RATIONAL MACKEY FUNCTORS FOR COMPACT LIE GROUPS I.

J.P.C.GREENLEES.

Abstract. A rational Mackey functor M for a compact Lie group G can be viewed as a collection of sheaves on spaces of conjugacy classes of subgroups, one for each closed subgroup H, related by restriction maps only. This in turn may be specified by giving a suitably continuous family consisting of a ℚπ_{0}(W_{G}(H))-module V (H) for each closed subgroup H with restriction maps V (K̂) ⟶ V (K) whenever K is normal in K̂ and K̂/K is a torus (a ‘continuous Weyl-toral module’). In Part I we show that the category of rational Mackey functors is equivalent to the category of rational continuous Weyl-toral modules. In Part II we use this result to give an algebraic analysis of the category of rational Mackey functors, showing in particular that it has homological dimension equal to the rank of the group.

1. Introduction.

A Mackey functor M is a structure analogous to the representation ring functor H ⟼ R(H) encoding good formal behaviour under induction and restriction. More explicitly, M associates an abelian group M(H) to each closed subgroup H of a fixed compact Lie group G, and to each inclusion K ⊆ H it associates a restriction map res_{K}^{H} : M(H) ⟶ M(K) and an induction map ind_{K}^{H} : M(K) ⟶ M(H). For finite groups Dress [2] introduced a neat algebraic formulation, which can be packaged as saying that a Mackey functor is an additive contravariant functor M : B ⟶ Ab. Here the Burnside category B is formed from the category of G-sets by adjoining reverse maps to supply the induction maps: if K ⊆ H there is a G-map π : G/K ⟶ G/H which induces the restriction map res_{K}^{H} : M(H) ⟶ M(K), and in B there is a transfer map ν : G/H ⟶ G/K inducing the induction map ind_{K}^{H} : M(K) ⟶ M(H). For compact Lie groups the definition of B involves stable homotopy theory, but B is still generated by G-maps π and related transfer maps, inducing restriction and induction, and a Mackey functor is still a contravariant additive functor M : B ⟶ Ab.

The structure of Mackey functors is very complicated in general, partly because of the existence of transfer maps increasing the size of subgroups as well as restriction maps decreasing them. However, if we require all values M(H) to be rational vector spaces there is a great simplification, since by use of idempotents in Burnside rings we may break up M as a sum of pieces for which the transfer maps can be deduced from the actions of the Weyl groups W_{G}(H) = N_{G}(H)/H on the vector spaces M(H).

For finite groups this is the well known [5, 7, 11] result stating that such a rational Mackey functor is equivalent to a collection of modules V (H) over the various group rings ℚW_{G}(H). The purpose of the present article is to obtain an analogous reduction for an arbitrary compact Lie group, and in the course of this we shall shed some light even on the case of finite groups. Our reduction takes takes the form of an equivalence of categories, and because of the simplicity of the target category we obtain in Part II the general result that the homological dimension of the category of rational Mackey functors is the rank of the group G. It also allows us to generalise the constructions for finite groups that take a module over a Weyl group and form an associated Mackey functor.

This shows that the algebraic complexity of equivariant rational cohomology theories is reasonably mild, but there is also topological complexity arising from the nontrivial rational cohomology of classifying spaces. This topological complexity may also be brought under control in favourable cases; for the circle group this is done in [6].

This is the third version of this work, and was begun whilst the author was visiting the University of Göttingen; the author is grateful for its hospitality, SFB 170 for its support and the Nuffield Foundation for making the visit possible.

To minimize confusion we adopt certain standard notational conventions. The letters H and L will be arbitrary closed subgroups of the fixed ambient group G; as a general rule they have subgroups K ⊆ H and P ⊆ L. Other notation associated with H and K will consist of letters alphabetically related in a similar way. The notation K ⊆_{G} H means that K is conjugate in G to a subgroup of H.

Letters M, N, … will denote Mackey functors. Letters e, f, … will be idempotents of Burnside rings.

On several occasions we need to consider the same space with various topologies. In that case we shall use subscripts to indicate this: for instance SG, S_{f}G, and S_{I}G denote the set of closed subgroups of G with various topologies.

2. Sketch of the reduction.

We shall work rationally for the rest of this paper: there are obvious integral versions of many definitions, but very little of the substantial analysis can be carried out integrally. Even for finite groups G there are integral Mackey functors of infinite projective dimension, although the analogue of Rim’s theorem holds [5]. One might hope for the analogue that if an integral Mackey functor has finite projective dimension then this is at most one more than the rank of G.

We shall be showing that the category of Mackey functors is equivalent to a suitable category of sheaves on the space of subgroups of G. However, in this brief sketch we shall simply explain what happens on objects, and leave all questions of naturality to the body of the paper.

Thus we begin with a Mackey functor M; the first step is to extend its domain to a larger category. In the usual way, if e is an idempotent self map of G/H_{+} we may define M on eG/H_{+} by M(eG/H_{+}) := eM(G/H_{+}), and we abbreviate this to M(H, e). One source of idempotents e is the Burnside ring A(H) which may be identified with the ring of continuous functions on the space ΦH of conjugacy classes of closed subgroups of H with finite Weyl groups. Thus an idempotent e corresponds to an open and closed subspace of ΦH, and in fact these open subsets form a basis for the topology on ΦH. We may therefore think of M(H) as a functor on basic open sets, which thus determines a sheaf on ΦH.

In order to compare these sheaves for different subgroups H we need to view M(H) as a sheaf on the space ΓH of all conjugacy classes of subgroups, and we do this by putting a new topology on ΓH: the smallest for which all idempotents of A(H) are continuous (we refer to this as the I-topology). Indeed for any subgroup K, the set of subgroups K̂ with finite Weyl group such that K is normal in K̂ and K̂/K is a torus are all conjugate; we may therefore define q = q_{H} : ΓH ⟶ ΦH by taking q(K) to be the conjugacy class of K̂. It turns out that any element of A(H) is constant on fibres of q, and the I-topology on ΓH is the smallest topology for which q is continuous; the I-topology is usually highly non-Hausdorff. Since q defines a bijection between the open sets of Γ_{I}H and those of ΦH, it is obvious that the sheaf M(H) on ΦH can equally be regarded as a sheaf on Γ_{I}H.

Now any G-map φ : G/K ⟶ G/H determines a continuous function φ : Γ_{I}K ⟶ Γ_{I}H, and a restriction map φ^{∗} : A(H) ⟶ A(K). Since the restriction φ^{∗} : M(H) ⟶ M(K) is a map of A(H)-modules we obtain a map φ^{∗} : M(H, e) ⟶ M(K, φ^{∗}e) for each idempotent e of A(H). These fit together to give a sheaf map φ^{∗} : M(H) ⟶ φ_{∗}M(K). Furthermore this depends only on the homotopy class of φ. Hence M determines an hO-sheaf, where O is the category of G-orbits and hO is its homotopy category. The main point is that M can be recovered from this hO-sheaf. Some of the transfer maps have been used in defining the stalks of the sheaves M(H), and tom Dieck Splitting shows that all other transfer maps can be recovered from the group actions on the stalks. This is the first step: a Mackey functor is equivalent to an hO-sheaf. Theorem A: There is an equivalence of categories

Mackey functors ≃ hO-Sheaves.

This is proved as Theorem 7.3 below.

The next step is to replace the hO-sheaf by a single equivariant sheaf V = V_{M} on the space SG of all closed subgroups of G with some extra structure corresponding to the inclusions of normal subgroups with torus quotients. The idea here is that whenever K has finite H-Weyl group the stalk M(H)_{(K)} is a retract of V (K) := M(K)_{(K)}, and the retraction is determined by the W_{H}(K)-module structure of M(K)_{(K)}; if K has infinite Weyl group then M(H)_{(K)} is equal to M(H)_{(q(K))} (since the I-topology does not separate (K) from (q(K)) ), and this is a retract of V (q(K)). Thus we define the sheaf V by giving it stalk V (K) at K. Of course, when K ⊆ H ⊆ L and K has a finite H-Weyl group but an infinite L-Weyl group we must include some additional structure to recover the restriction map map M(L)_{q_{L}(K)} = M(L)_{(K)} ⟶ M(H)_{(K)}. Accordingly we refer to V as a continuous Weyl-toral module and regard it as a functor V : T C ⟶ ℚ-mod where T C is the topological category whose objects are the closed subgroups of G and which has a single morphism K ⟶ K̂ whenever K is normal in K̂ and K̂/K is a torus. The Weyl group action is encoded by observing that G acts by conjugation on T C and requiring that V is an equivariant functor; we impose continuity conditions by saying (i) the action by G is continuous (ii) V is a sheaf on T C for a suitable topology (the f-topology) which only sees a subgroup K in those subgroups H with W_{H}(K) finite and (iii) restriction along T C-morphisms is continuous. The idea is that the information from infinite Weyl groups has been put into the morphisms of T C.

To recover the sheaf M(H) from V we first restrict V to the space FH of subgroups of H with finite H-Weyl groups. Now for a subgroup K of H with finite Weyl group we may consider the invariants V (K)^{W_{H}(K)}. This is precisely the subspace with trivial holonomy and so we have a canonical isomorphism between these spaces for the various representatives K for an H-conjugacy class (K). We may therefore take M(H)_{(K)} = V (K)^{W_{H}(K)}. The topology on S_{f}G is defined so that we obtain a sheaf on ΦH. The additional structure is exactly what is needed to supply the appropriate maps between these sheaves as H varies. This is the second step: an hO-sheaf is equivalent to an equivariant sheaf on T C.

Theorem B: There is an equivalence of categories

hO-Sheaves ≃ Continuous Weyl-toral modules.

This is proved below as Theorem 10.6.

Thus, composing these equivalences we have replaced the Mackey functor M : B ⟶ Ab by a continuous equivariant functor V : T C ⟶ ℚ-mod. Compared to the rather complicated category B, the toral chain category T C has very few morphisms and all of them increase the size of the subgroup. To show we have achieved something by this equivalence, and to give illustrations of the difficulties we sketch three of the simplest examples here. We shall discuss these and other examples in more detail in Part II.

Examples C (i): Finite groups. Let G be a finite group. Here all spaces have the discrete topology, and T C is also discrete as a category. If M is a Mackey functor, viewing M as an hO-sheaf simply corresponds to the decomposition

M(H) = ⨁_{(K)} e_{K}M(H),

where the sum is over H-conjugacy classes of subgroups K of H, and e_{K} is the idempotent corresponding to K-fixed points. The term e_{K}M(H) is the space of germs over (K). Now the value of V at H is

V (H) = e_{H}M(H),

and a continuous Weyl-toral module (which in this case is simply a ‘Weyl module’) is determined by giving a W_{G}(H)-module V (H) for each subgroup H with V (H^{g}) = g^{−1} · V (H) in the sense of representation theory. In particular V is determined by giving its value on a representative of each conjugacy class. We then recover e_{K}M(H) from V by

e_{K}M(H) = {⨁_{K′} V (K′)}^{H} ≅ V (K)^{W_{H}(K)},

where the sum is over subgroups K′ in the H-conjugacy class of K. The description using the sum is more natural; the description in terms of V (K) alone is more comprehensible. Altogether, we see

M(H) = ⨁_{(K)} V (K)^{W_{H}(K)}.

One way of thinking about this is to imagine the lattice of subgroups displayed with G at the top and 1 at the bottom. Thus each V (K) engenders a contribution to M at all subgroups H above K in the lattice. For example in the cyclic group C_{p^{3}} of order p^{3} we can describe M in terms of V in the diagram

V (C_{p^{3}}) ⊕ V (C_{p^{2}})^{C_{p^{3}}/C_{p^{2}}} ⊕ V (C_{p})^{C_{p^{3}}/C_{p}} ⊕ V (C_{1})^{C_{p^{3}}/C_{1}} = M(C_{p^{3}})
↓↑ ↓↑ ↓↑ ↓↑
V (C_{p^{2}})  ⊕ V (C_{p})^{C_{p^{2}}/C_{p}} ⊕ V (C_{1})^{C_{p^{2}}/C_{1}} = M(C_{p^{2}})
↓↑ ↓↑ ↓↑
V (C_{p})  ⊕ V (C_{1})^{C_{p}/C_{1}} = M(C_{p})
↓↑  ↓↑
V (C_{1})  = M(C_{1}).

Examples C (ii): The circle group. Let G be the circle group, SO(2). This example is discussed at greater length in Appendix A of [6]. Here ΦG is a single point, and the Burnside ring A(G) = ℚ, so that that the space Γ_{I}G of closed subgroups has the indiscrete topology. On the other hand set of closed subgroups as the objects of the category T C has the discrete topology, and its non-identity morphisms consist of a single map C_{n} ⟶ G for each positive integer n. All Weyl groups are connected. Thus, regarding M as an hO-sheaf, M(G) is the constant sheaf on a point, and if H is finite

M(H) = ⨁_{K⊆H} e_{K}M(K),

the two simplifications being that conjugacy classes are singletons, and e_{K}M(H) = e_{K}M(K). Finally, a continuous Weyl-toral module (which in this case is simply a ‘toral module’) is specified by a rational vector space V (H) for each subgroup H and a restriction map V (G) ⟶ V (H) for each finite H. Here it is easy to recover M as M(G) = V (G), and for finite subgroups H

M(H) = ⨁_{K⊆H} V (K).

We may represent V by a diagram of rational vector spaces as follows.

[figure: V (G); ⋮; V (C_{3}); V (C_{2}); V (C_{1})]

Examples C (iii): The group O(2). Let G be the orthogonal group O(2). In this case we have both finite Weyl groups and infinite Weyl groups; furthermore, many of the topologies are non-trivial.

The proper subgroups of O(2) are the circle SO(2) and its finite subgroups C_{n}, all of which are normal, together with the dihedral groups, each of index 2 in their normalizers. There is one conjugacy class of dihedral subgroups of order 2n for each n ≥ 1, and it is convenient to let D_{2n} denote the representative generated by C_{n} and reflection in the x-axis. The conjugacy class of D_{2n} is (D_{2n}) ≅ G/D_{4n}, and thus homeomorphic to a circle. Already in this one dimensional group we have an infinite set of conjugacy classes of subgroups with finite Weyl groups:

ΦG = {T, G} ∪ {(D_{2n})|n ≥ 1}.

As a topological space it has (G) as a cluster point. The spaces ΦH for proper subgroups are described above. Finally S_{f}G is homeomorphic a disjoint union of the discrete space S_{f}S^{1} and a union of concentric circles G/D_{4n} of radius 1/n and G/G at the centre.

Both steps now become non-trivial. The sheaves M(H) for proper subgroups H are all sheaves over discrete spaces ΦH, with stalk e_{K}M(H) over (K). However there are not enough genuine idempotents in A(G) = C(ΦG, ℚ) to give all stalks of the sheaf M(G). Over proper subgroups H the stalk is M(G)_{(H)} = e_{H}M(G), but the stalk at (G) is

M(G)_{(G)} = lim_{→}_{e(G)=0} M(G)/eM(G).

In this case, since there is no canonical extension of a germ at (G) over a neighbourhood, it becomes important to consider the sheaf structure.

Now to form V we have G/T -modules V (C_{n}) = e_{C_{n}}M(C_{n}), and V (T) = M(T) with restriction maps V (T) ⟶ V (C_{n}). There are no other morphisms between distinct conjugacy classes. For each n ≥ 1 we have a D_{4n}/D_{2n}-module e_{D_{2n}}M(D_{2n}) and finally we have the vector space V (G) = lim_{→}_{e(G)=0} M(G)/eM(G).

[figure: V (SO(2)); ⋮; V (C_{3}); V (C_{2}); V (C_{1}); V (G); ⋮; V (D_{6}); V (D_{4}); V (D_{2})]

Of course, we are more properly supposed to give stalks over the entire conjugacy class, and this is easily done by taking

G ×_{D_{4n}} V (D_{2n})

over the conjugacy class G/D_{4n} of D_{2n}.

To recover M(H) for a proper subgroup H we proceed exactly as before. However for M(G) we must take account of the topologies. Indeed M(G) is the vector space of G-equivariant sections of V |_{FG}. More explicitly for any n ≥ 0

M(G) = V (SO(2))^{G/SO(2)} ⊕ V (D_{2})^{D_{4}/D_{2}} ⊕ V (D_{4})^{D_{8}/D_{4}} ⊕ ⋯ ⊕ V (D_{2n})^{D_{4n}/D_{2n}} ⊕ R_{n}

where R_{n} is the space of G-equivariant sections over

{G} ∪ {H ⊆ G | H a dihedral group of order more than 2n}.

More informally R_{n} is the subspace of W -invariant sections over {G} ∪ {D_{2m} | m ≥ n + 1}. This description is natural, but we may obtain a more explicit isomorphism at the cost of choosing local sections to represent germs in V (G):

M(G) ≅ V (G) ⊕ V (SO(2))^{G/SO(2)} ⊕ ⨁_{n≥1} V (D_{2n})^{D_{4n}/D_{2n}}.

Examples C (iv): The 2-torus. Let G be the 2-torus, S^{1} × S^{1}. This example is necessary to counteract the misleading impression of combinatorial simplicity suggested by the rank 1 examples. The topology on T C is again discrete, all Weyl groups are connected and conjugacy classes are singletons. Thus V is simply a functor T C ⟶ ℚ-mod. Note however that there are many factorizations 1 ⟶ S ⟶ G of the inclusion 1 ⟶ G. It is also instructive to observe that there are many nonconjugate chains of closed subgroups whose union is dense in G.

The rest of the paper is arranged as follows.

Contents

1. Introduction. 1

2. Sketch of the reduction. 2

3. Spaces of subgroups and Burnside rings. 8

4. The idempotent topology and combinatorics of toral chains. 10

5. Morphisms in the rational orbit category. 13

6. Shifting to sheaves. 15

7. The equivalence of Mackey functors and hO-sheaves. 17

8. The topological category of toral chains. 19

9. Continuous Weyl-toral modules. 23

10. Recovering an hO-sheaf from a continuous Weyl-toral module. 28

References 32

3. Spaces of subgroups and Burnside rings.

We shall need various facts about the equivariant stable homotopy category of Lewis-May [8]. We use notation familiar from the category of spaces whenever possible, thus we write G_{+} ∧_{H} X for the extended object associated to the H-spectrum X (where Lewis-May would write G ⋉_{H} X) and whenever we refer to H-fixed points we intend geometric H-fixed points (Φ^{H} fixed points in the terminology of Lewis-May). This is because we work only in the Spanier-Whitehead category S of suspension spectra of finite dimensional based G-spaces and their suspensions; therefore we do not wish to distinguish notationally between a G-space and its suspension G-spectrum. We shall not need to make explicit use of Lewis-May fixed points.

For most purposes it is enough to describe the maps between G-spaces X and Y when X is finite. We take a complete G-universe U (i.e. an infinite dimensional G-inner product space containing all finite dimensional orthogonal representations of G), and take

[X, Y]^{G} := lim_{→}_{V} (S^{V} ∧ X, S^{V} ∧ Y)^{G} ⊗ ℚ

where (X, Y)^{G} means homotopy classes of unstable based G-maps and V runs through finite dimensional subspaces of U. A Mackey functor is then an additive contravariant functor M : B ⟶ Ab from the full subcategory B of homogeneous spaces G/H_{+} for closed subgroups H of G to abelian groups. For brevity we write M(H) for the value of M on the homogeneous space G/H_{+}.

If the rationalisation is omitted in the above we obtain the category of integral Mackey functors. It is not obvious that this definition corresponds to the algebraic one in the case of a finite group, but this is proved in [8, V.9.9].

In the sequel we shall make much use of the space SG of closed subgroups of G. This may be endowed with the Hausdorff metric, and G acts continuously on it. We shall repeatedly use certain properties of its topology.

Lemma 3.1. (a) The space SG is compact.

(b) For any closed subgroup H of G there is a number ϵ > 0 so that if d(H, L) < ϵ there is an element g ∈ G with L^{g} ⊆ H.

(c) If a sequence {K_{i}} converges to H in SG then there are group elements g_{i} so that g_{i}^{−1}K_{i}g_{i} ⊆ H for all sufficiently large i and g_{i} ⟶ e as i ⟶ ∞.

Proof: See [4, 5.6.2] and [1, II.5.6]. The fact that the conjugating elements may be taken close to the identity follows from the fact that the conjugacy class (K) of a subgroup K is homeomorphic to G/N_{G}(K) by the action map.

Remark: Perhaps the most useful consequence of Part (b) is that close implies conjugate for subgroups of the same ‘size’ (i.e. with the same dimension and the same number of components).

Much of our analysis will be conducted on the space ΓG = SG/G of conjugacy classes, which is compact, Hausdorff and totally disconnected [4, 5.6.1]. To any subgroup we may associate its Weyl group W_{G}(H) := N_{G}(H)/H, and we also need to consider the the closed G-subspace FG ⊆ SG consisting of subgroups H with finite Weyl groups (i.e. subgroups of finite index in their normalizers) and its space of conjugacy classes is denoted by ΦG = FG/G.

Lemma 3.2. The natural map p : ΓH ⟶ ΓG is both open and closed, and hence in particular its image ΓH_{G} is both open and closed in ΓG.

Proof: Since ΓH is compact and ΓG is Hausdorff the map is obviously closed. To see the map is open consider an open set U ⊆ ΓH; if p(U) is not open there is a G-conjugacy class (K) which is a limit of points outside p(U). However, given a sequence of subgroups K_{n} of G with K_{n} ⟶ K ⊆ H, we may suppose by 3.1 (b) that for all sufficiently large n K_{n} is conjugate to K′_{n} ⊆ K and K′_{n} ⟶ K. Since U is open, the H-conjugacy class of K′_{n} lies in U for all large enough n.

Lemma 3.3. The set Λ_{H}(K) ⊆ ΦH of subconjugates of K is open and closed.

Proof: Since any sequence of subgroups tending to a subconjugate of K eventually consists of subconjugates of K, the complement of Λ_{H}(K) is closed.

To see Λ_{H}(K) is closed, suppose (L_{i})_{i} is a sequence of subgroups of K, converging to a subgroup L. By 3.1(c), we may suppose that (L_{i})^{g_{i}} ⊆ L for all i with g_{i} ⟶ e. In this case L is the closure of ⋃_{i≥n}(L_{i})^{g_{i}} for any n. We claim this is a subgroup of K. Indeed, if l_{i} ∈ L_{i} is such that (l_{i}^{g_{i}}) is convergent to l, then (l_{i}) is also convergent to l and hence contained in K.

Now we shall be much concerned with idempotents of the rational Burnside rings A(H) = [S^{0}, S^{0}]^{H}, and indeed we shall use them to introduce a new coarser topology on ΓG, which will coincide with the existing topology on ΦG. First we recall the basic facts.

Proposition 3.4. (tom Dieck [4, 5.6.4, 5.9.13])

(a) If C(X, ℚ) denotes the ring of continuous functions from X to the discrete space ℚ, passage to fixed points induces a map

α : [S^{0}, S^{0}]^{G} ⟶ C(ΓG, ℚ)

where α(f)(H) is the degree of the H-fixed point map f^{H} : S^{0} ⟶ S^{0}.

(b) The composite of α and restriction is an isomorphism of rings

β : [S^{0}, S^{0}]^{G} ⟶^{≅} C(ΦG, ℚ).

Accordingly we may associate to any idempotent e ∈ A(H) its support S(e) = {(K) ∈ ΓH | α(e)(K) = 1} and its image S_{G}(e) in ΓG. The set S(e) is open and closed in ΓH since α(e) is continuous. We may also consider the intersection V (e) = S(e) ∩ ΦH; this is open and closed in ΦH, and we refer to it as the Φsupport.

We shall constantly appeal to this correspondence between idempotents and subsets of ΦH which are open and closed.

Lemma 3.5. The topology on ΦH is generated by the open and closed sets V (e).

Proof: This follows from the fact that ΦH is compact and totally disconnected.

4. The idempotent topology and combinatorics of toral chains.

The analogue of 3.5 is quite false for ΓH, so we define Γ_{I}H to be the same set ΓH, but with the idempotent topology, generated by declaring that for idempotents e ∈ A(H) the sets S(e) are to be open. Note that S(e) is also closed since S(1 − e) is its complement. By 3.4 the identity map ΓG ⟶ Γ_{I}G is continuous, but it is not a homeomorphism in general: for example Γ_{I}S^{1} is indiscrete since 0 and 1 are the only idempotents in A(S^{1}) = ℚ.

The following three lemmas are at the heart of the behaviour of Burnside rings; we shall appeal to them when we have set up the machinery to process them. The discussion of this section applies integrally, but for that case it is most appropriate to describe the topology on Γ_{I}H as the smallest for which all elements x ∈ A(H) are continuous.

Lemma 4.1. If K is normal in K̂ ⊆ H with T = K̂/K a torus then α(x)(K) = α(x)(K̂).

Proof: The K fixed point map x^{K} : S^{0} ⟶ S^{0} may be viewed as a T -map. But since A(T) ≅ ℚ is generated by the identity map the function α(x^{K}) is constant.

Lemma 4.2. The set S(e) has finitely many maximal elements, each of which is of finite index in its normalizer.

Proof: Suppose that {K_{i}} is a sequence of representatives of maximal classes. Since S(e) is compact the sequence {K_{i}} has a subsequence converging to an element of S(e), and by 3.1 (b) this represents a maximal element. This shows in turn that the set of maximal elements is both closed and finite.

Now if K is not of finite index in its normalizer there is a nontrivial torus T ⊆ N_{H}(K)/K with inverse image K̂ ⊇ K. If (K) ∈ S(e), by 4.1 we find 1 = α(e)(K) = α(e)(K̂), so that (K̂) ∈ S(e) and (K) is not maximal.

Lemma 4.3. If K ⊆ H then (K) ∈ S(e) if and only if there is a chain of subgroups

K = K_{0} ⊆ K_{1} ⊆ K_{2} ⊆ ⋯ ⊆ K_{n} ∈ V (e)

with K_{i} normal in K_{i+1} and K_{i+1}/K_{i} a torus.

We shall refer to such a chain of subgroups as a toral chain of length n from K_{0} to K_{n}. Proof: By 4.1 if L is normal in L̂ with T = L̂/L a torus for any idempotent e of A(L̂) we have e(L̂) = e(L). Thus the existence of a chain as in the statement guarantees (K) ∈ S(e).

Now suppose on the other hand that (K) ∈ S(e). If K is of infinite index in its H-normalizer, there is a subgroup K_{1} ⊆ H with K as a normal subgroup and a torus quotient. By 4.1, K_{1} ∈ S(e) and we may repeat this argument if K_{1} is of infinite index in its H-normalizer, and by dimension this process must eventually terminate.

In fact, it is quite easy to see from 4.3 that S(e) is the closure of V (e) in Γ_{I}H. To exploit this attractive reduction we need to understand toral chains a little

better. For the purposes of discussion suppose given a toral chain 4.4.

K = K_{0} ⊆ K_{1} ⊆ ⋯ ⊆ K_{n} = K̂.

First notice that if K is normal in K′ then the identity component K_{e} is normal in K′. Also (K′/K)_{e} is a quotient of K′_{e}/K_{e}. Therefore any toral chain may be replaced by a toral chain of the same length involving only connected groups. Note also that, if K is connected, all groups in the chain must be connected, but that connectedness of K̂ does not imply that of K.

Lemma 4.5. If A ◁ B ◁ C and C/B is connected then A is normal in C.

Proof: The group C acts on B by conjugation, and this induces a homomorphism from C/B to the outer automorphism group Out(B). But the outer automorphism group Out(B) of any compact Lie group is discrete, so that since C/B is connected, C acts through inner automorphisms on B. By hypothesis, A is preserved by inner automorphisms of B.

We may apply this to toral chains to see they are well behaved.

Corollary 4.6. Given any toral chain of the form 4.4, K is normal in K̂ and K̂/K is a torus.

Proof: It follows from 4.5 by induction that, in a toral chain of the form of 4.4, K is normal in K̂ and that the quotient is a group with a normal series with tori as sub-quotients. Hence K̂/K is a torus.

Corollary 4.7. If K̂ is at the top of a toral chain of length n then n ≤ rank(K̂).

Proof: By the structure theory of compact Lie groups K̂ has a finite cover which is a product of an n-dimensional torus and another group. Hence K̂ contains an n-torus.

Corollary 4.8. The length of the longest chain toral chain is the rank of the group.

Proof: The rank of a subgroup M is bounded by the rank of G so by, 4.8, no toral chain can be longer than the rank of G. It is easy to construct a chain with length the rank of G inside the maximal torus.

By 4.6 all toral chains from K lie in N_{G}(K); using conjugacy of maximal tori in W_{G}(K) we have a limited form of uniqueness.

Corollary 4.9. Given any maximal torus T of W_{G}(K) with inverse image T̃ ⊆ N_{G}(K) and a toral chain as in (3.5), K̂ is conjugate to a subgroup of T̃. In particular all top ends of maximal toral chains from K are conjugate.

This corollary has a very useful consequence. There is a retraction q : Γ_{I}H ⟶ ΦH defined by taking q(K) to be the conjugacy class of top ends of maximal toral chains from K. This allows a neat characterization of the topology on Γ_{I}H.

Corollary 4.10. The retraction q : Γ_{I}H ⟶ ΦH is continuous and the topology on Γ_{I}H is the smallest topology for which this is true.

Proof: The map q is continuous since q^{−1}(V (e)) = S(e) by 4.3, and the sets S(e) generate the I-topology.

It may be worth warning that 4.9 does not imply that all toral chains between the same groups are conjugate. For this reason we introduce the category T C of toral chains. Its objects are the closed subgroups of G and there is a single morphism K ⟶ K̂ whenever K is normal in K̂ and K̂/K is a torus. This is a category by 4.6. A version of Quillen’s Theorem A allows one to prove that conjugacy of toral chains principally depends on the degree of singularity of tori in the chain. In due course we shall make T C into a topological category, and this will be the central structure in our analysis.

5. Morphisms in the rational orbit category.

This section is fundamental to the feasability of the present approach. It shows that all morphisms of the stable orbit category are sums of composites of (a) unstable G-maps and (b) maps arising from idempotents of Burnside rings. The main point is that the only transfer maps required are the most classical norm maps, associated to coset averages, and hence explained by the actions of the Weyl groups. Evidently this needs a little interpretation.

Note that induction gives a ring homomorphism A(H) = [S^{0}, S^{0}]^{H} ⟶ [G/H_{+}, G/H_{+}]^{G}, and right multiplication gives a ring homomorphism ℚπ_{0}(W_{G}(H)) ⟶ [G/H_{+}, G/H_{+}]^{G}. Thus idempotents of the Burnside ring and idempotents of the group ring of the Weyl group both act on G/H_{+}. Moreover, all elements of the group ring are sums of multiples of group elements. The images of these two maps are related in the obvious way, with the action of W_{G}(H) on A(H) describing the effect of moving one past the other.

Next, if K ⊆ H has finite H-Weyl group we may define ν_{K}^{H} : G/H_{+} ⟶ G/K_{+} as follows. First, if N = N_{G}(H) we define the transfer ν′ : N/N_{+} ⟶ N/K_{+} to be the stable W_{G}(H)-map which is the desuspension of the Pontrjagin-Thom collapse S^{V} ⟶ S^{V} ∧ N/K_{+} map, where V is the regular representation of W_{G}(H). View it as an N-map, and take the composite H/H_{+} ⟶ H/N_{+} ⟶ H/K_{+} where the first map is also transfer and the second is the induction of ν′. Now induce this up to a G-map to obtain ν_{K}^{H}.

Now, if n_{K}^{H} is the average of the elements of W_{H}(K), it is apparent that ν_{K}^{H} factors through n_{K}^{H}G/K_{+}. In a neighbourhood of K this is a retract of G/H_{+}, and so we factor ν_{K}^{H} as the coset average, followed by collapse onto a group ring retract of G/K_{+}, identification of this with a Burnside ring retract of G/H_{+} and inclusion of the retract.

Lemma 5.1. If K has finite H-Weyl group, then in a ΦH-neighbourhood of K n_{K}^{H}G/K_{+} is a retract of G/H_{+}.

Proof: We may conduct the discussion H-equivariantly, and then induce up to G.

Consider the collapse map H/K_{+} ⟶ S^{0}. Applying n_{K}^{H} we see that n_{K}^{H}H/K_{+} ⟶ S^{0} is an equivalence at L whenever (H/K)^{L} is a single W_{H}(K)-orbit; this is true for L = K, and hence in a neighbourhood of K.

Remarks: (i) The element [H/K] ∈ A(H) is a composite S^{0} ⟶ H/K_{+} ⟶ S^{0}. It is of non-zero degree in L-fixed points throughout the support S of [H/K]. Thus e_{S}S^{0} is a retract of H/K_{+}: however the retraction may not be given by n_{K}^{H} throughout S. For example, if H is finite the trivial subgroup lies in S, but (H/K)^{1} = H/K. (ii) The proof of this lemma specifies a particular identification between these retracts, so that we do not need to be anxious about automorphisms.

To state the theorem it is convenient to let IB denote the full subcategory whose objects are the retracts of homogeneous spaces arising from idempotents in Burnside rings. Indeed, if e ∈ A(H) is an idempotent with support V ⊆ ΦH we write (H, e) or (H, V) for the object G_{+} ∧_{H} eS^{0}: the category IB is the full category of all such objects, as H and e vary.

Theorem 5.2. The category IB is generated as an additive category by

(i) right multiplications R_{g} : G/H_{+} ⟶ G/H_{+}^{g}, (ii) projections π_{K}^{H} : G/K_{+} ⟶ G/H_{+} when K ⊆ H, (iii) retractions G/H_{+} ⟶ G_{+} ∧_{H} eS^{0} for e an idempotent in A(H) (iv) inclusions G_{+} ∧_{H} eS^{0} ⟶ G/H_{+} for e an idempotent in A(H)

(v) the norm maps n_{K}^{H} : G/K_{+} ⟶ G/K_{+} for K ⊆ H with finite Weyl groups.

Proof: As remarked in [8, V.9], tom Dieck splitting shows that any stable map

G/H_{+} ⟶ G/L_{+} is a sum of composites G/H_{+} ⟶^{ν} G/K_{+} ⟶^{u} G/L_{+} with u an

unstable G-map. We refine this observation, starting from the observation that the splitting is natural for stable maps, and hence that it applies and is natural for stable retracts of spaces.

We wish to argue by induction on the size of subgroups of H. Thus we let the size of H be the pair (dimK, |π_{0}(K)|), and order sizes lexicographically. There are only finitely many conjugacy classes of subgroups of a given size, and we impose an order on the conjugacy classes of each size. Note that if K′ ⊂ K then (K′) is less than (K) in this ordering. Therefore, since all descending sequences of subgroups are finite, we may prove a statement about all conjugacy classes of subgroups by induction, showing that if it holds for all (K)′ less than (K) then it also holds for (K).

Consider the group of stable maps [G/H_{+}, G/L_{+}]^{G}. For each h : G/H_{+} ⟶ G/L_{+}, consider the set of subgroups K for which h^{K} is non-equivariantly essential; we refer to these as detecting subgroups. The number of maximal detecting conjugacy classes is finite by tom Dieck splitting, and there is therefore a unique largest detecting conjugacy class (K) in the order described above. We argue by induction on the largest detecting conjugacy class, that the theorem applies to all maps h : G/H_{+} ⟶ G/L_{+}. Thus we assume that for all (K′) < (K), the theorem is true for all maps with largest detecting conjugacy class (K′).

Since (K) is the largest detecting conjugacy class the contribution detected by it must arise as G/H_{+} ⟶^{ν} G/K_{+} ⟶^{u} G/L_{+} with u an unstable G-map. We may therefore restrict attention to ν, and we have remarked above that in a neighbourhood S of K it is induced up from the stable H-map

e_{S}H/H_{+} ≃ e_{S}n_{K}^{H}H/K_{+} ⟶ e_{S}H/K_{+},

where the second map is given by coset averages. Subtracting this contribution we obtain a map whose largest detecting conjugacy class is smaller than (K), to which the theorem applies by the inductive hypothesis.

We may now record a relationship we have previously noted informally.

Corollary 5.3. If K ⊆ H and W_{H}(K) is finite then M(H, (K)) = M(K, (K))^{W_{H}(K)}.

In one sense, the usefulness of introducing idempotents arises since one can separate sets between which there should be no morphisms. This again follows from tom Dieck splitting, but we omit the proof since we do not use the result.

Lemma 5.4. (Sparsity ) If (H, V) and (L, W) are objects with V ∩ cl W = ∅ (ie so that there is no toral chain from a conjugate of H to an element of W) then Hom((H, V), (L, W)) = 0.

6. Shifting to sheaves.

In this section we implement our declared intention of translating the discussion into sheaf theory.

We remarked in the introduction that a Mackey functor M can be extended to retracts of G/H_{+}, for if e is an idempotent self map of G/H_{+} we may take its value on the corresponding retract to be eM(H). Thus any Mackey functor has a natural extension to the category IB of objects (H, V) with H a subgroup and V an open and closed subset of ΦH.

We have already indicated that there are usually not enough idempotents in B: we really want one idempotent for each closed subgroup of G. To obtain substitutes for these extra idempotents there are two possible approaches. One can extend the category B to include pro-objects corresponding to idempotent refinements, and extend a B-module to this larger category. If we identify an idempotent e ∈ A(H) with the open and closed set V (e) of ΦH, the pro-objects consist of decreasing families of open and closed sets V_{α} thought of as representing the closed set V = ⋂_{α} V_{α}. A module M is extended by taking M(H, V) = lim_{→}_{α} M(H, V_{α}). This suggests the second alternative: we consider M(H) as a sheaf on ΦH, and then M(H, V) consists of germs around V. The special case when V = {(K)} is a singleton gives the stalk M(H, (K)), and makes sense of the expression \underline{e}_{K}M(H) where \underline{e}_{K} plays the part of an idempotent with support (K). Our main result may be viewed as the statement that M can be recovered from the values \underline{e}_{K}M(K) and the maps between them. We adopt the sheaf theoretic approach, since it uses more familiar terminology, and is slightly simpler technically.

For clarity we introduce the temporary notation M̃(H) when we consider M(H) as a sheaf on ΦH. Thus, we define the presheaf M̃(H) on ΦH by taking M̃(H)(U) := M(H, U) for an open and closed set U ⊆ ΦH.

Lemma 6.1. The presheaf M̃(H) is a sheaf.

Proof: In fact, we show that any presheaf on ΦH is a sheaf. To check a presheaf is a sheaf it is enough to check that unique patching holds for basic open sets. Since basic open sets are compact in our case, it is enough to deal with finite covers. Such covers correspond to finite direct sum decompositions and hence the sheaf condition is automatically satisfied.

Of course if V is closed M(H, V) is now the group of germs of M̃(H) at V, and we may write M̃(H)(V) or M̃(H)_{V} for it.

By 4.10 there is a bijective correspondence between open sets in ΦH and those in Γ_{I}H, so that sheaves on ΦH correspond to sheaves on Γ_{I}H. For the purpose of comparing the sheaves for various subgroups it is more convenient to think of M̃(H) as a sheaf on Γ_{I}H.

Lemma 6.2. An unstable G-map φ : G/H ⟶ G/L induces a continuous function φ_{Γ} : Γ_{I}H ⟶ Γ_{I}L. Furthermore φ_{Γ} only depends on the homotopy class of φ.

Proof: The G-map φ can be written as a composite of right multiplication R_{g} : G/H ⟶ G/H^{g} for some suitable element g ∈ G and projection π : G/H^{g} ⟶ G/L associated with the inclusion of a subgroup H^{g} ⊆ L. A subgroup K of H corresponds to the subgroup K^{g} of L, and the L-conjugacy class of K^{g} depends only on the H-conjugacy class of K. This defines a function φ_{Γ} : ΓH ⟶ ΓL.

To see that φ_{Γ} is continuous for the I-topology, first consider the case that φ is the projection for an inclusion H ⊆ L. It is enough to verify that the inverse image of a basic open set S(f) is open. But of course, (φ_{∗})^{−1}(S(f)) = S(res_{H}^{L}(f)). Now note that (R_{g})_{∗} is continuous since R_{g} induces an isomorphism A(H) ⟶ A(H^{g}).

Finally suppose given a homotopy φ_{0} ≃ φ_{1}. It is enough to observe that t ⟼ φ_{t}(K) gives a path in SL. Since ΓL is totally disconnected the path t ⟼ (φ_{t}(K)) ∈ ΓL is constant.

By stabilization φ gives a map φ : G/H_{+} ⟶ G/L_{+}, and hence a map φ^{∗} : M(G/L_{+}) ⟶ M(G/H_{+}). Furthermore if f is an idempotent in A(L) this gives a map M(L, S(f)) = fM(L) ⟶ φ^{∗}(f)M(H) = M(H, S(res_{H}^{K}(f))). Since φ_{Γ} is continuous by (7.2) we may use it to push the sheaf M̃(H) on Γ_{I}H to a sheaf φ_{∗}(M̃(H)) on Γ_{I}L by taking φ_{∗}(M̃(H))(U) := M̃(H)((φ_{Γ})^{−1}(U)).

Corollary 6.3. Any G-map φ : G/H ⟶ G/L induces a map

φ^{∗} : M̃(L) ⟶ φ_{∗}(M̃(H))

of sheaves on Γ_{I}L. Furthermore, this only depends on the homotopy class of φ.

We may restate 6.3 using the category Sh of sheaves of rational vector spaces, the orbit category O and its homotopy category hO.

Corollary 6.4. The Mackey functor M induces a functor M̃ : hO ⟶ Sh.

In fact, we want to go further and consider the category hO-Sh of contravariant functors F : hO ⟶ Sh for which F(H) is a sheaf on Γ_{I}H and for which F(φ) : F(L) ⟶ φ_{∗}F(H). It is easy to check that a map of Mackey functors induces a map of hO-sheaves, so that we have defined a functor

t : IB-mod ⟶ hO-Sh,

given on objects by t(M) = M̃.

In the next section we shall show that the process is reversible.

7. The equivalence of Mackey functors and hO-sheaves.

We show that t is an equivalence of categories by constructing an inverse. This is because, by 5.2, the information corresponding to induction maps is packaged in the decomposition of M̃(H) into stalks together with the action of the finite Weyl groups, whereas the restriction information comes from the maps φ^{∗} of 6.3.

Given an hO-sheaf F we construct an IB-module F′. Since an hO-sheaf includes the values F(H, V) at all idempotents (H, V) it remains to explain how to reconstruct the induced maps for any IB-map θ : (H, V) ⟶ (L, W).

For any object (H, V) we may consider the IB-module (H, V)^{∗} = Hom(·, (H, V)) and then the hO-sheaf t((H, V)^{∗}). Thus we may define F′ on objects by

7.1.

F′(H, V) = hO-Sh(t((H, V)^{∗}), F).

Similarly any IB-map θ : (H, V) ⟶ (L, W) induces a map θ : t((H, V)^{∗}) ⟶ t((L, W)^{∗}) of hO-sheaves. The induced map

θ^{∗} : hO-Sh(t((L, W)^{∗}), F) ⟶ hO-Sh(t((H, V)^{∗}), F)

provides the relevant structure maps. Note that it is not quite clear from this definition that F and F′ agree on objects; this is the main remaining point in proving Theorem A.

The essence is 5.2: all maps of IB are sums of composites of unstable G-maps and idempotents of the Burnside ring.

Proposition 7.2. The forgetful map

IB-mod(M, N) ⟶ hO-Sh(M̃, Ñ)

is an isomorphism for any Mackey functors M and N.

Proof: The map is obviously injective, and we show it is surjective since any transfer map at all can be deduced from those built into the structure of an hO-sheaf. More precisely we must show that if f : M̃ ⟶ Ñ is a map of hO-sheaves and θ : (H, V) ⟶ (L, W) is an IB-map then f commutes with θ.

Now we use the explicit description of maps, summarised in 5. By definition f commutes with G-maps, and with idempotents of Burnside rings (ie restrictions). It also commutes with norm maps, since they are finite sums of translations.

The Yoneda lemma now completes the proof of Theorem A.

Theorem 7.3. (First main reduction.) The functors M ⟼ M̃ and F ⟼ F′ give an equivalence of categories

IB-mod ≃ hO-Sh.

Proof: In fact, we show that there are natural isomorphisms (Ñ)′ ⟶^{≅} N and F̃′ ⟶^{≅} F for any Mackey functor M and hO-sheaf F.

The map (Ñ)′ ⟶ N is given at (H, V) by the evaluation

hO-Sh(t((H, V)^{∗}), Ñ) ⟶ N(H, V)

at the identity of (H, V). The Yoneda lemma states that evaluation is an isomorphism IB((H, V)^{∗}, N) ⟶^{≅} N(H, V), so the result follows from 7.2.

The map t(F′) ⟶ F is given at H on the sections over V using the map

hO-Sh(t((H, V)^{∗}), F) ⟶ F(H, V)

obtained by evaluating at the identity of (H, V). This result therefore follows from the Yoneda lemma argument once we know that t((H, V)^{∗}) represents evaluation at (H, V) in the category of hO-sheaves. This follows from the special case M = (H, V)^{∗}, N = (L, W)^{∗} of 7.2.

From now on we omit the t or tilde and simply write M(H) for the sheaf on Γ_{I}H associated to a Mackey functor M.

8. The topological category of toral chains.

In Section 4 we discussed toral chains at some length. We now want to give a global description and an appropriate notion of proximity.

When we discussed hO-sheaves we considered category O, in which the isomorphisms R_{g} : G/H ⟶ G/H^{g} have the same status as the projections π : G/K ⟶ G/H associated to the inclusion K ⊆ H of one subgroup in another. An alternative would have been to consider the category whose objects are closed subgroups and whose morphisms are inclusions. The isomorphisms are taken into account by using the action of G by conjugation. In some ways this is an easier way to think about an hO-sheaf, and this was reflected in our proofs, but convention and the naturality of hO swayed us to speak as we did.

We could now copy our approach to O, and consider the subcategory WT of O which is generated by the right multiplications R_{g} : G/H ⟶ G/H^{g} and by the projections G/H ⟶ G/Ĥ where H is normal in Ĥ and Ĥ/H is a torus. However it appears clearer to separate the two types of morphism, and we define the toral chain category T C to have as objects the closed subgroups of G, and to have one morphism H ⟶ Ĥ whenever H is normal in Ĥ and Ĥ/H is a torus. By 4.6 this is a category. Note also that G acts on the right of T C by conjugation.

Next we topologize T C. Its space of objects is simply SG, and we define a new topology on it, which we refer to as the f-topology since proximity is based on finite Weyl groups. A morphism in T C is determined by its endpoints so we topologize the space of morphisms as a subspace of S_{f}G × S_{f}G. It is therefore enough to discuss S_{f}G. The essential properties of the f-topology are (i) that the action of G should be continuous, (ii) that the map S_{f}G ⟶ SG should be continuous and (iii) that for every closed subgroup H the maps FH ⟶ S_{f}G should be continuous.

The idea is that the f-topology is constructed from the metric topology on the various spaces FH, by taking into account the conjugations. Thus for a closed subgroup H and ϵ > 0 consider the ball

O(H, ϵ) = {K ∈ FH | d(H, K) < ϵ}

in FH. It is worth emphasizing that subgroups close to H which have infinite Weyl groups are ignored, so that if H is a torus, O(H, ϵ) is a singleton. Now given also a neighbourhood A of the identity in G consider the set

8.1.

O(H, A, ϵ) = ⋃_{a∈A} O(H, ϵ)^{a},

where O(H, ϵ)^{a} is the set of a-conjugates of elements of O(H, ϵ). We take the ftopology to be generated by these sets as H, A and ϵ vary. To understand the topology we make some observations.

Lemma 8.2. If K is a subgroup of H there is a neighbourhood A of e in G (which depends on H and K) so that if a ∈ A, and the G-conjugate K^{a} lies in H then it is also an H-conjugate of K. Hence W_{H}(K) is finite if and only if W_{H}(K^{a}) is finite.

Proof: First we recall from 3.1 that a subgroup K′ which is close enough to K is conjugate in H to a subgroup of K. The first statement follows from the continuity of the conjugation action. The final statement follows immediately.

It is worth warning that in 8.2 the restriction of the conjugating element is necessary, since it is not hard to find two G-conjugate subgroups K and K′ of H with W_{H}(K) finite and W_{H}(K′) infinite. For example if H = O(2) the subgroups K = D_{2} and K′ = C_{2} of order 2 are conjugate in G = SO(3).

The following lemma is useful in several places.

Lemma 8.3. For any subgroup H there is a number ζ > 0 so that if d(H, H∩H^{n}) < ζ then H = H^{n}. In particular if K ⊆ H and d(H, K) < ζ then N_{G}(K) ⊆ N_{G}(H).

Proof: In fact, the subgroups H ∩ H^{n} are the isotropy groups of the action of H on G/H, and hence they fall into finitely many H-conjugacy classes. Each conjugacy class is a compact set in SH, so distinct conjugacy classes have a positive distance separating them.

The second statement follows since if n normalizes K then K ⊆ H ∩ H^{n} ⊆ H, so d(H, H ∩ H^{n}) ≤ d(H, K).

Lemma 8.4. Suppose given H ⊆ L with W_{L}(H) finite.

(i) There is a ζ > 0 so that if K ⊆ H and d(H, K) < ζ then if W_{H}(K) is finite so too is W_{L}(K). (ii) Consequently for any δ > 0 there is an ϵ > 0 with O(H, ϵ) ⊆ O(L, δ), and hence O(H, A, ϵ) ⊆ O(L, A, δ).

Proof: Part (ii) follows from Part (i) and the triangle equality, by taking ϵ ≤ min{ζ, δ}, for then any subgroup K of H with d(H, K) < ϵ has finite Weyl group.

For Part (i) we choose ζ as in 8.3 to guarantee that N_{L}(K) ⊆ N_{L}(H); then N_{L}(K)/N_{H}(K) = N_{L}(K)/(N_{L}(K) ∩ H) ≅ N_{L}(K).H/H is a subset of the finite group W_{L}(H) = N_{L}(H)/H.

Corollary 8.5. For a given subgroup H the sets O(H, A, ϵ) form a base of neighbourhoods of H in S_{f}G.

Proof: Suppose H ∈ O(L, A, δ); it is enough to find ϵ > 0 so that O(H, A, ϵ) ⊆ O(L, A, δ). By definition H = P^{a} for some a ∈ A and some P ∈ FL with d(L, P) < δ. Since translates of neighbourhoods A of the identity generate the topology of G it is enough to consider the case H = P. Now the result is immediate from 8.4 (ii).

Lemma 8.6. The f-topology has the desired properties:

(a) the action of G on T C is continuous and

(b) the map FH ⟶ S_{f}G is continuous.

(c) the map S_{f}G ⟶ SG is continuous.

Proof: (a) Consider the action map G×S_{f}G ⟶ S_{f}G. Suppose that H^{g} ∈ O(L, B, δ). Then O(H^{g}, B, ϵ) ⊆ O(L, B, δ) for suitably small ϵ by 8.4 (ii). Now we may find an open neighbourhood A of e so that A · A ⊆ B. Thus Ag × O(H, A, ϵ) is a neighbourhood of (g, H) mapping into O(L, B, δ) as required.

(b) is clear.

(c) The identity is continuous as a map S_{f}G ⟶ SG because G acts continuously on SG.

Lemma 8.7. If the object set of T C has the f-topology and the morphism set is topologized as a subspace of S_{f}G × S_{f}G then T C is a topological category.

Proof: Continuity of source, target and identity maps is immediate. Continuity of composition follows since morphisms are determined by their endpoints.

We shall want slightly more information about the topology of T C.

Proposition 8.8. The source and target maps are open functions T C_{1} ⟶ T C_{0} = S_{f}G.

Proof: Consider a fixed morphism H ⟶ Ĥ. If Ĥ/H = T, the structure of compact Lie groups shows that Ĥ = (H × T)/Z where Z is a finite central subgroup. A base of neighbourhoods of this morphism is given by the morphisms K ⟶ K̂ with (K, K̂) ∈ O(H, A, δ) × O(Ĥ, A, δ). We must show that the set of K’s occurring form a neighbourhood of H and that the set of K̂’s occurring form a neighbourhood of Ĥ.

We first show that the set of K̂ ⊆ Ĥ with a morphism from K ∈ O(H, δ) contain a set O(Ĥ, ϵ); since G acts continuously on T C this suffices. Since T is central in H ×T, its image \overline{T} in Ĥ is also central. Thus any subgroup K̂ of Ĥ with finite Weyl group must contain \overline{T}. Thus the inverse image of K̂ in H × T contains T and is therefore of the form K × T. Hence K̂ = (K × T)/Z with K ∈ FH; evidently K is normal in K̂ with torus quotient. We show below that K̂ is suitably close to Ĥ.

Conversely, we show that the set of K ⊆ H with a morphism to K̂ ∈ O(Ĥ, δ) contain a set O(H, ϵ); since G acts continuously on T C this suffices. Indeed, if K ∈ FH then we may take K̂ = [(K × T) · Z]/Z. This has normalizer (N_{H}(K) × T)/Z, and thus finite Ĥ-Weyl group. We show below that K is suitably close to H.

Finally we must observe that these two constructions preserve proximity. Indeed, if we use the product metric on H × T then d(K, K′) = d(K × T, K′ × T) and if we use the metric on H × T so that the covering H × T ⟶ (H × T)/Z = Ĥ is a local isometry then for subgroups A and B containing Z d(A, B) = d(A/Z, B/Z). Since these two metrics on H ×T define the same topology we can find ϵ_{1} > 0 to ensure that d(H, K) < ϵ_{1} implies d(Ĥ, K̂) < δ and ϵ_{2} > 0 to ensure that d(Ĥ, K̂) < ϵ_{2} implies d(H, K) < δ as required.

We can use 8.8 to lift the I-topology on Γ_{I}G to S_{I}G. Indeed given any f-open subset U of S_{f}G, we may form its T C-saturation by adjoining the end of any morphism which either starts or finishes in U.

I(U) = {H ⊆ G | H ⊴ L ∈ U with L/H a torus or H ⊵ L ∈ U with H/L a torus}.

The I-topology is the topology generated by the sets I(U). By 8.9, it is enough to take the f-open sets to be the tops of the tubes.

Lemma 8.9. For a fixed subgroup H the sets

IO(Ĥ, A, ϵ) = {H ⊆ G | H ⊴ L with L/H a torus W_{G}(L) finite and

d(L^{a}, Ĥ) ≤ ϵ for some a ∈ A}

form a base of I-neighbourhoods of H where H ⊴ Ĥ and Ĥ/H is a maximal torus in W_{G}(H).

Proof: The sets IO(Ĥ, A, ϵ) are neighbourhoods of H, so by 8.5 it suffices to show that every set I(O(H, B, δ)) contains a set of the form IO(Ĥ, A, ϵ).

This follows from 8.8. Indeed, t(s^{−1}O(H, B, δ)) is an open set containing Ĥ, and thus a set O(Ĥ, A, ϵ). Thus IO(Ĥ, A, ϵ) ⊆ I(O(H, B, δ)) as required.

The idea is really that if the map q could be defined without passage to conjugacy then the I-topology on S_{I}G would be induced from the topology on FG in the same way that that on Γ_{I}G was from ΦG.

Proposition 8.10. (i) The action of G on S_{I}G by conjugation is continuous. (ii) The I-topology on Γ_{I}G is the quotient topology induced by S_{I}G. (iii) The identity function gives a continuous map S_{f}G ⟶ S_{I}G .

Proof: (i) is built into the definition of the I-topology. (ii) Firstly, if U ⊆ Γ_{I}G is open we want to show its inverse image is also open. Thus, if (H) ∈ Γ_{I}G we must find a neighbourhood of H mapping into U. But we necessarily have (Ĥ) ∈ U where Ĥ/H is a maximal torus in W_{G}(H). By definition of the I-topology on Γ_{I}G, U ∩ ΦG is open and hence so is its inverse image in FG. This contains O(Ĥ, A, ϵ) for some A and ϵ > 0, and so the inverse image of U contains the neighbourhood IO(Ĥ, A, ϵ) of H.

Conversely, if the inverse image of U is open and (H) ∈ U we must find a neighbourhood of (H) in U. However, by 8.9, the inverse image contains IO(Ĥ, A, ϵ) for some A and ϵ > 0, and hence IO(Ĥ, G, ϵ). Now, (IO(Ĥ, G, ϵ)/G) ∩ ΦG is open, since ΦG has the quotient topology of the metric topology. Furthermore, IO(Ĥ, G, ϵ) contains all toral chains that end in it, and hence a neighbourhood of (H), by 8.8. (iii) By 8.8 the set of H admitting a morphism into an open set U is open, as is the set of H admitting a morphism from U. It follows that if U is f-open then so too is I(U).

9. Continuous Weyl-toral modules.

In Section 7 we eliminated many redundant transfer maps to replace a Mackey functor M by an hO-sheaf F. In the present section we eliminate redundancy in the sheaves to replace F by a single sheaf with additional structure. The point is that if K has finite H-Weyl group then the stalk F(H)_{(K)} is simply the W_{H}(K) invariant part of the principal stalk V (K) = F(K)_{(K)}. Thus each sheaf F(H) can be recovered provided we record the principal stalks together with information about the actions of finite Weyl groups. To recover the structure maps of an hO-sheaf a little extra information is required. Consider the unstable G-map φ : G/H ⟶ G/L corresponding to an inclusion H ⊆ L, with associated structure map φ^{∗} : F(L) ⟶ φ_{∗}F(H). This induces a map of stalks φ^{∗} : F(L)_{p} ⟶ (φ_{∗}F(H))_{p}. Now if p = (P) is the L-conjugacy class of a subgroup P then the image stalk is (φ_{∗}F(H))_{p} = F(H)((φ_{∗})^{−1}(p)). Since φ_{Γ} is finite to one this is either zero (if p does not represent a subgroup of H) or the sum of the stalks F(H)_{p_{i}} where p_{1}, …, p_{n} are the H-conjugacy classes making up the L-conjugacy class (P). Suppose then that p = (K) for some subgroup K of H. If W_{L}(K) is finite then F(L)_{p} and F(H)_{p} are both retracts of V (K), and so the Weyl group action is enough. However if q_{L}(p) = (K̂) where K ◁ K̂ ⊆ L and K̂/K is a torus then we neeed extra information to give us the induced map V (K̂) ⟶ V (K).

With this motivation we explain how to replace an hO-sheaf F by a single sheaf V = V_{F} on the space of all closed subgroups of G, with additional structure from the action of the Weyl groups and from the restrictions associated to normal inclusions with torus quotients. More precisely V will be an equivariant sheaf on T C. To explain the force of this definition we think of V as an equivariant functor V : T C ⟶ ℚ-mod with the sheaf structure as a continuity condition. In fact, it is continuous in three senses. Firstly we require that the G action is continuous in the sense that if there is a path (g_{t})_{t∈[0,1]} in G with H^{g_{t}} = L for all t then g_{0} = g_{1} as maps V (H) ⟶ V (L). Secondly we require continuity for the topology on the objects of T C in the sense that we give V the structure of a sheaf on S_{f}G, and finally we require continuity for the morphisms in the sense that the maps V (Ĥ) ⟶ V (H) take some representative section in a neighbourhood of Ĥ to a representative section in a neighbourhood of H.

Note that the isotropy group W_{G}(H) of the subgroup H acts on the left of the vector space V (H), and since the G-action is continuous, the action of the Weyl group on V (H) is through π_{0}(W_{G}(H)). We also have restrictions corresponding to each normal inclusion with torus quotient. Accordingly we shall refer to these functors as continuous Weyl-toral modules, and denote the category of them by cts-ℚWT -mod.

Before proceeding more formally, we recall the definition of a sheaf V on a topological category ℂ with object space ℂ_{0} and morphism space ℂ_{1}. The idea is that V is a sheaf on ℂ_{0} together with restriction maps c(f) : V_{q} ⟶ V_{p} on stalks for each morphism f : p ⟶ q. More precisely the restriction maps are given by a map c : d_{0}^{∗}V ⟶ d_{1}^{∗}V of sheaves on ℂ_{1} satisfying the cocyle condition d_{1}^{∗}(c) = d_{0}^{∗}(c)d_{2}^{∗}(c) codifying the transitivity of restriction (ie c(gf) = c(f)c(g)). We are using the notation of the nerve of ℂ, so that d_{0} and d_{1} : ℂ_{1} ⟶ ℂ_{0} are the target and source respectively.

We define a pair of adjoint equivalences

v : hO-Sh ⟶ cts-ℚWT -mod and f : cts-ℚWT -mod ⟶ hO-Sh.

In this section we define v, and in the next we construct f and show that it is a quasi-inverse to v

Construction 9.1. Given an hO-sheaf F, we define an equivariant functor V = v(F) : T C ⟶ ℚ-mod as follows. The values of V are given by

V (H) := F(H)_{(H)},

the left G-action is given by letting g act via R_{g}^{∗} : V (H^{g}) ⟶ V (H), and the restriction map associated to the inclusion H ⟶ Ĥ comes from the stalk of an hO-sheaf structure map:

V (Ĥ) = F(Ĥ)_{(Ĥ)} = F(Ĥ)_{(H)} ⟶ F(H)_{(H)} = V (H).

The continuity of the G-action in the sense described above follows immediately from the fact that an hO-sheaf is defined on the homotopy category of O.

We may now give V the structure of a sheaf on S_{f}G by describing the associated étale space E(V). We then take V (U) to be the space of sections of E(V) over U. First we need a means of extending a germ of F(H) to a germ of F(H′) for any subgroup H′ which is close to H and conjugate to it.

Lemma 9.2. There is a neighbourhood A of e in G so that if H′ = H^{a} = H^{b} for a, b ∈ A then R_{a} ≃ R_{b} : G/H ⟶ G/H′.

Proof: Suppose H′ = H^{a}; the set of group elements b ∈ G so that H′ = H^{b} is the coset N_{G}(H)a. Since N_{G}(H) is a submanifold of G we may find a smooth open ball A′ about e so that A′ ∩ N_{G}(H) is also a ball. Choose A so that A · A^{−1} ⊆ A′. Now if H′ = H^{a} = H^{b} with a, b ∈ A then ab^{−1} ∈ N_{G}(H) ∩ A′, so that R_{a^{−1}b} ≃ R_{e} = 1 : G/H ⟶ G/H. Since R_{a} = R_{b}R_{ab^{−1}} the result follows.

In particular for a, b ∈ A with H′ = H^{a} = H^{b} we have R_{a}^{∗} = R_{b}^{∗} : V (H) ⟶ V (H′), which gives a canonical identification between them.

More generally R_{a} = R_{b} : ΦH ⟶ ΦH′, so that we have canonical identifications between F(H)_{(K)} and F(H′)_{(K′)} where H′ = H^{a} and K′ = K^{a}.

Corollary 9.3. If A is a sufficiently small neighbourhood of e ∈ G, a, b ∈ A and K ∈ O(H, ϵ)^{a} ∩ O(H, ϵ)^{b} then for any section s of F(H) defined on O(H, ϵ) we have

(R_{a^{−1}}^{∗}(s))(K) = (R_{b^{−1}}^{∗}(s))(K),

viewed as an equality in V (K).

We may now describe the topology on the space E(V) = ∐_{H} V (H). We describe the basic open neighbourhoods of points (H, x) with x ∈ V (H). Let s be a section of F(H) defined in a neighbourhood of (H); we may choose ϵ > 0 so that s is defined at (K) whenever W_{H}(K) is finite and d(H, K) < ϵ. We thus have s(K) ∈ V (K)^{W_{H}(K)}, and as we have remarked, since monodromy is trivial we may regard s as defined on subgroups in O(H, ϵ) rather than on conjugacy classes. Now choose A as above to allow canonical identification for A-conjugates of H and define a basic open set for E(V) by

9.4.

U(H, A, ϵ, s) := {(K, s′(K)) | K ∈ O(H, A, ϵ)}.

In this definition if K ∈ O(H, ϵ)^{a} we take H′ = H^{a} and s′ = R_{a^{−1}}^{∗}(s) ∈ F(H′)(O(H′, ϵ)), and use the fact that by 9.3 this is idependent of the choice of a. This is a basis by 8.5.

It is clear from the definition that the projection E(V) ⟶ S_{f}G is continuous. By construction, this étale space has the stalks we are seeking. Before we can verify that the restriction maps of stalks fit together to give a structure map c we need a couple more lemmas.

Lemma 9.5. If we have a diagram

\begin{aligned}Ĥ & ⊇ & K̂ \\ ▽ & & ▽ \\ H & ⊇ & K \end{aligned}

of subgroups with Ĥ/H and K̂/K being tori, and if WĤ(K̂) and W_{H}(K) are finite, then the diagram

F(Ĥ)_{(K̂)} ⊆ F(K̂)_{(K̂)} = V (K̂)
↓  ↓
F(H)_{(K)} ⊆ F(K)_{(K)} = V (K)

commutes.

Proof: We simply apply F to the diagram of subgroups and consider stalks at (K).

Lemma 9.6. Given a subgroup H let ζ > 0 denote the number from 8.3. Then there is a neighbourhood A of e ∈ G, so that if a, b ∈ A and K ∈ O(H, ζ)^{a} ∩ O(H, ζ)^{b}

then the composites G/K ⟶ G/H^{a} ⟶^{R_{a^{−1}}} G/H and G/K ⟶ G/H^{b} ⟶^{R_{b^{−1}}} G/H are

homotopic.

Proof: It is sufficient to show that the two maps G/K ⟶ G/H corresponding to a^{−1}, b^{−1} ∈ (G/H)^{K} are homotopic. Consider the case a = e; the general case is similar. Now gH is fixed by K only if K^{g} ⊆ H; if K ⊆ H this means K ⊆ H ∩ H^{g^{−1}}, and hence if K is close enough to H it follows from 8.3 that g ∈ N_{G}(H). In other words if d(H, K) < ζ then (G/H)^{K} = W_{G}(H). Now, if b is small enough, it lies in the identity component so there is a path from the image of e = a to the image of b in W_{G}(H), which may be lifted to a path from e = a to b in N_{G}(H) since the quotient map N_{G}(H) ⟶ W_{G}(H) is a fibre bundle.

Now let us be explicit about the sheaves d_{i}^{∗}V on the morphism space of T C. The étale space of d_{0}^{∗}V consists of points (H ⊆ Ĥ, x̂) with x̂ ∈ V (Ĥ), and that of d_{1}^{∗}V consists of points (H ⊆ Ĥ, x) with x ∈ V (H). It is thus natural that we should consider the map of stalks given by restriction:

c : (H ⊆ Ĥ, x̂) ⟼ (H ⊆ Ĥ, resĤ ; H x̂).

Theorem 9.7. The maps of stalks give a map c : d_{0}^{∗}V ⟶ d_{1}^{∗}V of sheaves on the morphism space of T C, which satisfies the cocycle condition. Accordingly V and c specify a sheaf on T C. Furthermore the construction is natural in the sense that it extends to a functor v : hO-Sh ⟶ cts-ℚWT -mod.

Proof: The only statement that is not immediate from the construction is that the map c defined on stalks is in fact a continuous map on étale spaces.

Since pullback is defined by taking d_{1}^{∗}V to be the sheaf of sections of d_{1}^{∗}E(V) it is enough, by the universal property of the pullback, to show that the function c′ : d_{0}^{∗}E(V) ⟶ V over d_{1} given by (H ⊆ Ĥ, x̂) ⟼ (H, resĤ ; H(x̂)) is continuous. Indeed, for any section s of F(H) representing x := resĤ ; H(x̂), which we may suppose is defined over O(H, ϵ), we shall find a neighbourhood of (H ⊆ Ĥ, x̂) which maps into U(H, A, ϵ, s). This is sufficient, since the sets U(H, A, ϵ) form a basis for the I-topology.

Let ŝ be a section of F(Ĥ) representing the germ x̂ in a neighbourhood of Ĥ. Now the quotient map πĤ ; H : G/H ⟶ G/Ĥ induces the structure map

resĤ ; H = (πĤ ; H)^{∗} : F(Ĥ) ⟶ (πĤ ; H)_{∗}F(H),

and hence in particular a map of sections over O(Ĥ, ϵ). Thus we obtain a section resĤ ; H(ŝ) whose germ at (H) is x. Thus resĤ ; H(ŝ) = s in some neighbourhood of (H), and, we may choose δ > 0 so that they are both defined and equal throughout O(H, δ).

Now d_{0}^{∗}E(V) consists of pairs (K ⊆ K̂, (K̂, y)) with y ∈ V (K̂), and the topology is defined as a subspace of the product. Choose a neighbourhood Â ⊆ A of e in G small enough for canonical identification of conjugates of Ĥ as in 9.2. Now a point of the basic open set

{O(H, Â, ϵ) × O(Ĥ, Â, ϵ) × U(Ĥ, Â, ϵ, ŝ)} ∩ d_{0}^{∗}E(V)

is of form (K ⊆ K̂, (K̂, ŝ′(K̂))) where K ⊆ K̂ ⊆ Ĥ′ = Ĥ^{b}, and K ⊆ H^{a} with a, b ∈ Â. Now let ŝ′ = R_{b^{−1}}^{∗}(ŝ) and observe that

c′(K ⊆ K̂, (K̂, ŝ′(K̂))) = (K, resK̂ ; Kŝ′(K̂)).

We need to show this is (K, s′(K)), or, more precisely, that

resK̂ ; K((R_{b^{−1}}^{∗}ŝ)(K̂)) = (R_{a^{−1}}^{∗}resĤ ; H ŝ)(K).

By continuity of the conjugation action, it suffices to deal with the case b = e.

We first put ourselves in a position to apply 9.5, by observing that by 3.1 we may choose ϵ > 0 (depending only on H and Ĥ) so that for subgroups L of Ĥ the condition d(L, H) < 2ϵ implies that L is Ĥ-conjugate to a subgroup of H. Also choose Â so that d(H^{a}, H^{a′}) < ϵ for all a, a′ ∈ Â. In particular d(K, H) ≤ d(K, H^{a}) + d(H^{a}, H) < 2ϵ and so K^{n} ⊆ H for some n ∈ Ĥ. Thus we have a diagram of subgroups

Ĥ =  Ĥ  ⊇ K̂
▽  ▽  ▽
H ⟶^{R_{n^{−1}}}  H^{n^{−1}} ⊇ K.

Now we may calculate as follows:

resK̂ ; K{ŝ′(K̂)} = resK̂ ; K(ŝ(K̂)) (since b = e)
= R_{n}^{∗}resK̂^{n} ; K^{n}((R_{n^{−1}}^{∗}ŝ)(K̂^{n}))
= R_{n}^{∗}{resĤ ; H(R_{n^{−1}}^{∗}ŝ)(K^{n})} (by 9.5)
= {resĤ^{n^{−1}} ; H^{n^{−1}}(ŝ)}(K)
= {resĤ^{a} ; H^{a}(R_{a^{−1}}^{∗}ŝ)}(K) (by 9.6)
= {R_{a^{−1}}^{∗}resĤ ; H(ŝ)}(K)
= s′(K).

10. Recovering an hO-sheaf from a continuous Weyl-toral module.

In this section we define the inverse functor f : cts-ℚWT -mod ⟶ hO-Sh. Before giving the definition, we note what we are trying to achieve at the level of stalks. Indeed if V = v(F) then F(H)_{(K)} = V (K)^{W_{H}(K)}; since H acts on the H-conjugacy class (K) and W_{H}(K) is the isotropy group of K we could write this in a more invariant way as

F(H)_{(K)} = {V ( (K) )}^{H},

where V ( (K) ) denotes the set of germs of V about the H-conjugacy class (K) regarded as a closed subset of S_{f}H. This should make the following definition appear natural.

Construction 10.1. Given a continuous Weyl-toral module V we define an hO-sheaf F := f(V). The sheaf F(H) on ΦH is defined on an open subset U ⊆ ΦH by

F(H)(U) := {V (π^{−1}(U))}^{H},

where π = π_{H} : FH ⟶ ΦH denotes passage to H-conjugacy classes. More compactly

F(H) := {π_{∗}(V |_{FH})}^{H}.

The structure maps for R_{g} : G/H ⟶ G/H^{g} are easy to define, since we have a diagram

FH ⟶^{R_{∗}^{g}} FH^{g}
↓  ↓
ΦH ⟶^{R_{∗}^{g}} ΦH^{g},

which gives an isomorphism

R_{g}^{∗} : π_{∗}(V |_{FH^{g}}) ⟶^{≅} R_{∗}^{g}π_{∗}(V |_{FH}).

Furthermore this is equivariant over the homomorphism R_{g} : H ⟶ H^{g}. It is equally easy to define the induced maps of a projection φ : G/H ⟶ G/L on the space of classes (P) where W_{L}(P) is finite. However to describe the complete map we retreat into formalism; we shall show in 10.5 that it achieves the desired maps on stalks.

We have routinely regarded F(H) as a sheaf on Γ_{I}H, but for the present discussion we need a notation for this process. Accordingly we let i denote the inclusions FH ⟶ S_{I}H and also ΦH ⟶ Γ_{I}H: since we have given the space of subgroups the idempotent topology i_{∗} and i^{∗} give the inverse isomorphisms of categories of sheaves on ΦH and Γ_{I}H. The letter j will play the analogous role for spaces associated with L. We need a map

φ^{∗} : j_{∗}F(L) = j_{∗}({π_{∗}(V |_{FL})}^{L}) ⟶ φ_{∗}i_{∗}({π_{∗}(V |_{FH})}^{H}) = φ_{∗}i_{∗}F(H).

The difficulty is that passage to invariants does not commute with i_{∗} since the isotropy groups of various points of q_{H}^{−1}((K)) are different (whereas we know that the sheaves on Γ_{I}H have the same stalk at all points of q_{H}^{−1}((K))). Nonetheless we can construct a map

φ^{∗} : j_{∗}π_{∗}(V |_{FL}) ⟶ φ_{∗}i_{∗}π_{∗}(V |_{FH})

between the sheaves which do not refer to invariants, and check that the relevant subspaces are mapped to each other.

We saw in 8.10 that the identity function is continuous as a map S_{f}G ⟶ S_{I}G, and hence we may form the diagram

10.2.

FH ⟶^{i′} S_{f}H \underrightarrow{φ′} S_{f}L ⟵^{j′} FL
=↓ α ↓ ↓ β ↓=
FH ⟶^{i} S_{I}H \underrightarrow{φ} S_{I}L ⟵^{j} FL
π ↓ π ↓ ↓ π ↓ π
ΦH ⟶^{i} Γ_{I}H \underrightarrow{φ} Γ_{I}L ⟵^{j} ΦL.

Now all relevant sheaves are obtained from V′ = V |_{S_{f} L} in the sense that V |_{FL} = (j′)^{∗}V′ and V |_{FH} = (φ′i′)^{∗}V′. In fact, we can avoid reference to the F spaces and concentrate on the central top square. For this we need to use a construction for sheaves on categories. First recall that if V is a sheaf on the topological category ℂ we may consider the sections compatible with the restriction maps. More precisely, for any open set U ⊆ ℂ_{0} we may define a presheaf V^{ℂ} by taking

V^{ℂ}(U) = {s ∈ V (U) | c(f)(s(q)) = s(p) for all morphisms f : p ⟶ q with p, q ∈ U},

where the notation is suggested by the case where ℂ is the category associated with a topological group. This will not generally give a sheaf on ℂ_{0} since there are ‘long morphisms’, so that locally morphism-compatible sections need not be globally morphismcompatible. In any case, if we have a continuous function α : ℂ_{0} ⟶ X we may form a presheaf α_{#}V := α_{∗}(V^{ℂ}) on X by taking (α_{#}V)(U) = V^{ℂ}(α^{−1}(U)). We apply this to ℂ = T CL and β : S_{f}L ⟶ S_{I}L.

Lemma 10.3. For any sheaf W on T CL

(i) β_{#}W is a sheaf on S_{I}L and (ii) there is a canonical isomorphism

β_{#}W = j_{∗}(j′)^{∗}W.

Proof: (i) The I-open sets are unions of categorical components of T CL, so that sections which are morphism-compatible on each set of an open cover are globally morphism-compatible. The unique patching property follows from that of W. (ii) Indeed, since j_{∗} is inverse to j^{∗} it is enough to observe that j^{∗}β_{#}W = (j′)^{∗}W. Because FL has the subspace topology for both S_{f}L and S_{I}L this simply states that β_{#} does not change stalks in FL. Now, by definition, the stalk of β_{#}W at P is the limit of the spaces W^{T CL}(U) where U runs over I-open neighbourhoods of P, whereas for the stalk of W the limit is over all open neighbourhoods of P; there is thus an evident comparison map. If K and K̂ both lie in U the value of a section at K is determined by that at K̂ using c, so if P is terminal in T CL (i.e. P ∈ FL) the limits agree. Indeed, by 8.9, P has a base of I-open neighbourhoods U = I(U′) so that the sets U′ = {K ∈ I(U′) | K is terminal in U} are f-open and give a neighbourhood base of P in the f-topology.

We need a map

10.4.

j_{∗}(V |_{FL}) = β_{#}V′ ⟶ φ_{∗}α_{#}(φ′)^{∗}V′ = φ_{∗}i_{∗}(V |_{FH}).

Using the unit 1 ⟶ φ_{∗}φ^{∗} of the adjunction it is sufficient to give a map

φ^{∗}β_{#}V′ ⟶ α_{#}(φ′)^{∗}V′.

Now for any sheaf V′′ on S_{I}L, by definition φ^{∗}V′′(U) = lim_{→}Ũ V′′(Ũ), where Ũ runs through open sets of S_{I}L containing U. In particular

φ^{∗}β_{#}V′(U) = lim_{→}Ũ(V′)^{T CL}(Ũ),

whereas α_{#}(φ′)^{∗}V′(U) = lim_{→}Ũ(V′)^{T CH}(Ũ); evidently the inclusions

(V′)^{T CL}(Ũ) ⟶ (V′)^{T CH}(Ũ)

fit together to give the required map of sections.

Applying π_{∗} to 10.4 and using the commutativity of the bottom half of the diagram, we obtain the required map φ^{∗} : j_{∗}π_{∗}(V |_{FL}) ⟶ φ_{∗}i_{∗}π_{∗}(V |_{FH}).

Proposition 10.5. The above map behaves on stalks as described at the start of the section, and hence induces the required structure map

φ^{∗} : F(L) ⟶ φ_{∗}F(H).

Proof: Since the stalks of any sheaf over ΓL over points of q_{L}^{−1}(K) are equal, and since a germ at any one such stalk extends uniquely over q_{L}^{−1}(K), it is enough to check the result at points (K) of ΦL. Since φ_{∗}F(H) is concentrated over points (K) with K ⊆ H, it is enough to check the result at stalks (K) with K ⊆ H ⊆ L and with W_{H}(K) and W_{L}(K) both finite.

Since the I-topology on S_{I}H is the topology as a subspace of S_{I}L the map 1 ⟶ φ_{∗}φ^{∗} is an isomorphism on S_{I}H, and in particular, the map of stalks j_{∗}(V |_{FL})_{K} ⟶ φ_{∗}i_{∗}(V |_{FH})_{K} is the identity. This applies equally well to all H-conjugates of K. Applying π_{∗} we see that any L-equivariant germ around the L-conjugacy class (K) is represented uniquely by an element of V (K)^{W_{L}(K)} which is mapped into V (K)^{W_{H}(K)}; this in turn uniquely represents an H-equivariant germ around the H-conjugacy class (K).

This completes the definition of the functors f and v, and it is now an easy matter to show they define a pair of inverse equivalences.

Theorem 10.6. The functors v and f give an equivalence of categories

hO-Sh ≃ cts-ℚWT -mod.

Proof: We define natural transformations

μ : 1 ⟶ fv and ν : vf ⟶ 1;

it will be clear from the definition that they are both isomorphisms.

By the nature of hO-sheaves, the map F ⟶ fv(F) is determined by the corresponding maps

μ(H) : F(H) ⟶ fv(F)(H) = {π_{∗}v(F)|_{FH})}^{H}

of sheaves on ΦH for the various subgroups H. Indeed, since H acts trivially on F(H), it is enough to construct an H-equivariant map into π_{∗}(v(F)|_{FH}), or equivalently a map

π^{∗}F(H) ⟶ v(F)

of sheaves on FH. Since F(H)_{(K)} is a submodule of V (K) = F(K)_{(K)} we may just use the inclusion. By construction μ(H) is an isomorphism on stalks, and hence an isomorphism of sheaves. To see the various maps μ(H) give a morphism of hOsheaves, it is enough to check diagrams commute on stalks, and this is easily verified from the constructions.

Since v is defined at the level of étale spaces, we may define ν by giving a continuous function ν : E(vf(V)) ⟶ E(V) of étale spaces. But the stalk of vf(V) over H is f(V)(H)_{(H)} = (π_{∗}(V |_{FH})^{H})_{(H)}, and since W_{H}(H) = 1 this is simply V (H); we may thus use the identity on stalks. We must now verify that a basic open set of E(V) is open in E(vf(V)). Suppose then that t is a section of V over the neighbourhood O(H, A, ϵ). The set U′ = U′(H, A, ϵ, t) := {(K, t(K)) | K ∈ O(H, A, ϵ)} is a basic open set of E(V). The point is that this is very similar to 9.4, but here V does not arise from an hO-sheaf, and therefore it makes no sense to require that the section t arise from a section s. By 8.5 it is enough to show such an open set contains a E(vf(V))-neighbourhood of (H, t(H)).

In fact we find a section s of f(V)(H) in a neighbourhood O(H, δ) of (H) so that U(H, B, δ, s) ⊆ U′, where U(H, B, δ, s) is the basic open set of E(vf(V)) defined as in 9.4. Indeed f(V)(H)_{(H)} = V (H)^{W_{H}(H)} = V (H), so we may choose s to be a section representing the germ of t. Finally to verify this works we need to know that the extension of s to a section of E(vf(V)) in a neighbourhood of H has the same germ as t i.e. that s′(K) = t(K), or more precisely (R_{b^{−1}}^{∗}t)(K) = t(K), in some neighbourhood of H. This follows from the continuity of the G action on V which implies that there is an open neighbourhood B × U′(H, B, δ, u) of (e, (H, t(H))) in G × E(V) mapping into U′ = U′(H, A, ϵ, t); thus u and t define the same germ at H so we may suppose u = t and we have precisely the desired relation.

References

1. G.E.Bredon “Introduction to compact transformation groups.” Academic Press (1972).

2. A.W.M.Dress “Contributions to the theory of induced representations.” Lecture notes in maths. 342 (Springer-Verlag) (1973) 183-240

3. T. tom Dieck “Orbittypen und äquivariante Homolgie II.” Arch. Math. 26 (1975) 650-662.

4. T. tom Dieck “Transformation groups and representation theory.” Lecture notes in maths. 766 Springer-Verlag (1979)

5. J.P.C.Greenlees “Some remarks on projective Mackey functors.” J. Pure and Applied Algebra 81 (1992) 17-38.

6. J.P.C.Greenlees “Rational S^{1}-equivariant cohomology theories.” Preprint (1996) 225pp (submitted for publication).

7. J.P.C.Greenlees and J.P.May “Generalized Tate cohomology.” Mem. American Math. Soc. 543 (1995) 176pp.

8. L.G.Lewis, J.P.May and M.Steinberger (with contributions by J.E.McClure) Equivariant stable homotopy theory.” Lecture notes in maths. 1213 Springer-Verlag (1986)

9. H.Lindner “A remark on Mackey functors.” Manuscripta Math. 18 (1976) 273-278.

10. J.Slominska “On the equivariant Chern homomorphism.” Bull. de l’Acad. Pol. des Sci. 24 (1976), 909-913.

11. J.Thevenaz and P.Webb “Simple Mackey functors” Proc 2nd International Group Theory Coference, Bressanone (1989), Rendiconti del Circolo Matematico di Palermo 23 (1990) (Supplement), 299-319.