form of the observation that inner automorphisms of a group induce the identity on a central subgroup). Thus, as an element of W_{X}, the map h carries f to g and so provides a conjugacy in N(T_{X}) between jf and jg. □

3.5 Remark. Let f : Y ⟶ X be a homomorphism of connected p-compact groups. It follows from [5, 8.11] and 3.4 that there is an induced homomorphism f_{T} : T_{Y} ⟶ T_{X}, unique up to the action of W_{X} on the set of homotopy classes of homomorphisms T_{Y} ⟶ T_{X}, such that the diagram

T_{Y} ⟶^{f_{T}} T_{X}
↓  ↓
Y ⟶ X

commutes up to homotopy. For convenience we will refer to such a homomorphism f_{T} as “the homomorphism induced by f”; it is unique, however, only up to the action of W_{X}. The corresponding map π_{1}(f_{T}) : L_{Y} ⟶ L_{X} is denoted f_{L}.

3.6 Proposition. Suppose that X, Y, and Z are connected p-compact groups and that f : Y ⟶ X and g : Z ⟶ X are homomorphisms. Let f̄_{L} : L̄_{Y} ⟶ L̄_{X} and ḡ_{L} : L̄_{Z} ⟶ L̄_{X} be the maps obtained by tensoring f_{L} and g_{L} (3.5) with Z/p. Then Y\X /Z is F_{p}-finite if and only if

(1) f̄_{L} and ḡ_{L} are monomorphisms, and

(2) for each element w ∈ W_{X} the groups f̄_{L}(L̄_{Y}) and w · g_{L}(L̄_{Z}) intersect trivially in L̄_{X}.

Proof. By 3.3, Y\X /Z is F_{p}-finite if and only if T_{Y}\X /T_{Z} is F_{p}-finite. By 3.2, this latter space is not F_{p}-finite if and only if there are homomorphisms i : Z/p ⟶ T_{Y} and j : Z/p → T_{Z}, at least one of which is nontrivial, such that fi is conjugate to gj. By 3.4, this last is the case if and only if there are homomorphisms i : Z/p ⟶ T_{Y} and j : Z/p → T_{Z}, at least one of which is nontrivial, and some element w ∈ W_{X} which carries f_{T} ·i to g_{T} ·j. The proof is finished by observing that if T is a p-compact torus, the set of conjugacy classes of homomorphisms Z/p ⟶ T corresponds bijectively in a natural way to Z/p⊗π_{1}T. The correspondence assigns to a map Bi : BZ/p ⟶ BT the image under the induced homology map of a chosen generator of H_{2} BZ/p. □

Recall that if X is a space with suitable finiteness properties, the (cohomological) dimension cd_{F_{p}}(X) (resp. cd_{Q_{p}}(X)) is defined to be the largest degree in which H^{∗}X (resp. H_{Q_{p}}^{∗}X) is nonzero. We end this section with a short discussion of the relationship between the dimension of a double coset space and the dimensions of the associated p-compact groups.

3.7 Proposition. Suppose that f : Y ⟶ X and g : Z ⟶ X are homomorphisms of p-compact groups, and assume that the associated double coset space Y\X /Z F_{p}-finite. Then

cd_{F_{p}}(Y\X /Z) = cd_{F_{p}}(X) − (cd_{F_{p}}(Y) + cd_{F_{p}}(Z)) .

Proof. This follows from applying the argument of [5, proof of 6.14] to the principal fibration Y × Z ⟶ X ⟶ Y\X /Z. The fundamental group of the base acts nilpotently on the homology of the fibre here, because π_{0}(Y × Z) is a finite p-group [5, 11.6]. □

In applying 3.7, it is useful to have a way of calculating the dimension of a pcompact group X in terms of W_{X}. Suppose that M is a vector space over Q_{p} of rank r and that W is a finite subgroup of Aut_{Q_{p}}(M) generated by reflections (2.4). Let M^{#} denote the Q_{p}-dual of M, so that the symmetric algebra Sym(M^{#}) is the algebra of polynomial functions on M. We regard Sym(M^{#}) as a graded algebra in which M^{#} itself has degree 2. The action of W on M induces an action of W on M^{#} and a degree-preserving action of W on Sym(M^{#}). It is known [3] that the fixed point subalgebra Sym(M^{#})^{W} is isomorphic to a graded polynomial algebra on r generators of degree, say, d_{1}, …, d_{r}. The sequence of integers d_{1}, …, d_{r} is independent, up to permutation, of the choice of polynomial generators for Sym(M^{#})^{W}. We define the dimension associated to W, denoted δ(W), by the formula

δ(W) = ∑_{i=1}^{r}(d_{i} − 1) = (∑_{i=1}^{r} d_{i}) − r .

3.8 Lemma. Suppose that X is a connected p-compact group with Weyl group W_{X} ⊂ Aut_{Q_{p}}(Q ⊗ L_{X}) (2.4). Then cd_{F_{p}}(X) = δ(W_{X}).

Proof. As in [6, 4.5], cd_{F_{p}}(X) = cd_{Q_{p}}(X). Set M = Q⊗L_{X} and let r = rank_{Q_{p}}(M). The graded algebras H_{Q_{p}}^{∗}(BT_{X}) and Sym(M^{#}) are isomorphic in a way which respects the actions of W_{X} on the two of them. By [5, 9.7] there is an isomorphism H_{Q_{p}}^{∗}(BX) ≅ Sym(M^{#})^{W_{X}}, and so H_{Q_{p}}^{∗}(BX) is isomorphic to a polynomial algebra on generators of dimension d_{1}, …, d_{r}, where these integers are the ones associated to the reflection group W_{X} above. It follows from the Eilenberg-Moore spectral sequence that H_{Q_{p}}^{∗}(X) is an exterior algebra on generators of dimension d_{1} − 1, …, d_{r} − 1. Thus

cd_{F_{p}}(X) = cd_{Q_{p}}(X) = ∑_{i=1}^{r}(d_{i} − 1) = δ(W_{X}) . □

3.9 Lemma. Suppose that M_{1} and M_{2} are finite dimensional vector spaces over Q_{p}, and that W_{i} (i = 1, 2) is a finite subgroup of Aut_{Q_{p}}(M_{i}) generated by reflections. Let W = W_{1} × W_{2} ⊂ Aut_{Q_{p}}(M_{1} × M_{2}). Then W is generated by reflections (2.4), and δ(W) = δ(W_{1}) + δ(W_{2}).

Proof. It is obvious that W is generated by reflections. The formula for δ(W) follows from the calculation

Sym((M_{1} × M_{2})^{#})^{W} = Sym(M_{1}^{#})^{W_{1}} ⊗_{Q_{p}} Sym(M_{2}^{#})^{W_{2}} . □

§4. Normalizers

Suppose that f : Y ⟶ X is a homomorphism of p-compact groups. We can assume after making an adjustment up to homotopy that Bf : BY ⟶ BX is a fibration.

4.1 Definition. The Weyl space of f, denoted W_{X} (f(Y)) or W_{X} (Y), is the space of maps BY ⟶ BY over BX.

4.2 Remark. The Weyl space W_{X} (Y) is a topological monoid under composition of maps, and acts in a natural way on BY. The above definition of W_{X} (Y) generalizes the one given previously (2.4) in the special case Y = T_{X}.

Let q : f(Y)\X /f(Y) ⟶ BY be the natural fibration (3.1). It is clear that W_{X} (Y) can be interpreted as the space Γ(q) of sections of q. In the language of [5, 10.8] it is natural to write W_{X} (Y) = (X /f(Y))^{hf(Y)} or simply W_{X} (Y) = (X /Y)^{hY}. Evaluation at the basepoint of BY gives a natural map

W_{X} (Y) = (X /Y)^{hY} ⟶ X /Y .

The Weyl space W_{X} (f(Y)) is particularly useful when f is a monomorphism. One reason is the following.

4.3 Proposition. Suppose f : Y ⟶ X is a monomorphism of p-compact groups. Then π_{0}W_{X} (Y) is a group. Equivalently, every self map of BY over BX is an equivalence.

Proof. Let i : BY ⟶ BY be a self map of BY over BX. By 3.2, i is a monomorphism. By dimension counting (3.7) the coset space Y/i(Y) is homotopically discrete, and so i is an equivalence on identity components [5, 6.16]. A fibration argument now shows that the homotopy fibre of the map Bπ_{0}Y ⟶ Bπ_{0}Y induced by i is the same as Y/i(Y) and in particular is homotopically discrete. This implies that π_{0}(i) is injective, and hence an isomorphism because π_{0}Y is finite. □

4.4 Definition. Suppose that f : Y ⟶ X is a monomorphism of p-compact groups. The normalizer of f(Y) in X, denoted N_{X} (f(Y)) or N_{X} (Y), is the loop space whose classifying space is the Borel construction of the action (4.2) of W_{X} (Y) on BY.

4.5 Remark. This definition of N_{X} (Y) generalizes one given previously (2.4) in the special case Y = T_{X}. In general (even if Y = T_{X}) N_{X} (Y) is not a p-compact group. Since the action of W_{X} (Y) on BY commutes with the projection Bf : BY ⟶ BX, there is a natural map BN_{X} (Y) ⟶ BX or equivalently a natural loop space homomorphism N_{X} (Y) ⟶ X.

4.6 Remark. There is a fibration sequence of spaces (corresponding to a short exact sequence [5, 3.2] of loop spaces)

BY ⟶ BN_{X} (Y) ⟶ BW_{X} (Y) .

The space W_{X} (Y) should therefore be interpreted as a quotient N_{X} (Y)/Y. In fact, the map W_{X} (Y) ⟶ X /Y of 4.2 corresponds to the usual map N_{X} (Y)/Y ⟶ X /Y if, say, the p-compact groups involved are finite p-groups. Observe that the restriction to Y of the homomorphism N_{X} (Y) ⟶ X is the original homomorphism f : Y ⟶ X.

4.7 Definition. Let f : Y ⟶ X be a monomorphism of p-compact groups. The monomorphism f is said to be normal (or Y is said to be a normal subgroup of X) if the natural homomorphism (4.5) N_{X} (Y) ⟶ X is an equivalence.

4.8 Proposition. If f : Y ⟶ X is a monomorphism of p-compact groups, the following three conditions are equivalent.

(1) f(Y) is a normal subgroup of X.

(2) The standard map (4.2) W_{X} (Y) ⟶ X /Y is an equivalence.

(3) The fibration Y\X /Y ⟶ BY with fibre X /Y is trivial.

4.9 Remark. As usual, a fibration E ⟶ B with fibre F is said to be trivial if it is fibre homotopy trivial, that is, if there is a map E ⟶ F which restricts on F ⊂ E to the identity map.

Proof of 4.8. The equivalence between the first two conditions results from constructing a commutative diagram

W_{X} (Y) ⟶ BY ⟶ BN_{X} (Y)
↓  ^{=}↓  ↓
Y/X ⟶ BY ⟶ BX

and comparing the upper fibration sequence to the lower one. The equivalence between the second two follows from [6, 9.3]; in general, a fibration q over BY with fibre a space F which is both F_{p}-complete and F_{p}-finite is trivial if and only if evaluation at the basepoint of BY gives an equivalence Γ(q) ⟶ F. □

4.10 Proposition. If f : A ⟶ X is a central monomorphism of p-compact groups, then f(A) is a normal subgroup of X.

Proof. The map Bf : BA ⟶ BX is up to homotopy the principal fibration constructed by pulling back to BX the path fibration over B(X /A). The fibration q : A\X /A ⟶ BA is thus also principle, since it is a further pullback of Bf. Since the fibration q has an obvious section given by the identity map of BA, it follows that q is a trivial fibration, and the desired result is a consequence of 4.8. □

§5 Bundles over BX

In this section we study bundles over the classifying space of a p-compact group, with the goal of finding situations in which 4.8(3) can be used to show that some given monomorphism is normal. The basic idea in this section is due to Notbohm [11]. The main theorem is the following one.

5.1 Theorem. Let X be a p-compact group, and q : E ⟶ BX a fibration in which the fibre F is F_{p}-finite and F_{p}-complete. Then q is a trivial fibration (4.9) if and only if its pullback over BN_{p}(T_{X}) is trivial. If X is connected, then q is a trivial fibration if and only if its pullback over BT_{X} is trivial.

By Theorem 4.8, this has an immediate corollary.

5.2 Corollary. Suppose that f : Y ⟶ X is a monomorphism of p-compact groups. Then f is normal (4.7) if and only if the fibration N_{p}(T_{Y})\X /Y ⟶ BN_{p}(T_{Y}) is trivial. If Y is connected, then f is normal if and only if the fibration T_{Y}\X /Y ⟶ BT_{Y} is trivial.

The rest of this section is devoted to proving 5.1.

5.3 Definition. A space Z is said to be BZ/p-null if evaluation at the basepoint of BZ/p gives an equivalence

Map(BZ/p, Z) ⟶^{≃} Z.

5.4 Example. By Miller’s theorem [8], if F is a space which is F_{p}-finite and F_{p}- complete, then F is BZ/p-null.

If G is a finite p-group and f : G ⟶ Z is a homomorphism of loop spaces, define the kernel ker(f) of f to be the set of all elements g ∈ G such that the restriction of f to the cyclic subgroup ⟨g⟩ of G is trivial (2.2). In general ker(f) is not necessarily a subgroup of G, but there is one key situation in which the kernel is well behaved.

5.5 Proposition. Suppose that G is a finite p-group, Z is a loop space which is BZ/p-null, and f : G ⟶ Z is a homomorphism. Then ker(f) is a normal subgroup of G, and f factors uniquely up to homotopy through a homomorphism G/ ker(f) ⟶ Z. In particular, f is trivial if and only if ker(f) = G.

Proof. This is essentially the same as the proof given in [5, §7] for the special case in which Z is a p-compact group. It is only necessary to replace the reference to [5, 5.4] in the proof of [5, 7.2] by the assumption that Z is BZ/p-null, and observe that the cyclic group K in the proof of [5, 7.7] can be assumed to have order p. □

Recall that a space U is F_{p}-local (in the sense of Bousfield [1]) if any F_{p}- equivalence A ⟶ B induces an equivalence Map(B, U) ⟶ Map(A, U). Any F_{p}- complete space is F_{p}-local [1].

5.6 Theorem. Suppose that X is a p-compact group and Z is a loop space such that Z is BZ/p-null and BZ is F_{p}-local. Then a homomorphism f : X ⟶ Z is trivial (2.2) if and only if the restriction of f to N_{p}(T_{X}) is trivial. If X is connected, a homomorphism f : X ⟶ Z is trivial if and only if the restriction of f to T_{X} is trivial.

Proof. The statement involving N_{p}(T_{X}) appears in a slightly different form in [11, 1.4]. One way to prove this statement is to copy the inductive argument of [4, §5] (compare [4, 5.4]), being careful to rely on the homology decomposition theorem for p-compact groups from [6, §8] instead of on the theorem of Jackowski and McClure used in [4]. The effect of this line of reasoning is to show that f : X ⟶ Z is trivial if and only if it is null on finite p-groups, i.e., if and only if the composite fρ is trivial for every homomorphism ρ : G ⟶ X with G a finite p-group. The desired result now follows from the fact that every such homomorphism ρ : G ⟶ X lifts up to conjugacy to a homomorphism G ⟶ N_{p}(T_{X}) [6, 2.14], so that fρ is trivial if the restriction of f to N_{p}(T_{X}) is.

Suppose now that X is connected. We will prove the remaining statement in the theorem by using the idea in the proof of [9, 5.7]. As noted above, we have proved that f : X ⟶ Z is trivial if and only if the composite fρ is trivial for every homomorphism ρ : G ⟶ X with G a finite p-group. For such a homomorphism ρ, the composite fρ is trivial if and only if ker(fρ) = G (5.5). However, any homomorphism from a cyclic p-group ⟨g⟩ to X lifts up to conjugacy to a homomorphism ⟨g⟩ ⟶ T_{X} (see the proof of [6, 2.14]), and so ker(fρ) = G if the restriction of f to T_{X} is trivial. □

5.7 Proposition. Suppose that F is a space which is F_{p}-finite and F_{p}-complete. Let Aut^{h}(F) denote the the topological monoid of self homotopy equivalences of F, and Z some union of components of Aut^{h}(F) such that Z is closed under composition and π_{0}Z is a finite p-group. Then Z is BZ/p-null and BZ is F_{p}-local.

Remark. Since Z is a topological monoid such that π_{0}Z is a group, Z is equivalent to Ω BZ, i.e., Z is a loop space.

Proof of 5.7. If g : A ⟶ B is an F_{p}-equivalence, it is clear that the induced map

Map(F, Map(B, F)) ⟶^{Map(Id,Map(g,Id))} Map(F, Map(A, F))

is an equivalence. and hence from an adjointness argument that the induced map

Map(B, Map(F, F)_{Id}) ⟶^{Map(g,Id)} Map(A, Map(F, F)_{Id})

is also an equivalence. This in effect shows that Z is F_{p}-local, since a space is F_{p}-local if and only if each of its components is. The space Z is F_{p}-good [5, 1.3] because it is a loop space, and therefore [5, 11.3] F_{p}-complete. An application of the fibre lemma [5, 11.7] to the path fibration over BZ now shows that BZ is F_{p}-complete and hence F_{p}-local.

In order to show that Z is BZ/p-null, consider the map

Map(BZ/p, Map(F, F)) ≅ Map(F, Map(BZ/p, F)) ⟶ Map(F, F) ,

given by evaluation at the basepoint of BZ/p. This map is an equivalence by 5.4. The fact that Z is BZ/p-null follows from the fact that a space is BZ/p-null if and only if each of its components is. □

Proof of 5.1. Let Bf : BX ⟶ BAut^{h}(F) be the classifying map for the fibration q, and f = Ω Bf : X ⟶ Aut^{h}(F) the corresponding loop space homomorphism. By bundle theory the fibration q is trivial if and only if the homomorphism f is trivial. Let Z ⊂ Aut^{h}(F) be the union of those components of Aut^{h}(F) which intersect the image of f nontrivially. Then Z is closed under composition, and π_{0}Z, which is a quotient of π_{0}X, is a finite p-group. By 5.7, Z is BZ/p-null and BZ is F_{p}-local. The homomorphism f clearly lifts to a homomorphism f̃: X ⟶ Z, and f̃ is trivial if and only if f is. The desired result now follows from 5.6. □

§6. Splittings of p-compact groups

In this section we will prove Theorem 1.4. By induction it is enough to treat the case in which there are only two factors in the product decomposition of the dual weight lattice. The goal of this section is thus to prove the following.

6.1 Theorem. If X is a connected p-compact group, then any product decomposition L_{X} = L_{1} × L_{2} of modules over W_{X} can realized by a p-compact group splitting X ≃ X_{1} × X_{2}

We will now set up some notation which will be used for the rest of the section. Let X be a p-compact group, and L_{X} = L_{1} × L_{2} a product splitting of modules over W_{X}. For i = 1, 2 denote by k_{i} : T_{i} ⟶ T_{X} the central subgroup (2.3) of T_{X} obtained by taking the p-closure (2.5) of L̆_{i} ⟶ L̆_{X} = T̆_{X}. Let Y′ = C_{X} (T_{2}) and Z′ = C_{X} (T_{1}). By [6, 4.3] the maximal torus T_{X} ⟶ X lifts to maximal tori T_{X} ⟶ Y′ and T_{X} ⟶ Z′. Denote the quotients Y′/T_{2} and Z′/T_{1} by Y and Z respectively. These p-compact groups can then be arranged into a commutative diagram

(6.2)  \begin{aligned}T_{2} & ⟶^{k_{2}} & T_{X} & ⟶^{=} & T_{X} & ⟵^{=} & T_{X} & ⟵_{k_{1}} & T_{1} \\ ↓^{=} & & ^{i_{Y′}} ↓ & ^{i_{X}} ↓ & & ^{i_{Z′}} ↓ & & ^{=}↓ \\ T_{2} & ⟶^{k′_{2}} & Y′ & ⟶ & X & ⟵ & Z′ & ⟵^{k′_{1}} & T_{1} \\ & & ^{q′_{Y}} ↓ & & & ^{q′_{Z}} ↓ \\ & & Y = Y′/T_{2} & & & Z = Z′/T_{1} \end{aligned}

in which the upper vertical maps are maximal torus monomorphisms and the lower vertical maps are quotient projections. The fact that the outer squares commute, where k′_{2}, for instance, is the natural map (2.3) from T_{2} to its centralizer, follows either from inspection or from [6, 6.5]. All of these p-compact groups are connected, since the centralizer of a torus inside a connected p-compact group is connected [6, 7.8], and any central quotient of a connected p-compact group is connected.

Let W_{1} ⊂ W_{X} be the subgroup which pointwise fixes L_{2}, and W_{2} ⊂ W_{X} the subgroup which pointwise fixes L_{1}.

6.3 Lemma. The subgroups W_{1} and W_{2} of W_{X} commute, and the product map W_{1} × W_{2} ⟶ W_{X} is an isomorphism.

Proof. It is clear from the definitions that if x ∈ W_{1} and y ∈ W_{2} then the commutator [x, y] belongs to W_{1} ∩ W_{2}. Since X is connected the action of W_{X} on L_{X} is faithful [5, 9.7]. This shows that W_{1} ∩ W_{2} is trivial, thus that x commutes with y, and thus that the product map W_{1} × W_{2} ⟶ W_{X} is a monomorphism. By elementary rank considerations, if s ∈ W_{X} is a reflection (2.4) then either s ∈ W_{1} or s ∈ W_{2}. Since W_{X} is generated by reflections, the indicated product map is onto. □

We will now concentrate briefly on Y′ and Y. By [6, 7.6], the action of W_{Y′} on π_{1}T_{Y′} = π_{1}T_{X} = L_{X} (see above) gives an isomorphism between W_{Y′} and W_{1}. The calculation (Y′/T_{2})/(T_{X} /T_{2}) ≃ Y′/T_{X} shows that the quotient map T_{X} /T_{2} ⟶ Y is a maximal torus for Y (see the proof of [6, 6.3]). The composite map

(6.4)  T_{1} ⟶ T_{X} ⟶ T_{X} /T_{2}

is an equivalence because on fundamental groups it induces the quotient isomorphism L_{X} /L_{2} ≅ L_{1}. It follows that we can choose a maximal torus T_{Y} by setting T_{Y} = T_{1} and letting i_{Y} : T_{Y} ⟶ Y be the result of following the composite 6.4 by the monomorphism T_{X} /T_{2} ⟶ Y. The group W_{Y} is then isomorphic in its action on L_{Y} = π_{2}T_{1} = L_{1} to W_{1}, so from now on we denote L_{1} by L_{Y} and W_{1} by W_{Y}. The map k′_{1} above (6.2) thus gives a commutative diagram

\begin{matrix}T_{Y} & ⟶^{k′_{1}} & Y′ \\ ^{=}↓ & ^{q′_{Y}} ↓ \\ T_{Y} & ⟶^{i_{Y}} & Y \end{matrix} .

6.5 Lemma. There is up to homotopy a unique homomorphism s_{Y} : Y ⟶ Y′ such that s_{Y} is a section for q′_{Y} and agrees on T_{Y} with k′_{2}, i.e., such that q′_{Y} · s_{Y} is homotopic to Id_{Y} and s_{Y} · i_{Y} is homotopic to k′_{2}.

The proof of this will be given below.

6.6 Remark. By construction the map B(q′_{Y}) : BY′ ⟶ B(Y′/T_{Z}) = BY is a principal fibration [5, pf. of 8.3]. The existence of the section B(s_{Y}) implies that this fibration is trivial, or in other words that Y′ is equivalent as a p-compact group to the product T_{Z} × Y.

The above considerations and their “Z” analogues lead to a diagram

Y ⟶^{s_{Y}} Y′ ⟶ X ⟵ Z′ ⟵^{s_{Z}} Z .

6.7 Proposition. The double coset space (3.1) Y\X /Z is contractible, i.e., the loop space multiplication map Y × Z ⟶ X is an equivalence of spaces.

Proof. Since Z/p ⊗ L_{X} is isomorphic to (Z/p ⊗ L_{1}) × (Z/p ⊗ L_{2}) = (Z/p ⊗ L_{Y}) × (Z/p⊗L_{Z}) as a module over W_{X}, Proposition 3.6 implies that Y\X /Z is F_{p}-finite. By 6.3, W_{X} is isomorphic to W_{Y} × W_{Z} as a reflection group, so by 3.9 and 3.8, cd_{F_{p}}(X) = cd_{F_{p}}(Y) + cd_{F_{p}}(Z). Therefore (3.7), cd_{F_{p}}(Y\X /Z) = 0. As a F_{p}- complete space which has the mod p cohomology of a point, the double coset space is contractible. □

6.8 Remark. Exactly the same argument as in the proof of 6.7 shows that the double coset space Y\Y′/T_{Z} is contractible. This is clear in any case from 6.6.

6.9 Proposition. The square

BT_{Z} ⟶ BZ
↓  ↓
BY′ ⟶ BX

is a homotopy fibre square.

Proof. By 6.7 and 6.8 the natural map

Y\Y′/T_{Z} ⟶ Y\X /Z

is an equivalence; in fact, both spaces are contractible. It follows that the natural map Y′/T_{Z} ⟶ X /Z is also an equivalence; this, however, is the induced map on vertical fibres in the above square. □

Proof of 6.1. By 4.10 and 4.8(3), the fibration T_{Z}\Y′/T_{Z} ⟶ BT_{Z} is trivial. By 6.9, this fibration is the same up to homotopy as the fibration T_{Z}\X /Z ⟶ BT_{Z}, and so this second fibration is also trivial. According to 5.2, then, the monomorphism Z ⟶ X is normal. By 6.7, the sum map

(6.10)  π_{∗}(BY) × π_{∗}(BZ) ⟶ π_{∗}(BX)

is an isomorphism. Now consider the diagram

BY ; ↓ ; BZ ⟶ BX ≃ BN_{X} (Z) ⟶ BW_{X} (Z) ,

in which the lower line is a fibration sequence. Calculating with homotopy groups shows that the composite map BY ⟶ BW_{X} (Z) is an equivalence and that, under this equivalence, the induced map

π_{∗}(BY) × π_{∗}(BZ) ≅ π_{∗}(BX) ⟶ π_{∗}BW_{X} (Z) ≅ π_{∗} BY

amounts to projection on the second factor. Repeating the construction with Y and Z interchanged gives an analogous map

BX ≃ BN_{X} (Y) ⟶ BW_{X} (Y) ≃ BZ

which on homotopy groups has the effect of projecting onto the second summand (6.10) of π_{∗} BX. It is now easy to see that the product map

BX ⟶ BW_{X} (Z) × BW_{X} (Y) ≃ BY × BZ

is an equivalence which realizes the product decomposition L_{X} ≅ L_{1} × L_{2} = L_{Y} × L_{Z}. □

All that remains is to give a proof of 6.5. This depends on a lemma.

6.11 Lemma. For any connected p-compact group U the natural map

H_{0}(W_{U}, L_{U}) = H_{0}(W_{U}, H_{2}(BT_{U})) ⟶ H_{2}(BU)

is an epimorphism with finite kernel.

Proof. To prove surjectivity, it is enough by reducing mod p and dualizing to show that the restriction map H^{2}(BU) ⟶ H^{2}(BT_{U}) is injective. Pick a nonzero element k ∈ H^{2}(BU). The principal fibration sequence

(6.12)  BZ/p ⟶ BV ⟶ BU

classified by k expresses U as the quotient of a connected p-compact group V by a central subgroup equivalent to Z/p. Parallel to 6.12 is another sequence

BZ/p ⟶ BT_{V} ⟶ BT_{U}

in which the middle space BT_{V} is of course simply connected (see 6.14); this second fibration sequence is classified by the image of k in H^{2}(BT_{V}) under the restriction map. This image of k must be nonzero because the second sequence is not a product.

Finiteness of the kernel is equivalent to the statement that the natural map H_{0}(W_{U}, Q ⊗ H_{2}(BT_{U})) ⟶ Q ⊗ H_{2}(BU) is injective. This injectivity follows from the fact that the dual map H_{Q_{p}}^{2}(BU) ⟶ H^{0}(W_{U}, H_{Q_{p}}^{2}(BT_{U})) is an isomorphism [5, 9.7]. □

Proof of 6.5. Consider the map of exact sequences

(6.13)  \begin{aligned}0 & ⟶ & L_{Z} & ⟶ & H_{0}(W_{Y}, L_{X}) & ⟶ & H_{0}(W_{Y}, L_{Y}) & ⟶ & 0 \\ & & ↓ = & & ↓ & & ↓ u \\ 0 & ⟶ & L_{Z} & ⟶ & H_{2} BY′ & ⟶ & H_{2} BY & ⟶ & 0 \end{aligned} .

The upper sequence here comes from applying H_{∗}(W_{Y}, –) to the exact sequence

0 ⟶ L_{Z} ⟶ L_{X} ⟶ L_{Y} ⟶ 0

and noticing that the appropriate connecting homomorphism is zero because the group H_{1}(W_{X} , L_{C}) is torsion. The lower line in 6.13 is derived by looking at the Serre spectral sequence of 6.14 and using the fact that the differential terminating at E_{0,2}^{2} is trivial because π_{3} BY and hence H_{3} BY are torsion groups [5, 9.7]. The vertical maps in 6.14 are surjective with finite kernel (6.11). The Yoneda class of the lower exact sequence determines an element k in the group

Ext_{Z_{p}}^{1}(H_{2} BY, L_{Z}) ≅ H^{3}(BY, L_{Z})

and by a naturality argument k can be identified with the classifying map

BY ⟶ B^{2} T_{Z} ≃ K(L_{Z}, 3)

of the fibration sequence

(6.14)  BT_{Z} ⟶ BY′ ⟶^{B(q′_{Y})} BY .

Let j be the Yoneda class of the upper exact sequence in 6.13, so that j is the image of k under the map

u^{∗} : Ext_{Z_{p}}^{1}(H_{2} BY, L_{Z}) ⟶ Ext_{Z_{p}}^{1}(H_{0}(W_{Y}, L_{Y}), L_{Z}) .

Since L_{X} splits as the product L_{Z} ×L_{Y} of W_{Y}-modules, it is clear that H_{0}(W_{Y}, L_{X}) splits as a product L_{Z} × H_{0}(W_{Y}, L_{Y}) of abelian groups, and so the extension class j is zero. Since the abelian group L_{Z} is torsion free and the kernel of u is finite, the map u^{∗} is a monomorphism and hence the extension class k is also zero. This shows that the fibration 6.14 is trivial, and that the map B(q′_{Y}) has a right inverse up to homotopy. Now any two sections of B(q′_{Y}) differ up to homotopy by an element of H^{2}(BY, L_{Z}), whereas the map L_{Y} ⟶ L_{X} induced (3.5) by such a section differs from the one induced by the original map T_{Y} ⟶ T_{X} by an element of Hom_{W_{Y}} (L_{Y}, L_{Z}). Since L_{Z} is torsion free and has a trivial action of W_{Y}, Lemma 6.11 implies that the restriction map

H^{2}(BY, L_{Z}) ⟶ Hom_{W_{Y}} (L_{Y}, L_{Z})

is an isomorphism. Thus any section of Bq′_{Y} can be adjusted uniquely up to homotopy to have the desired restriction to BT_{Y}. □

§7. p-compact groups with trivial center

In this section we will prove 1.5.

7.1 Proposition. Suppose that X is a connected p-compact group and that

Q ⊗ L_{X} ≅ M_{1} ⊕ ⋯ M_{n}

is a direct sum decomposition of L_{X} as a module over W_{X}. Let W_{i} (i = 1, …, n) be the subgroup of W_{X} given by the elements which act trivially on M_{1} ⊕ ⋯ M̂_{i} ⊕ ⋯ ⊕ M_{n}. Then

(1) W_{i} commutes with W_{j} if i ≠ j, and

(2) the product map W_{1} × ⋯ × W_{n} ⟶ W is a group isomorphism.

Proof. This is essentially the same as the proof of 6.3. □

7.2 Calculating the center. We now recall from [6, §7] how the center of a p-compact group X is computed. The loop space N(T_{X}) has in its own right a “discrete approximation”; this is a group N̆(T_{X}) which lies in a short exact sequence

1 ⟶ T̆_{X} ⟶ N̆(T_{X}) ⟶ W_{X} ⟶ 1 .

The conjugation action of W_{X} on T̆_{X} = L̆_{X} derived from this short exact sequence is the same as the obvious one induced by the action of W_{X} on L_{X}. If X is connected the p-discrete approximation C̆ to the center C of X is a subgroup of T̆_{X} = L̆_{X} which can be described in the following way. Let rank(X) denote the rank of L_{X} over Z_{p}. For each reflection (2.4) s ∈ W_{X},

(1) the fixed point set F (s) of s is the fixed point set of the conjugation action of s on T̆_{X},

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

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

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

Then C̆ is the intersection ∩_{s}σ(s), where s runs through the reflections in W_{X}. Note that if the prime p is odd then this intersection is just the fixed point set of the action of W_{X} on T̆_{X}; if p = 2, the intersection has finite index in the fixed point set.

Proof of 1.5. By definition, the map α is a map of modules over W_{X}. Since α fits into a commutative diagram

L_{1} × ⋯ × L_{n} ⟶ M_{1} × ⋯ × M_{n}
α↓  ≅↓
L_{X}  ⟶  Q ⊗ L_{X}

in which the other maps are injective, it is clear that α is injective. Let D denote coker(α). If x is any element of Q⊗L_{X}, there exists an integer k such that p^{k}x ∈ L_{X}; this implies that each quotient group M_{i}/L_{i} is a p-primary torsion group, hence that their direct sum is torsion, and hence that D is torsion. Homological algebra now gives a short exact sequence

0 ⟶ Tor(Z/p^{∞}, D) ⟶ ⊕_{i}(Z/p^{∞} ⊗ L_{i}) ⟶^{Z/p^{∞}⊗α} Z/p^{∞} ⊗ L_{X} ⟶ 0

in which the kernel Tor(Z/p^{∞}, D) is isomorphic to D, so in order to show that D is zero it is enough to show that Z/p^{∞} ⊗ α is injective. Note that, by the definition of L_{i}, the cokernel of the inclusion L_{i} ⟶ L_{X} is torsion free; this implies that the map Z/p^{∞} ⊗ L_{i} ⟶ Z/p^{∞} ⊗ L_{X} is a monomorphism and allows Z/p^{∞} ⊗ L_{i} to be treated as a subgroup of Z/p^{∞} ⊗ L_{X}.

Let W_{i} (i = 1, …, n) be as in 7.1, so that each reflection in W_{X} lies in W_{i} for some i. Choose x ∈ ker(Z/p^{∞} ⊗ α), so that x = (x_{1}, …, x_{n}) with x_{i} ∈ Z/p^{∞} ⊗ L_{i} and Σx_{i} = 0. We will be done if we can prove that x is zero. Consider some component x_{k} of x, and pick a reflection s ∈ W_{X} with s ∈ W_{i}. If i ≠ k, then x_{k} ∈ Z/p^{∞} ⊗ L_{k}, which is a divisible subgroup of L̆_{X} = Z/p^{∞} ⊗ L_{X} fixed by s, and so x_{k} belongs to H(s) (7.2). If i = k, then x_{k} = − ∑_{j≠k} x_{j} belongs to ∑_{j≠k} Z/p^{∞} ⊗ L_{k}, which is a divisible subgroup of L̆_{X} fixed by s, and so in this case x_{k} ∈ H(s) too. The conclusion is that x_{k} ∈ ∩_{s}H(s), where the intersection is taken over all reflections in W_{X}. This intersection is contained in the p-discrete approximation C̆ to the center C of X, and is trivial because C is trivial. Thus each component x_{k} of x is zero, and so x itself is zero. □

References

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[3] C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.

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

[7] 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.

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

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[10] J. M. Moller and D. Notbohm, Centers and finite coverings of finite loop spaces, J. reine u. angew. Math (to appear).

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University of Notre Dame, Notre Dame, Indiana 46556

Purdue University, West Lafayette, Indiana 47907

Processed December 14, 1994