TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx XXXX, Pages 000–000 S 0002-9947(XX)0000-0

HOMOLOGY DECOMPOSITIONS FOR CLASSIFYING SPACES OF COMPACT LIE GROUPS

ALEXEI STROUNINE

Abstract. Let p be a prime number and G be a compact Lie group. A homology decomposition for the classifying space BG is a way of building BG up to mod p homology as a homotopy colimit of classifying spaces of subgroups of G. In this paper we develop techniques for constructing such homology decompositions. In [6] Jackowski, McClure and Oliver construct a homology decomposition of BG by classifying spaces of p-stubborn subgroups of G. Their decomposition is based on the existence of a finite-dimensional mod p acyclic G − CW -complex with restricted set of orbit types. We apply our techniques to give a parallel proof of the p-stubborn decomposition of BG which does not use this geometric construction.

1. Introduction

Let p be a fixed prime number and G be a compact Lie group. A homology decomposition for the classifying space BG is a mod p homology isomorphism

hocolim_{D}F ˜→_{p}BG

where D is a small category, F is functor from D to the category of spaces, and for each object d of D, F (d) has the homotopy type of BH for some subgroup H of G. Jackowski, McClure and Oliver [6] construct a homology decomposition of BG by classifying spaces of p-stubborn subgroups of G. Their decomposition is based on the existence of a finite-dimensional mod p acyclic G − CW -complex with restricted set of orbit types [6, 2.14]. In this paper we give a parallel proof of the p-stubborn decomposition of BG which does not use [6, 2.14] and has a potential to be extended to p-compact groups.

We will call a set C of closed subgroups of G a collection if it is closed under the process of taking conjugates in G. Let C be a collection. We define the C-orbit category O(C) whose objects are the G-sets G/H, H ∈ C, and whose morphisms are G-maps. We regard O(C) as a topological category, where the set of objects is discrete and where the morphism sets have compact-open topology. By J_{O(C)}, or simply J, we will denote the inclusion functor from O(C) to the category of G-spaces. Composing J with the Borel construction EG ×_{G} − gives a functor

EG ×_{G} J : O(C) → Spaces
\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

1991 Mathematics Subject Classification. Primary 55R35; Secondary 55R40.

.

whose value EG ×_{G} G/H at an object G/H has the homotopy type of BH. Let ∗ denote the one-point space with the trivial action of G. The natural maps EG ×_{G} G/H → EG ×_{G} ∗ = BG induce a map

hocolim_{O(C)}EG ×_{G} J → BG (1.1)

Here the operator hocolim is the homotopy colimit defined as a special case of the bar construction 2.2.

Definition 1.1. Call a collection C of subgroups of G ample if the map (1.1) induces an isomorphism on mod p homology.

Since the Borel construction commutes with taking homotopy colimits

EG ×_{G} hocolim_{O(C)}J ≅ hocolim_{O(C)}EG ×_{G} J

to prove that a collection C is ample, it suffice to show that in the fibration

hocolim_{O(C)}J → EG ×_{G} hocolim_{O(C)}J → BG

the fiber hocolim_{O(C)}J is F_{p}-acyclic (i.e. the map hocolim_{O(C)}J → ∗ induces an isomorphism on mod p homology). For a collection of subgroups C, we write EO(C) = hocolim_{O(C)}J. The action of G on the values of the functor J induces an obvious G-action on this space. In section 3 we consider sufficient conditions that imply the F_{p}-acyclicity of EO(C).

Suppose that the category O(C), for some collection C, has finite morphism sets. Then the standard Bousfield-Kan cohomology spectral sequence of a homotopy colimit [1, XII.5.8] associated to hocolim_{O(C)}EG ×_{G} J has the form:

(1.2) E_{2}^{i,j} = lim _{O(C)}^{i}H^{j}(EG ×_{G} J, F_{p}) ⇒ H^{i+j}(hocolim_{O(C)}EG ×_{G} J, F_{p})

Definition 1.2. An ample collection C of subgroups of G is said to be sharp if O(C) is a discrete category (in the sense that the morphism sets of O(C) have discrete topology) and the spectral sequence (1.2) collapses, that is E_{2}^{i,j} = 0 for i > 0 and E_{2}^{0,j} ≈ H^{j}(BG, F_{p}).

The E_{2}-term of the above spectral sequence can be identified with the ordinary equivariant cohomology groups of the G-space EO(C) with certain coefficient systems. In fact a collection C is sharp if and only if EO(C) has equivariant cohomology of a point. In section 4 we obtain a refinement of the equivariant transfer map constructed by Oliver [9] which is used to show that, for suitable collections C, the E_{2}-term of the spectral sequence described above has appropriate vanishing properties.

Recall that a compact Lie group P is called p-toral if its identity component P_{0} is a torus and P/P_{0} is a finite p-group. A p-toral subgroup P of G is called p-stubborn if N_{G}(P)/P is finite and has no nontrivial normal p-subgroups.

The main result of this work is the following

Theorem 1.3. Suppose that G is a compact Lie group that contains an element of order p. Then the collection of nontrivial p-toral subgroups of G is ample and the collection of nontrivial p-stubborn subgroups of G is sharp.

Throughout the paper p is a fixed prime number. The symbol F_{p} denotes the field with p elements. A map f : X → Y between two spaces is said to be an F_{p}- equivalence or mod p equivalence and is denoted X ˜→_{p}Y if it induces an isomorphism f_{∗} : H_{∗}(X, F_{p}) → H_{∗}(Y, F_{p}). A space X is said to be F_{p}-acyclic or mod p acyclic if the projection into a one point space X → ∗ is an F_{p}-equivalence.

2. Bar constructions, homotopy colimits and G-spaces

Throughout this paper a topological category is a category D with topologized morphism sets such that composition is continuous and for each object d of D the inclusion id_{d} → Mor(d, d) is a closed cofibration.

Definition 2.1. Let D be a small topological category, Φ : D^{op} → T op and Ψ : D → T op be continuous functors. The bar construction B(Φ, D, Ψ) is the topological realization |B_{•}| of the simplicial space B_{•}, which in dimension k is the following disjoint union of spaces indexed by ordered sets of k + 1 objects of D:

B_{k} = ∐_{(d_{0},... ,d_{k})} Ψ(d_{k}) × Mor_{D}(d_{k,}d_{k−1}) × … × Mor_{D}(d_{1,}d_{0}) × Φ(d_{0})

The face operators δ_{i} are induced by composite maps

Mor_{D}(d_{i+1,}d_{i}) × Mor_{D}(d_{i,}d_{i−1}) → Mor_{D}(d_{i+1,}d_{i−1})

when 0 < i < k, and by evaluation maps

Mor_{D}(d_{1,}d_{0}) × Φ(d_{0}) → Φ(d_{1}),
Ψ(d_{k}) × Mor_{D}(d_{k,}d_{k−1}) → Ψ(d_{k−1})

when i is respectively 0 and k.

The degeneracies are given by appropriate inclusions of identities.

Definition 2.2. Let ∗ denote the constant one point valued functor on D, considered as necessary to be either covariant or contravariant. With the above notation the homotopy colimits of functors Φ and Ψ are

hocolimΦ = B(Φ, D, ∗); hocolimΨ = B(∗, D, Ψ)

Recall that the n-skeleton B_{•}^{n} of a simplicial space is the smallest subobject that contains B_{k} for all k ≤ n. The mod p homology spectral sequence associated to the filtration

|B_{•}^{0}| ⊂ |B_{•}^{1}| ⊂ … ⊂ |B_{•}^{n}| ⊂ …

of |B_{•}| has the E^{1}-term E_{i,j}^{1} = H_{j+i}(|B_{•}^{i}| , |B_{•}^{i−1}|; F_{p}) ≈ H_{j}(B_{i}/sB_{i}; F_{p}) and is first quadrant strongly convergent. Here sB_{i} denotes the subspace of B_{i} which is the union of images of B_{i−1} under the degeneracy operators. A simple application of this spectral sequence is the following:

Lemma 2.3. Suppose that D is a small topological category, Φ, Φ′ : D^{op} → T op and Ψ, Ψ′ : D → T op are continuous functors. If u : Φ → Φ′ and v : Ψ → Ψ are natural transformations such that for each object d of D u(d) and v(d) are mod p-equivalences, then the map B(u, D, v) : B(Φ, D, Ψ) → B(Φ′, D, Ψ′) is a mod p equivalence.

Proof. Let B_{•} and B′_{•} be the simplicial spaces used to define the bar constructions B(Φ, D, Ψ) and B(Φ′, D, Ψ′) respectively. The transformations u and v induce a map (u, v) : B_{•} → B′_{•} of simplicial spaces such that the corresponding maps B_{i} → B′_{i} and sB_{i} → sB′_{i} are mod p homology isomorphisms in each dimension. Since the inclusions sB_{i} → B_{i} and sB′_{i} → B′_{i} are cofibrations, (u, v) induces an isomorphism of the spectral sequences described above.

Suppose that C is a collection of subgroups of G and X is a G-space. Define X_{C}, the C resolution of X, to be the bar construction B(Map_{G}(J, X), O(C), J). There is an obvious G-map φ_{C}(X) : X_{C} → X induced by the evaluation maps in each dimension of the simplicial space

B_{k} = ∐_{(G/H_{0},... ,G/H_{k})} G/H_{k} × Map_{G}(G/H_{k,}G/H_{k−1}) × … × Map_{G}(G/H_{0}, X) → X

Essentially the space X_{C} contains all the equivariant homotopy information of X which is detectable by the subgroups of G in C. It is clear form the construction that the isotropy subgroups of X_{C} are those subgroups H in C for which the fixed-point space X^{H} = Map_{G}(G/H, X) is nonempty.

A map f : X → Y of G-spaces is said to be a weak G-equivalence (respectively a G-mod-p-equivalence) if for every subgroup H of G the induced map on the fixed point sets f^{H} : X^{H} → Y^{H} is a weak equivalence (respectively mod p equivalence). We will be interested in finding out to what extent the weak G-homotopy (homology) type of a G space X is determined by the fixed-point sets X^{H} for some restricted type of subgroups H of G. The following facts are simple consequences of general properties of the bar construction:

Theorem 2.4. Let G be a compact Lie group, C a collection of subgroups of G and X a G-space. Then

(a) for every H in C, the map on the fixed-point sets (X_{C})^{H} → X^{H} is a homotopy equivalence;

(b) if H is in C then φ_{C}(G/H) : (G/H)_{C} → G/H is a G-equivalence;

(c) if X is a G-CW -complex such that for each orbit type G/H of X, φ_{C}(G/H) is a G-equivalence then so is φ_{C}(X) : X_{C} → X.

Proof. Note that for a topological category D, functors Φ : D^{op} → T op and Ψ : D → T op, and an object d of D, the evaluation maps induce natural maps

(2.1)  B(Φ, D, Mor_{D}(d, −)) ˜→ Φ(d)
(2.2)  B(Mor_{D}(−, d), D, Ψ) ˜→ Ψ(d)

which are strong deformation retractions [4, 3.1(5)].

(a) For H in C

(X_{C})^{H} = B(Map_{G}(J, X), O(C), J)^{H} =
= B(Map_{G}(J, X), O(C), Mor_{O(C)}(G/H, −)) ˜→Map_{G}(G/H, X) ≈ X^{H}

(b) For H in C and Q any subgroup of G

(G/H_{C})^{Q} = B(Map_{G}(J, G/H), O(C), J)^{Q} =
= B(Mor_{O(C)}(−, G/H), O(C), Map_{G}(G/Q, J)) ˜→
˜→ Map_{G}(G/Q, G/H) ≈ (G/H)^{Q}

(c) A G-CW -complex X is constructed from a collection of “G-cells” {G/H×D^{n}} by taking disjoint unions, pushouts in which one of the maps is a cofibration, and sequential colimits. Clearly for a subgroup Q of G the fixed point functor Map_{G}(G/Q, −) preserves this three operations and the bar construction commutes with colimits. Hence φ_{C}(X) is a weak G-equivalence if it is one for each G-cell of X.

Corollary 2.5. If X is a G-space and C contains all isotropy subgroups of X then φ_{C}(X) is a weak G-equivalence.

Corollary 2.6. If f : X → Y is a G-map such that f^{H} : X^{H} → Y^{H} is a mod p equivalence whenever H is an isotropy subgroup of X or Y, then f is a G-mod-p-equivalence.

Proof. Let C be a collection of isotropy subgroups of X and Y. By 2.4 φ_{C}(X) : X_{C} → X and φ_{C}(Y) : Y_{C} → Y are weak G-equivalences, and by 2.3 f_{C} : X_{C} → Y_{C} is a G-mod-p-equivalence.

Remark 2.7. Note that the G-space EO(C) associated to a collection C is the C resolution of a point ∗_{C}, so its isotropy subgroups are those in C and for any H in C the fixed point set (EO(C))^{H} is contractible. We will study the space EO(C) in more detail in the next section.

We conclude this section by a lemma that considers C resolutions of transitive G-spaces.

Lemma 2.8. Suppose that H < G is a pair of compact Lie groups and C is a collection of subgroups of G. Let C|_{H} be the collection of those subgroups of H which are in C and ξ : G ×_{H} EO(C|_{H}) → G/H be the H-bundle induced by projection EO(C|_{H}) → ∗. Then there is a weak G-equivalence f : G ×_{H} EO(C|_{H}) → G/H_{C} such that ξ = φ_{C} ∘ f.

Proof. The map f is defined on bar constructions in the following way: First we identify the constant functor ∗ : O(C|_{H}) → T op with Map_{H}(J, ∗) and note that the functor G ×_{H} − commutes with the bar construction, then we obtain a map of bar constructions induced by the functor G ×_{H} − : O(C|_{H}) → O(C) :

G ×_{H} B(∗, O(C|_{H}), J) = B(Map_{H}(J, ∗), O(C|_{H}), G ×_{H} J) →
→ B(Map_{G}(J, G ×_{H} ∗), O(C), J)

Note that the G-spaces G ×_{H} EO(C|_{H}) and G/H_{C} have the same set of isotropy subgroups, namely those subgroups Q in C which are subconjugates of H. For any such Q consider the maps on the fixed point spaces

ξ^{Q} : (G ×_{H} EO(C|_{H}))^{Q} →^{f^{Q}} (G/H_{C})^{Q} →^{φ_{C}^{Q}} (G/H)^{Q}.

The map φ_{C}^{Q} is a weak equivalence by 2.4. The composite ξ^{Q} is a fiber bundle with the fiber over a point gH in (G/H)^{Q} being equal to (EO(C|_{H}))^{g^{−1}Qg} which is contractible since g^{−1}Qg ∈ C|_{H}. Thus f is a weak equivalence on the fixed point sets for isotropy subgroups of G ×_{H} EO(C|_{H}) and G/H_{C}, and hence it is a weak G-equivalence by 2.6.

3. The G-space EO(C)

As mentioned before a collection of subgroups C is ample if the space EO(C) has mod p homology of a point. In this section we state some conditions that imply the mod p-acyclicity of this space. We first note that this is a necessary condition for ampleness of C if π_{0}(G) is a p-group:

Lemma 3.1. Let G be a compact Lie group such that π_{0}(G) is a finite p-group. Then C is an ample collection of subgroups of G if and only if EO(C) ˜→_{p}∗.

Proof. If EO(C) ˜→_{p}∗ the conclusion is obvious. Suppose now that EG×_{G}EO(C) ˜→_{p}BG.

Let π = π_{1}(G) ≈ G/G_{0}. Since π is a p-group it acts nilpotently on F_{p}[π] and the homology isomorphism EG×_{G}EO(C) → EG×_{G}∗ with coefficients in F_{p} implies the isomorphism with twisted coefficients H_{∗}(EG ×_{G} EO(C), F_{p}[G/G_{0}]) → H_{∗}(EG ×_{G} ∗, F_{p}[G/G_{0}]). By Shapiro’s lemma H_{∗}(EG×_{G} X, F_{p}[G/G_{0}]) ≈ H_{∗}(EG×_{G_{0}} X, F_{p}), thus we have a fibration EO(C) → EG ×_{G_{0}} EO(C) → BG_{0} in which the base space is simply-connected and the projection map is a homology isomorphism. Serre long exact sequence on homology of a fibration implies the acyclicity of the fiber.

Definition 3.2. Call a subgroup H in C a core subgroup of the collection C if H is normal in G and HP is in C for every P in C.

The following is a theorem of Quillen in a slightly more general form:

Lemma 3.3. If a collection of subgroups C has a core subgroup then EO(C) is contractible.

Proof. Let P be a core subgroup of C. Consider the following subcollection of C

C_{P} = {H ∈ C : P ⊂ H}.

Note that P acts trivially on every object of the associated orbit category O(C_{P}), therefore we can identify the functor J_{O(C_{P})} with Mor_{O(C_{P})}(G/P, −). By (2.1) the space EO(C_{P}) = B(∗, O(C_{P}), Mor_{O(C_{P})}(G/P, −)) is contractible.

The map EO(C_{P}) → EO(C) induced by inclusion of categories is a homotopy equivalence: The inverse is induced by the functor (−)_{P} : O(C) → O(C_{P}) that sends every object of G/H of O(C) its P -orbit space (G/H)_{P}. Since P is a normal subgroup of G, (G/H)_{P} is a transitive G space of type G/HP. The projection G/H → (G/H)_{P} defines a natural transformation between the identity functor on

O(C) and the composite O(C) →^{(−)_{P}} O(C_{P}) → O(C). By [4, 3.1(7)] this transformation gives a homotopy between the maps on EO(C) induced by these functors.

We denote by O_{p}(G) (respectively R_{p}(G)) the orbit category associated to the collection of all nontrivial p-toral subgroups (respectively all nontrivial p-stubborn subgroups).

Remark 3.4. If G has a nontrivial normal p-subgroup P, then P is a core subgroup of the collection of nontrivial p-toral subgroups, hence EO_{p}(G) ≃ ∗. This is the original result of Quillen.

The following theorem will allow us to reduce the proof of the homology decomposition theorem to the case of compact Lie groups with simpler structure (e.g. groups of smaller dimension or groups with π_{0} a p-group).

Theorem 3.5. Let G be a compact Lie group and p be a prime number. Suppose that there exists a normal subgroup K of G such that for any p-toral subgroup Q of G (including the trivial subgroup) EO_{p}(QK) ˜→_{p}∗. Then EO_{p}(G) ˜→_{p}∗.

Proof. Denote the collection of nontrivial p-toral subgroups of G by C. Let D = C ∪ {KQ : Q ∈ C} ∪ {K}. Clearly D is a collection of subgroups. Consider the following diagram of functors

[figure: O(D); F_{h^{∗}}J; J; O_{p}(G); J; G-spaces; F]

Here F is the inclusion functor and F_{h^{∗}}J is the Kan extension of J along F as defined in [4, 5]. Then

EO_{p}(G) ≅ hocolim_{O(D)}F_{h^{∗}}J ˜→_{p}hocolim_{O(D)}J ≅ ∗

The first map is an equivalence by the homotopy pushdown theorem [4, 5.5], the third one is the equivalence of lemma 3.3 (Note that K is a core subgroup of D). The map in the middle is constructed in the following way:

For each object G/H of D, by definition the Kan extension is the bar construction F_{h^{∗}}J(G/H) = B(Mor_{D}(F, G/H), O_{p}(G), J), which is in fact the C resolution of G/H. There is a natural transformation F_{h^{∗}}J → J defined on each object G/H of D to be the map φ_{C}(G/H) : G/H_{C} → G/H. We will show that this map is mod p equivalence for each object G/H of D which will imply the mod p equivalence of the map on the homotopy colimits by 2.3.

Note that the projection of the fiber bundle G ×_{H} EO_{p}(H) → G/H is a mod p equivalence since its fiber EO_{p}(H) is mod p acyclic by assumption when H is in the form KQ for a p-toral Q or it is contractible when H is nontrivial p-toral. By 2.8 this projection factors through a weak G-equivalence:

G ×_{H} EO_{p}(H) ˜→G/H_{C} →^{φ_{C}} G/H

Therefore φ_{C} is a mod p equivalence and this finishes the proof.

Remark 3.6. Let G be an infinite compact Lie group, we let K = G_{0}, the identity component of G. For any p-toral Q, π_{0}(QG_{0}) is a p-group, thus Theorem 3.5 combined with lemma 3.1 allows us to reduce the proof of the homology decomposition theorem to the case of a compact Lie group π_{0} of which is a p-group.

4. The transfer on equivariant cohomology

Let G be a compact Lie group. We will denote by O(G) the orbit category O(C) when C is the collection of all subgroups of G.

Definition 4.1. A contravariant coefficient system M for G is a contravariant functor from O(G) to Ab, the category of abelian groups, such that if f and g are G-homotopic maps in O(G), M(f) = M(g).

Let X be a G-space and M a coefficient system. By H_{G}^{∗}(X, M) we will denote the ordinary equivariant cohomology of X with coefficients in M as defined in [5] and [12].

Suppose F is a functor from a small discrete (in the sense that the set of morphisms has the discrete topology) category D to the category of G-spaces such that F (d) is a transitive G-space for each object d of D . There is an equivariant cohomology spectral sequence of hocolim_{D}F associated to the skeleton filtration of hocolim_{D}F:

(4.1) E_{2}^{i,j} = lim _{D}^{i}H_{G}^{j}(F, M) ⇒ H_{G}^{i+j}(hocolim_{D}F, M)

Note that by the dimension axiom of the ordinary equivariant cohomology H_{G}^{j}(F, M) = 0 when j > 0 and H_{G}^{0}(F, M) = M(F). Thus the spectral sequence (4.1) collapses and

(4.2) H_{G}^{i}(hocolim_{D}F, M) ≈ lim _{D}^{i}M(F).

The mod p cohomology Bousfield-Kan spectral sequence for hocolim_{D}EG ×_{G} F has the form

E_{2}^{i,j} = lim _{D}^{i}H^{j}(EG ×_{G} F, F_{p}) ⇒ H^{i+j}(hocolim_{D}EG ×_{G} F, F_{p})

Clearly its E_{2} term can be identified with (4.2): M^{j} = H^{j}(EG ×_{G} −, F_{p}) defines a contravariant coefficient system and E_{2}^{i,j} ≈ H_{G}^{i}(hocolim_{D}F, M^{j}). The next proposition is obvious.

Proposition 4.2. Let G be a compact Lie group and M^{j} = H^{j}(EG ×_{G} −, F_{p}) a contravariant coefficient system. A collection C of subgroups of G, for which O(C) is discrete, is sharp if and only if EO(C) has M^{j} equivariant cohomology of a point:

H_{G}^{i}(EO(C), M^{j}) ˜←H_{G}^{i}(∗, M^{j}).

We now describe sufficient conditions that imply the equivariant acyclicity of a G-space X. First we make a few definitions. Let P be a maximal p-toral subgroup of G, M a coefficient system for G. Define M|_{P}, the restriction of M to P, to be the composite O(P) ⟶^{G×_{P} −} O(G) →^{M} Ab. Following the terminology of [7] we will call a subgroup Q of G sub-p-toral if it is contained in a p-toral subgroup of G. We call a sub-p-toral subgroup Q of G p-relevant if it contains an element of order p (equivalently, it is not a finite group of order prime to p). For a G-space X we let X_{s} = {x ∈ X : G_{x} ∩ P is p-relevant˙}, a P -invariant subspace of X. Recall that a G-space X is said to have finitely many orbit types if there are finitely many conjugacy classes of isotropy subgroups of X. The following is a result of Slomińska:

Proposition 4.3. ([10, 1.1]) Let G be a Lie group and P be a maximal p-toral subgroup of G. Suppose that X is a G-CW complex with finitely many orbit types. With the notation above, if X_{s} is P -mod-p-equivalent to a point, then, for j > 0

H_{G}^{∗}(X, M^{j}) ˜←H_{G}^{∗}(∗, M^{j})

Proof. The main idea behind the proof, which can be found in [10], is that the functors M^{j} are restrictions of Mackey functors defined on stable orbit category of G as described in [8], for which a natural equivariant cohomology transfer map is constructed. The transfer exhibits the map H_{G}^{∗}(X, M^{j}) ˜←H_{G}^{∗}(∗, M^{j}) as a retract of H_{P}^{∗} (X, M^{j}|_{P}) ˜←H_{P}^{∗} (∗, M^{j}|_{P}). Since the coefficients M^{j}|_{P} (j > 0) vanish on the P -orbits of X outside X_{s}, the inclusion map X_{s} → X induces an isomorphism H_{P}^{∗} (X, M^{j}|_{P}) ˜→H_{P}^{∗} (X_{s}, M^{j}|_{P}). Finally H_{P}^{∗} (X_{s}, M^{j}|_{P}) is isomorphic to H_{P}^{∗} (∗, M^{j}|_{P}) by [6, A.13].

Remark 4.4. The problem one faces trying to get a result analogous to 4.3 when j = 0 is that the functor M^{0}|_{P} does not vanish on outside of X_{s} and the map H_{P}^{∗} (X, M^{j}|_{P}) → H_{P}^{∗} (X_{s}, M^{j}|_{P}) induced by inclusion is not necessarily an isomorphism. Another result of Slomińska [10, 1.2] covers the case j = 0, however it does depend on Jackowski-McClure-Oliver construction. We will give an independent proof, which is based on ideas we are planning to exploit later.

Proposition 4.5. Let G be a compact Lie group, P be a maximal p-toral subgroup of G, and X be a G-CW -complex with finitely many orbit types such that each point in X is fixed by an element in G of order p. If X_{s}/P is mod-p-acyclic then so is the orbit space X/G.

Note that the coefficient system M^{0} is a constant system,

M^{0}(G/H) ≈ H^{0}(BH, F_{p}) = F_{p}

and the equivariant cohomology H_{G}^{∗}(X, F_{p}) is the ordinary cohomology of the orbit space H^{∗}(X/G, F_{p}). Our proof of 4.5 is analogous to the proof of 4.3 and is based on the construction of the transfer map given by Oliver [9]. For the rest of this section H^{∗} denotes the Čech cohomology.

Theorem 4.6. ([9]) Let G be a compact Lie group, P a subgroup of G and R any coefficient group then for any paracompact and perfectly normal G-space X, there is a transfer homomorphism

trf : H^{∗}(X/P; R) → H^{∗}(X/G; R), (4.3)

natural in X, and such that the composite

trf ∘ π^{∗} : H^{∗}(X/G; R) → H^{∗}(X/G; R),

where π denotes the projection X/P → X/G, is multiplication by the Euler characteristic χ(G/P).

Note that any G − CW -complex is paracompact and perfectly normal. We will show that

Lemma 4.7. If P is a maximal p-toral subgroup of G, X is a G-space such that each point in X is fixed by an element in G of order p, and R = F_{p}, then the

transfer map (4.3) factors as H^{∗}(X/P; F_{p}) →^{i^{∗}} H^{∗}(X_{s}/P; F_{p}) → H^{∗}(X/G; F_{p}), where i^{∗} is induced by inclusion X_{s} → X.

First we recall basic properties of the Dold fixed point index [2] and the key constructions form [9]:

The fixed point index assigns an integer Ind(f) to any map f : U → M, where M is an ENR (Euclidean neighborhood retract), U an open subset of M, and F ix(f) = {u ∈ U : f(u) = u} is compact. We will use the following properties of the index:

Lemma 4.8. ([2, 1.3-1.8]) Let f : U → M be as above.

(a) If V ⊆ U is open and F ix(f) ⊆ V, then Ind(f|_{V}) = Ind(f).

(b) If U = ∐_{i=0}^{n} U_{i}, a finite disjoint union of open sets, then Ind(f) = ∑_{i=0}^{n} Ind(f|_{U_{i}}).

(c) If φ : U × I → M is a homotopy such that F ix(φ) = {u ∈ U : φ(u, t) = u for some t ∈ I} is compact, then Ind(φ_{0}) = Ind(φ_{1}) (φ_{t} = φ(−, t), t ∈ I).

(d) If M and M′ are ENR’s, U ⊆ M and U′ ⊆ M′ are open sets, and f : U → M′, g : U′ → M are maps, then the two composites gf : f^{−1}(U′) → M and fg : g^{−1}(U) → M′ have homeomorphic fixed point sets and, if these sets are compact, Ind(gf) = Ind(fg).

As a simple consequence of 4.8 we have the following:

Lemma 4.9. Suppose that M is an ENR, V an open subset of M, f : V → M is a compactly fixed map and q : M → M is an automorphism of M. Then the map

qfq^{−1} : q(V) →^{q^{−1}} V →^{f} M →^{q} M has the same index as f.

Proof. Note that qf : V → M and q^{−1}|_{q(V)} : q(V) → M are maps whose two composites have the same index by 4.8 (d).

Definition 4.10. Let X be a space and Y be a G-space. A homotopy

φ : X × I → Y/G

is called stratum decreasing if for any x in X and t ≥ s in I the isotropy subgroups of the orbit φ(x, t) in Y are subconjugates to the ones of φ(x, s).

Lemma 4.11. ([9, Theorem 2]) Let G be a compact Lie group acting on spaces X and Y, where X is paracompact and perfectly normal and Y is completely regular, then, for any G-map f : X → Y, any stratum decreasing homotopy of f/G can be lifted to a G-homotopy of f.

Lemma 4.12. ([9, Theorem 3]) For any pair P ⊂ G of compact Lie groups and any paracompact perfectly normal G-space X there is a stratum decreasing homotopy φ : X/P × I → X/P of the identity on X/P, commuting with the projection π : X/P → X/G; and such that the restriction of π :

π̂ : F ix(φ_{1}) → X/G (φ_{1} = φ(−, 1))

has finite fibers.

The “stratum decreasing” property of φ implies that it restricts to a homotopy φ_{s} : X_{s}/P × I → X_{s}/P with analogous properties. We will denote by π̂_{s} the restriction of π̂ to F ix_{s}(φ_{1}) = F ix(φ_{1}) ∩ X_{s}/P.

Let H (respectively H_{s}) denote the sheaf over X/G induced by the presheaf U → H^{0}(π̂^{−1}(U); F_{p}) (respectively U → H^{0}(π̂_{s}^{−1}(U); F_{p})) (U open in X/G). Since π̂ and π̂_{s} are closed maps, H (respectively H_{s}) has stalks H_{x} = H^{0}(π̂^{−1}(x); F_{p}) (respectively (H_{s})_{x} = H^{0}(π̂_{s}^{−1}(x); F_{p})) (x ∈ X/G). Since each of the sets π̂^{−1}(x), π̂_{s}^{−1}(x) is zero dimensional, H^{∗}(F ix(φ_{1}), F_{p}) ≅ H^{∗}(X/G, H) and H^{∗}(F ix_{s}(φ_{1}), F_{p}) ≅ H^{∗}(X/G, H_{s}).

For any x ∈ X/G, choose x̃ in the orbit x. Let K denote the isotropy subgroup Gx̃ of x̃. The fiber π^{−1}(x) can be identified with the orbit space (G/P)_{K} of the Kaction on G/P through the homeomorphism (G/P)_{K} → π^{−1}(x) ⊆ X/P that sends the equivalence class [gP] in (G/P)_{K} to [g^{−1}x̃] in X/P. Under this identification φ restricts to a homotopy φ|_{(G/P)_{K}×I} : (G/P)_{K} × I → (G/P)_{K}, which is stratum decreasing with respect to the action of K by [9, Lemma 5 (iii)]. By 4.11 it can be lifted to a K-equivariant homotopy ψ of the identity map on G/P. Note that the fixed point set of the resulting map ψ_{1} = ψ(−, 1) projects into π̂^{−1}(x) ⊂ (G/P)_{K}. For any point y ∈ π̂^{−1}(x) ⊂ (G/P)_{K} the corresponding K-orbit ȳ ⊂ G/P has a K-equivariant tubular neighborhood U that does not contain any fixed point sets of ψ_{1} other than those in ȳ. Let I(y) denote the Dold fixed point index of ψ_{1}|_{U} : U → G/P. This index is independent of the choice of x̃, ψ and U. Oliver defines a map J(φ) from the sheaf H to the constant sheaf X/G × F_{p}, which is a homomorphism

J(φ)_{x} : H^{0}(π̂^{−1}(x); F_{p}) ≅ ⨁_{y∈π̂^{−1}(x)} H^{0}(y; F_{p}) \underrightarrow{∑ I(y)} F_{p}

on stalks. The map J(φ) is continuous [9, 14], and hence a sheaf homomorphism. Our proof of 4.7 is based on the following fact:

Lemma 4.13. With the notation above, I(y) is divisible by p for every y ∈ π̂^{−1}(x), which is not in π̂_{s}^{−1}(x).

Proof. Let y ∈ π̂^{−1}(x) − π̂_{s}^{−1}(x) ⊆ (G/P)_{K}. It follows from the definition of X_{s} that the K-isotropy subgroups of any point of the corresponding orbit ȳ in G/P are finite of order prime to p. Since ψ_{1} is a K-equivariant map it either fixes the entire K-orbit ȳ or has no fixed points in ȳ. In the latter case I(y) = 0 and the conclusion is obvious. For the rest of the proof we assume that ȳ ⊆ F ix(ψ_{1}). Let Q be a cyclic subgroup of K of order p, and U be a K equivariant tubular neighborhood of ȳ in G/P that does not contain fixed point sets of ψ_{1} others than those in ȳ. Clearly Q acts freely on U. Consider projections U →^{α} U/Q →^{β} U/K. Since any subspace of a paracompact and perfectly normal space is paracompact and perfectly normal ([9, Lemma 1 (2)]), by 4.12 there exist a stratum decreasing homotopy of the identity ω : U/Q × I → U/Q which commutes with the projection U/Q → U/K, and such that the resulting map ω_{1} has finite fixed-point set on each fiber of U/Q → U/K. Composing ω with ψ_{1}/Q we get a homotopy

U/Q × I →^{ω} U/Q →^{ψ_{1}/Q} (G/P)_{Q}

which, by 4.11, can be lifted to a Q-equivariant homotopy ω̄ : U × I → G/P of ψ_{1}|_{U}. Note that the fixed point set of ω̄ projects to the fixed point set of φ_{1}|_{U/K} : U/K → (G/P)_{K}, which consists of one point y, therefore F ix(ω̄) is compact. By 4.8 (c) Ind(ψ_{1}|_{U}) = Ind(ω̄_{1}). Now the p-fold covering α projects F ix(ω̄_{1}) into a finite set F ix(ω̄_{1}/Q) = F ix(ω_{1}) ∩ β^{−1}(y). For all z ∈ F ix(ω̄_{1}/Q) chose disjoint neighborhoods V_{z} ∋ z in U/Q which are evenly covered by α. Note that {α^{−1}V_{z}}_{z∈F ix(ω̄_{1}/Q)} is an open covering of F ix(ω̄_{1}) by disjoint sets, by 4.8 (b) Ind(ω̄_{1}) = ∑_{z∈F ix(ω̄_{1}/Q)} Ind(ω̄_{1}|_{α^{−1}V_{z}}). Each of the open sets α^{−1}V_{z} is a disjoint

union of p open sets each of which is homeomorphic to V_{z}. Let V denote one of them,

then α^{−1}V_{z} = ∐_{q∈Q} qV and Ind(ω̄_{1}|_{α^{−1}V_{z}}) = ∑_{q∈Q} Ind(ω̄_{1}|_{qV} ˙). By 4.9 Ind(ω̄_{1}|_{qV}) =

Ind(ω̄_{1}|_{V}) for each q ∈ Q (note that ω̄_{1} is a Q-equivariant map and ω̄_{1}|_{qV} = (qω̄_{1}q^{−1}|_{qV}).

Proof of 4.7. Note that the sheaf map i^{∗} : H → H_{s} induced by inclusions π̂_{s}^{−1}(U) → π̂^{−1}(U) (U open in X/G) is a quotient map, and 4.13 implies that J(φ)|_{Ker(i^{∗})} = 0.

Therefore J(φ) can be factored as H →^{i^{∗}} H_{s} →^{J(φ)_{s}} X/G × F_{p}. Consider the following commutative diagram

[figure: H^{∗}(X/H, F_{p}) → H^{∗}(F ix(φ_{1}), F_{p}) ≅ H^{∗}(X/G, H) →^{J} H^{∗}(X/G, F_{p}); J_{s}; H^{∗}(X_{s}/H, F_{p}) → H^{∗}(F ix_{s}(φ_{1}), F_{p}) ≅ H^{∗}(X/G, H_{s})]

Here J and J_{s} are the maps induced by J(φ) and J(φ)_{s} respectively and the unlabeled maps are those induced by inclusions. The bottom row composite is Oliver’s transfer trf : H^{∗}(X/P; F_{p}) → H^{∗}(X/G; F_{p}) which factors through the top row.

5. Relation between two decompositions

In this section we establish a relation between the p-toral and p-stubborn decompositions. The following properties of p-toral subgroups will be used:

Lemma 5.1. ([7, 3.1, 3.2]) Let G be a compact Lie group and Q be a sub-p-toral subgroup of G.

(a) For any p-toral subgroup P ⊃_{≠} Q in G, N_{P} (Q)/Q is p-toral and nontrivial.

(b) If G acts smoothly on a compact manifold M, then χ(M^{Q}) ≡ χ(M) mod p.

(c) If N_{G}(Q)/Q is finite and has no nontrivial normal p-subgroup, then Q is p-toral.

Lemma 5.2. ([6, 1.6]) The category R_{p}(G) is equivalent to a finite category.

Lemma 5.3. Let G be a compact Lie group which is not a finite group of order prime to p. Suppose that for any p-relevant subgroup Q of G such that N_{G}(Q)/Q is infinite, EO_{p}(N_{G}(Q)/Q) ˜→_{p}∗. Then for any p-relevant subgroup Q the fixed point

space ER_{p}(G)^{Q} is mod p acyclic.

Proof. To simplify the notation denote ER_{p}(G) by X. Note that X has finitely many orbit types by 5.2. Let S denote the set of p-relevant subgroups Q such that X^{Q} is not mod p acyclic. We will prove that S is empty. Assuming the contrary, we order the elements of S by inclusion and use Zorn’s lemma to prove that S has a maximal element:

Let Q_{1} ⊂ Q_{2} ⊂ … ⊂ Q_{i} ⊂ … be a chain in S and Q be the closure of the union of Q_{i}’s. Note that a subgroup H of G is sub-p-toral if and only if (G/P)^{H} ≠ ∅, where P is a maximal p-toral subgroup of G. For i sufficiently large, (G/P)^{Q_{i}} = (G/P)^{Q} and X^{Q_{i}} = X^{Q} [11, Proposition IV.3.4], therefore Q is in S.

Now let Q be a maximal element of S. We prove that X^{Q} is mod p acyclic and get a contradiction.

If N_{G}(Q)/Q is finite and has no nontrivial normal p subgroups then Q is ptoral and therefore X^{Q} ≅ ∗. Otherwise consider X^{Q} and (G/P)^{Q} as a W = N_{G}(Q)/Q spaces. The isotropy subgroups of the action of W on X^{Q} are of the form (H ∩ N_{G}(Q))/Q = N_{P} (Q)/Q for H a p-stubborn subgroup containing Q. By 5.1 (a) these subgroups of W are p-toral and nontrivial. Let H̄ be a nontrivial ptoral subgroup of W and H ⊂ N_{G}(Q) its preimage. Note that H properly contains Q and is a sub-p-toral subgroup of G since (G/P)^{H}is nonempty:

χ((G/P)^{H}) = χ(((G/P)^{Q})H̄) ≡ χ((G/P)^{Q}) ≡ χ(G/P) ≢ 0 (mod p)

By the maximality of Q the fixed point set (X^{Q})H̄ = X^{H} is mod p acyclic. Therefore X^{Q} is mod p equivalent to EO_{p}(W) which is either mod p acyclic by the hypothesis if W is infinite or contractible if W has a nontrivial normal psubgroup.

Theorem 5.4. Under the hypothesis of 5.3, the map ER_{p}(G) → EO_{p}(G) induced by the inclusion of categories R_{p}(G) → O_{p}(G) is a G-mod-p-equivalence, which implies that one of the collections is ample if and only if the other is.

Proof. The map induces mod p homology isomorphism on the fixed point sets ER_{p}(G)^{H} → EO_{p}(G)^{H} for every isotropy subgroup H of ER_{p}(G) or EO_{p}(G). The result follows from 2.6.

Theorem 5.5. Suppose that G is a compact Lie group that contains an element of order p. Under the hypothesis of 5.3, the collection of nontrivial p-stubborn subgroups of G is sharp.

Proof. Let P be a maximal p-toral subgroup of G. Consider the P space (ER_{p}(G))_{s}. By definition, isotropy subgroups of (ER_{p}(G))_{s} are p-relevant and for every isotropy subgroup Q of (ER_{p}(G))_{s}, including P, (ER_{p}(G))_{s}^{Q} = ER_{p}(G)^{Q} is mod p acyclic by 5.3. Therefore the projection map ER_{p}(G) → ∗ is an equivalence by 2.6. The result follows from 4.3 and 4.5.

6. Proof of the Homology Decomposition Theorem

In this section we complete our proof of 1.3. First we observe the following elementary facts:

Lemma 6.1. Suppose that G is a compact Lie group and π_{0}(G) is a p-group. If N is a proper subgroup of G that contains G_{0}, then N is a proper subgroup of its normalizer in G.

Proof. The p-group N/G_{0} acts on G/N by left translation. The fixed point set of this action contains a nontrivial coset gN. The element g is in the normalizer of N and not in N.

Lemma 6.2. Let G be a compact Lie group and N be a normal subgroup of G that contains G_{0}. If P is a maximal p-toral subgroup of G, then P ∩ N is a maximal p-toral subgroup of N.

Proof. Clearly P_{0} is the identity component of P ∩N and (P ∩N)/P_{0} is a subgroup of the p-group P/P_{0}, hence P ∩N is p-toral. Since N is normal in G the coset space N/(N ∩ P) is homeomorphic to NP/P. Therefore χ(N/(N ∩ P)) = χ(NP/P) ≢ 0 mod p which implies the maximality of N ∩ P in N.

Lemma 6.3. Suppose that G is a compact Lie group, π_{0}(G) is a p-group, N is a subgroup of G that contains G_{0}. If P is a maximal p-toral subgroup of G, then P ∩ N is a maximal p-toral subgroup of N.

Proof. By 6.1 there exists a normal series N = N_{1} ◁ N_{2} ◁ … ◁ N_{m} = G. The proof is by downward induction on the subscript of N_{i}. The inductive step is provided by 6.2.

Now we are ready to prove the main result. Note that under the hypothesis of 5.3 the conclusion of 1.3 has been proven in the previous section (5.5 and 5.4). The following theorem implies this hypothesis.

Theorem 6.4. For any infinite compact Lie group G, the G-space EO_{p}(G) is mod p acyclic.

Proof. The proof is by induction on the dimension of G.

When dim G = 0 the statement of the theorem is vacuously true.

Let n be a positive integer. Suppose the theorem holds for any infinite G whose dimension is less than n. First we prove that EO_{p}(G) ˜→_{p}∗ for compact Lie group

G for which dim G = n and π_{0}(G) is a p-group and then proceed to an arbitrary compact Lie group of dimension n.

Suppose π_{0}(G) is a finite p-group. We consider two cases:

Case 1. For any p-relevant subgroup Q of G, dim N_{G}(Q)/Q < dim G. Then by the inductive assumption the hypothesis of lemma 5.3 is satisfied and hence EG ×_{G} EO_{p}(G) ˜→_{p}BG. Now an application of 3.1 shows that EO_{p}(G) ˜→_{p}∗

Case 2. There is a p-relevant subgroup Q of G with dim N_{G}(Q)/Q = dim G. Then obviously Q is a finite group and its normalizer N = N_{G}(Q) contains the identity component of G. Let K denote the normal subgroup of G which is the intersection of all maximal p-toral subgroups of G. We will show that K contains an element of order p.

Let P be a maximal p-toral subgroup of G that contains Q. The subgroup Q is contained in P ∩ N, which is a maximal p-toral subgroup of N by 6.3. Since Q is normal in N, it is contained in the intersection of maximal p-toral subgroups of N, which is, again by 6.3, K ∩ N. Therefore K contains Q and hence an element of order p.

If G is itself a p-toral group then EO_{p}(G) ≃ ∗. If G is not p-toral then for any p-toral subgroup H of G dim H < n and, since

KH is contained in some maximal p-toral subgroup of G, dim KH < n. If K is infinite then by the inductive assumption EO_{p}(KH) ˜→_{p}∗ and EO_{p}(G) ˜→_{p}∗ by 3.5.

If K is finite, then the group of its inner automorphisms A = G/C_{G}(K) is also finite and is a p-group. The action of A on K fixes a subgroup K^{A} of K and |K^{A}| ≡ |K| ≡ 0. The Sylow p-subgroup of K^{A} is a nontrivial normal p-subgroup of G and EO_{p}(G) ≃ ∗ by 3.3.

This finishes the proof of the case when π_{0}(G) is a p-group. The case of an arbitrary compact Lie group of dimension n now follows by an

application of 3.5.

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Mathematics Department, University of Notre Dame, Notre Dame, IN 46556

E-mail address: alexei.strounine.1@nd.edu