Splitting rational G-spectra.

David Barnes March 1, 2007

Abstract We give a monoidal Quillen equivalence between the category of rational Gspectra and the product of two ‘simpler’ categories.

Contents

1 Introduction and Main Result 1 2 Rational spectra 3 3 Bousfield localisations of GI𝒮 4 4 The category GI𝒮 _{ℚ} 7 5 The categories F𝒮 and F̃𝒮 10 6 The Splitting 12 Index 19 References 20

1 Introduction and Main Result

Recall the definition of the space FG of subgroups of G with finite index in their normalizer (equipped with the Hausdroff topology, see [LMSM86, Notations 2.3]). Letting G act on this space by conjugation we obtain the orbit space FG/G. Definition 1.1 Let ℱ be a family of subgroups of G and let S be the set of conjugacy classes of subgroups in ℱ with finite index in their normalizer. Define e_{ℱ} : FG/G → ℚ to be characteristic function of S (so e_{ℱ} ([H]) = 1 if H ∈ ℱ and H has finite index in its normalizer and e_{ℱ} is zero on all other points of FG/G). Then, equipping ℚ with the discrete topology, we call such a collection an idempotent family if e_{ℱ} is a continuous map.

An idempotent family ℱ of G corresponds via tom Dieck’s isomorphism

A(G) ⊗ ℚ ≅ C(FG/G, ℚ)

to an idempotent e_{ℱ} ∈ A(G)⊗ℚ of the rational Burnside ring of G and the associated cofamily corresponds to the complement 1 − e_{ℱ} .

Theorem 1.2 For G a compact Lie group and ℱ an idempotent family of subgroups of G, we have a strong monoidal Quillen equivalence of cofibrantly generated, proper, monoidal model categories satisfying the monoid axiom

Δ : GI𝒮 _{ℚ}\overrightarrow{⟵}F𝒮 _{ℚ} × F̃𝒮 _{ℚ} : ×.

In particular we have the following natural isomorphism for any G-spectra X and Y

[X, Y]^{G} ⊗ ℚ ≅ [X ∧ Eℱ_{+}, Y ∧ Eℱ_{+}]^{G} ⊗ ℚ ⊕ [X ∧ Eℱ̃, Y ∧ Eℱ̃]^{G} ⊗ ℚ.

Thus in terms of cohomology theories we have a natural isomorphism

E^{∗} ≅ E_{ℱ}^{∗} ⊕ E∗ ; ℱ̃

for E_{ℱ}^{∗} an Eℱ_{+}-local orthogonal spectrum and E∗ ; ℱ̃ an Eℱ̃-local orthogonal spectrum. The proof also shows that E_{ℱ} ∨ Eℱ̃ is rationally weakly equivalent to E and hence in terms of cohomology theories

E_{∗} ≅ (E_{ℱ} )_{∗} ⊕ (Eℱ̃)_{∗}.

We give some explanation of the result: any idempotent in [𝕊, 𝕊]^{G} will allow us to split the homotopy category, but we have a poor understanding of such idempotents. Thus we rationalise the category so we can apply tom Dieck’s isomorphism and understand these idempotent maps and relate them to subgroups of G. We want to take this splitting of the homotopy category and extend it to a splitting of the model category: this requires the use of families and cofamilies to obtain suitable model categories (either before or after rationalisation). When these ideas coincide, i.e. the family is an idempotent family we have the Quillen adjunction (Δ, ×) and a splitting of the homotopy categories, from here it is a short step to show that these are compatible and give a Quillen equivalence. Any Lie group G has an idempotent family given by the subgroups of the identity component G_{e} as we prove in the lemma below.

For the purposes of this paper we use GI𝒮 , the category of G-equivariant orthogonal spectra, one could use the Quillen equivalent category of G-equivariant S-modules and prove the same result. In fact from knowing the result for one of these categories it is easy to see that it holds for the other. We give a construction of a rational sphere spectrum which works in either category, but the only condition neccesary is a natural isomorphism π_{∗}^{H}(X ∧𝕊_{ℚ}) ≅ π_{∗}^{H}(X)⊗ℚ for any spectrum X and subgroup H, so the reader can use whichever construction they prefer. Furthermore if one uses a model category of rational G-spectra constructed by considering the category of modules over a suitable strict ring spectrum, then our result implies the splitting in that case also.

Lemma 1.3 Consider a short exact sequence of compact Lie groups

1 → G_{e} → G → F → 1

then ℱ, the set of subgroups of G_{e}, is an idempotent family.

Proof First note that since G is compact, F is finite and that ℱ is clearly closed under conjugation (G_{e} is normal) and taking subgroups and hence is a family. Let ℱ′ be the set of conjugacy classes of groups in ℱ with finite index in their normalizer. We first show that ℱ′ is open in FG/G, take H ∈ ℱ′ by [Bre72, Corollary 5.6, Chapter II] we know that if K is in some sufficiently small neighbourhood of H in the space Γ(G), then K is subconjugate to H and so K is a subgroup of G_{e}. It follows that ℱ′ is open in FG/G. Now take (K) to be in FG/G \ ℱ, so there is a g ∈ G \ G_{e} such that K ∩ gG_{e} is non-empty. Then any subgroup L of G that is sufficiently close to K also has a non-trivial intersection with gG_{e} so L is not a subgroup of G_{e}. It follows that ℱ′ is also closed, hence e_{ℱ} is a continuous map FG/G → ℚ which is clearly an idempotent (it has image {0, 1}).

Remark 1.4 In [Gre98] a space S_{f}G is constructed, as a set it consists of the closed subgroups of G, but it does not have the Hausdorff topology. We can relate it to the space FG via the equivalence relation ∼ on S_{f}G which is generated by the following: two subgroups H ⊴ H′ are related by ∼ if the quotient H′/H is a torus. We then have (S_{f}G/ ∼)/G ≅ FG and we use tom Dieck’s isomorphism

A(G) ⊗ ℚ ≅ C(FG/G, ℚ)

to see that an idempotent of A(G)⊗ℚ corresponds to an open and closed G-invariant subspace of S_{f}G that is a union of ∼ classes.

2 Rational spectra

We construct a ‘rational sphere spectrum’ which in turn provides a method to rationalise any spectrum X, by taking the smash product of X with the rational sphere spectrum. Our constructions is a based on the group structure of ℚ, we construct ℚ as a group and translate this into spectra. Begin with a free resolution of the rational numbers as an abelian group

0 → R → F → ℚ → 0.

Since a free abelian groups is a direct sum of copies of ℤ we can rewrite this short exact sequence as

0 → ⊕_{i}ℤ →^{f} ⊕_{j}ℤ → ℚ → 0

and we can think of f as an integer valued matrix. Rationally we have a good understanding of the self maps of the sphere spectrum in the homotopy category:

[𝕊, 𝕊]^{H} ⊗ ℚ = A(H) ⊗ ℚ ≅ C(FH/H; ℚ)

and we know that the identity map is sent to the constant map with value one in C(FH/H; ℚ). It follows that we have an injective ring map ℤ → [𝕊, 𝕊]^{H}. We now construct a map g: ∨_{i} 𝕊 → ∨_{j}𝕊 such that the map of zeroth H-equivariant homotopy groups induced by g is the map f ⊗ 1: R ⊗ A(H) → F ⊗ A(H). For each n ∈ ℤ we have a self map of f̂𝕊 which induces multiplication by n on homotopy groups. We combine these using the pinch map 𝕊 → 𝕊 ∨ 𝕊 to obtain the map written in matrix form below. Taking these as our factors we have our map g.

(n_{1}, n_{2}, …, n_{m}): f̂𝕊 → ⋁_{i=1}^{m} f̂𝕊

Definition 2.1 For the map g as constructed above, the cofibre of g, C_{g} is the rational sphere spectrum and we have a cofibre sequence

∨_{i}f̂𝕊 →^{g} ∨_{j}f̂𝕊 → 𝕊_{ℚ}.

One can show directly that the choices made in this definition are unimportant up to weak equivalence directly. For our purposes we need a weaker, yet more general result which we will prove later

Proposition 2.2 The rational sphere spectrum 𝕊_{ℚ} is cofibrant.

Proof First we draw the picture which gives the definition of a cofibre

[figure: ∨_{i}𝕊; g; j; ∨_{j}𝕊; I ∧ ∨_{i}𝕊; ⌜; C_{g}; ·]

then we note that since a wedge of cofibrant objects is again cofibrant the map j is a cofibration. Cofibrations are preserved by pushouts, so ∨_{j}𝕊 → C_{g} is a cofibration from a cofibrant object to C_{g}.

3 Bousfield localisations of GI𝒮

This section describes how to change the model structure using Bousfield localisation. The following theorem does all the hard work for us and is [MM02, IV, Theorem 6.3].

Definition 3.1 Let E be a cofibrant object of GI𝒮 or a cofibrant based G-space and let X, Y and Z be orthogonal spectra.

(i). A map f: X → Y is an E-equivalence if id ∧ f: E ∧ X → E ∧ Y is a weak equivalence.

(ii). Z is E-local if f^{∗}: [Y, Z]^{G} → [X, Z]^{G} is an isomorphism for all E-equivalences f: X → Y.

(iii). An E-localisation of X is an E-equivalence λ: X → Y from X to an E-local object Y.

(iv). A is E-acyclic if the map ∗ → A is an E-equivalence.

Lemma 3.2 An E-equivalence between E-local objects is a weak equivalence.

Proof Take an E-equivalence between E-local objects: f: A → B. Then f^{∗} as above is an isomorphism, hence there is a [g] ∈ [B, A]^{G} such that [g][f] = f^{∗}[g] = [id]. Thus [f] is an isomorphism in the homotopy category and we can conclude that f is a weak equivalence.

Theorem 3.3 Let E be a cofibrant object of GI𝒮 or a cofibrant based G-space. Then GI𝒮 has an E-model structure whose equivalences are the E-equivalences and whose E-cofibrations are the q-cofibrations of GI𝒮 . The E-fibrant objects are precisely the fibrant E-local objects and E-fibrant approximation constructs a Bousfield localisation f_{X}: X → f̂_{E}X of X at E. The notation for E-model structure on the underlying category of GI𝒮 is L_{E}GI𝒮 or GI𝒮 _{E}.

A set of generators [SS03b, Definition 2.1.2] for GI𝒮 _{E} are the spectra 𝕊 ∧ G/H_{+} where H runs over all subgroups H of G.

Proposition 3.4 The identity map is the left adjoint of a Quillen pair GI𝒮 → L_{E}GI𝒮 .

Lemma 3.5 A spectrum C is E-local if and only if C is a fibrant object of GI𝒮 and [A, C]^{G} = 0 for all E-acyclic spectra A.

Proof We tackle the only if part first. The E-local objects are the fibrant objects of L_{E}GI𝒮 and an E-fibration is a fibration of GI𝒮 , so C is a fibrant object of GI𝒮 . Now take an E-acyclic spectrum A, then since the map ∗ → A is an E-equivalence [A, C]^{G} ≅ [∗, C]^{G} = 0. For the converse, since smashing with E preserves cofibre sequences we can use [MM02, III, Theorem 3.5(vi)] to see that the cofibre of an E-equivalence f: A → B is E-acyclic. Noting that C is fibrant allows us to use the long exact sequence of a cofibre sequence ([MMSS01, Theorem 8.12(vi)]) to see that [B, C]^{G} ≅ [A, C]^{G} as required.

Proposition 3.6 The fibrant replacement functor f̂_{E} of the E-model structure on GI𝒮 is a localisation functor.

Proof We must show the following:

(i). there is a natural transformation f: 1 → f̂_{E}

(ii). f̂_{E}f: f̂_{E} → f̂_{E}^{2} is a weak equivalence

(iii). for any X and Y f_{X}^{∗}: [f̂_{E}X, f̂_{E}Y]^{G} → [X, f̂_{E}Y]^{G} is an isomorphism.

(iv). if f̂_{E}X is weakly equivalent to ∗ then so is f̂_{E}(X ∧ Y) for any cofibrant Y.

The first is easy: f is defined by the functorial factorisation which gives fibrant replacement. The second statement is Lemma 3.2 since f̂_{E}X is E-local for any X. The third is follows from the definitions of E-local and E-equivalence. To prove the last statement we note that since Y is cofibrant it preserves weak equivalences (and hence E-equivalences) thus X ∧ Y is E-equivalent to ∗. We also have an Eequivalence between X ∧ Y and f̂_{E}(X ∧ Y) hence the unique map f̂_{E}(X ∧ Y) → ∗ is an E-equivalence between E-local objects and thus a weak equivalence.

Proposition 3.7 (Pushout product axiom) For two q-cofibrations, f: U → V and g: W → X, the induced map

V ∧ W ⋁_{U∧W} U ∧ X → V ∧ X

is a q-cofibration which is an E-acylic cofibration if either f or g is.

Proof The q-cofibrations are unchanged so we only need consider the case when one of f or g is an E-acylic cofibration. Apply the left-adjoint − ∧ E and conisder the relevant pushout diagram. The result then follows from the the pushout product axiom for GI𝒮 and the fact that E is cofibrant.

Recall that 𝕊, the unit of GI𝒮 , is cofibrant and since the q-cofibrations are the same in GI𝒮 _{E}, 𝕊 is cofibrant in this model category also. This removes the need to show a technical condition that is often listed as part of the pushout product axiom.

Proposition 3.8 For a cofibrant orthogonal G-spectrum X, the functor X ∧ (−) preserves E-equivalences.

Proof Recall that the cofibrations of GI𝒮 are precisely the cofibrations of GI𝒮 _{E}, hence the cofibrant objects are the same also. Begin with f: A → B, an Eequivalence, smashing with E makes f ∧ 1: A ∧ E → B ∧ E a π_{∗}-isomorphism, hence by [MM02, III, Proposition 7.3] f ∧1∧1: A∧E∧X → B∧E∧X is a π_{∗}-isomorphism. Since the smash product is commutative we see that f ∧ 1: A ∧ X → B ∧ X is an E-equivalence.

Proposition 3.9 (Monoid axiom) For i: A → X, an E-acyclic cofibration and any spectrum Y the map i ∧ 1: A ∧ Y → X ∧ Y is an E-equivalence and an hcofibration. Moreover cobase changes and sequential colimits of such maps are Eequivalences and h-cofibrations.

Proof Recall that an h-cofibration is a map with a certain lifting property see [MM02, page 38]. First use [MM02, III, Lemma 7.1] to see that i ∧ 1_{Y} is a hcofibration, to see it is also an E-equivalence we smash with E to obtain i ∧ 1_{E}, a q-cofibration and a π_{∗}-isomorphism. By the monoid axiom for GI𝒮 ([MM02, III, Proposition 7.4]) i ∧ 1_{E} ∧ 1_{Y} is a h-cofibration and a π_{∗}-isomorphism and thus i ∧ 1_{Y} is an E-equivalence as desired.

We have proven the first statement of the proposition, the second follows from [MM02, III, Theorem 3.5] and the fact that h-cofibrations are closed under pushouts and sequential colimits (easily seen from the description of h-cofibrations as maps with a lifting property).

Theorem 3.10 For V ∈ 𝒱 , a map of G-prespectra f: X → Y is an E-equivalence if and only if Σ^{V} f: Σ^{V} X → Σ^{V} Y is an E-equivalence.

Proof This is immediate from Lemma 4.3, [MM02, III, Theorem 3.6] and the fact that the smash product of spectra commutes with smashing with spaces.

Proposition 3.11 The model category GI𝒮 _{E} is both left and right proper.

Proof We show a stronger result: that in the diagrams below, with α a h-cofibration and β a level fibration, the maps labelled l and r are E-equivalences (∼_{E}). The left hand diagram then implies left properness and the right implies right properness.

[figure: A; ⌜; ·; α; ∼_{E}; B; l; W; δ; r; X; ∼_{E}; C; γ; D; Y; β; Z; ⌟; ·]

Begin with the left hand diagram and apply the functor − ∧ E. This preserves h-cofibrations, pushouts and takes E-equivalences to π_{∗}-isomorphisms. Now use [MM02, III, Lemma 4.13] to see that l ∧ 1 is a π_{∗}-isomorphism and hence l is an E-equivalence.

For the right hand diagram we use the proof of [MMSS01, Lemma 9.10] (we could have done the same for the left hand diagram, but we preferred to use our standard method of proof). Let F β be the pullback of β over a point (the dual construction to Z/Y; it is the pre-image of the basepoint of Z). This pullback is constructed levelwise and for each level V, F δ(V) ≅ F β(V) (by simply writing down the definitions of these spaces), thus the map of spectra F δ → F β is an isomorphism.

Now we use [Hat02, 4.65] to see that the fibre of a fibration of spaces is homotopy equivalent to the pre-image of the basepoint of the codomain. The map of spectra β is a level G-fibration, that is for each level V and subgroup H of G, β(V)^{H} is a fibration. The fixed point functor (−)^{H} is a right adjoint ([MM02, III, Equation 1.5]) and F (I, B^{H}) ≅ F (I, B)^{H} for any spectrum B ([MM02, III, Lemma 1.6]). So it follows that the fibre of β(V) is G-equivalent to F (β(V)) = (F β)(V) and thus we have a level G equivalence between the fibre of β and F β. Similarly we have a level G equivalence between the fibre of δ and F δ and thus a level G equivalence between F δ and F β. Now use the natural π_{∗}-isomorphism ν_{δ}: F δ → ΩCδ and ν_{β}: F β → ΩCβ to see that there is a π_{∗}-isomorphism between Cβ and Cδ. Finally apply − ∧ E and compare the long exact cofibre sequences (which are preserved by − ∧ E) to see that r is an E-equivalence.

Remark 3.12 The model category on GI𝒮 is cofibrantly generated, with generating cofibrations I and generating acyclic cofibrations J. The E-local model structure is also cofibrantly generated, with generating cofibrations I and generating acyclic cofibrations J_{E}, see [Hir03, Proposition 4.5.1]. The defining property of J_{E} is that any cofibration that is an E-equivalence is in J_{E}-cof.

One can repeat localisations finitely many times as the theorem below implies.

Theorem 3.13 Let F be a cofibrant object of L_{E}GI𝒮 or a cofibrant based Gspace. Then L_{E}GI𝒮 has a Bousfield F -model structure with weak equivalences the (F ∧ E)-equivalences and cofibrations the q-cofibrations of GI𝒮 . The fibrant objects are precisely the fibrant (F ∧ E)-local objects of GI𝒮 and (F ∧ E)-fibrant approximation constructs a Bousfield localisation f_{X}: X → f̂_{F ∧E}X of X at F ∧ E. The notation for the F -model structure on L_{E}GI𝒮 is L_{F} L_{E}GI𝒮 . Furthermore we have the following identifications of model categories

L_{F} L_{E}GI𝒮 = L_{F ∧E}GI𝒮 = L_{E∧F} GI𝒮 = L_{E}L_{F} GI𝒮 .

4 The category GI𝒮 _{ℚ}

In our case we will take E = 𝕊_{ℚ} we will call this model structure the rational model structure and write L_{𝕊_{ℚ}}GI𝒮 or GI𝒮 _{ℚ} or GI𝒮 /ℚ. We call E-equivalences rational equivalences and E-fibrations will be called rational fibrations. We will avoid the temptation to call a q-cofibration in this new model category a ℚcofibration and continue to call them q-cofibrations. The rational model category of orthogonal G-spectra will be denoted G𝒮_{ℚ} and the set of rational homotopy classes of maps X to Y will be written [X, Y]_{ℚ} Note that since the q-cofibrations agree factorising a map into a q-cofibration followed by a acyclic fibration is the same operation in both GI𝒮 and GI𝒮 _{ℚ}.

Proposition 4.1 Let X be an orthogonal G-spectrum, then for any subgroup H of G, π_{∗}^{H}(X ∧ 𝕊_{ℚ}) ≅ π_{∗}^{H}(X) ⊗ ℚ.

Proof Using the cofibre sequence which defines 𝕊_{ℚ} we have a long exact sequence

⋯ → π_{n}^{H}(X ∧ ∨_{i}𝕊) ⟶^{(1∧g)_{∗}} π_{n}^{H}(X ∧ ∨_{j}𝕊) → π_{n}^{H}(X ∧ 𝕊_{ℚ}) → …

But this is the same as

⋯ → π_{n}^{H}(∨_{i}X) ⟶^{(1∧g)_{∗}} π_{n}^{H}(∨_{j}X) → π_{n}^{H}(X ∧ 𝕊_{ℚ}) → …

which in turn gives

⋯ → ⊕_{i}π_{n}^{H}(X) → ⊕_{j}π_{n}^{H}(X) → π_{n}^{H}(X ∧ 𝕊_{ℚ}) → …

and then we see that this is isomorphic to

⋯ → ⊕_{i}ℤ ⊗ π_{n}^{H}(X) → ⊕_{j}ℤ ⊗ π_{n}^{H}(X) → π_{n}^{H}(X ∧ 𝕊_{ℚ}) → ….

Since the map ⊕_{i}ℤ ⊗ π_{n}^{H}(X) → ⊕_{j}ℤ ⊗ π_{n}^{H}(X) is injective for all n, this long exact sequence splits into short exact sequences and we can conclude that π_{∗}^{H}(X ∧ 𝕊_{ℚ}) ≅ π_{∗}^{H}(X) ⊗ ℚ.

Remark 4.2 There is a similar result for maps in the homotopy category, X is H-small ([SS03b, Definition 2.1.2]) if

[X, ∨_{i}Y_{i}]^{H} ≅ ⊕_{i}[X, Y_{i}]^{H}

for any collection of objects {Y_{i}} it is worthwhile to look at how the change of groups adjunctions interact with this definition. Given the cofibre sequence

∨_{i}𝕊 →^{f} ∨_{j}𝕊 → 𝕊_{ℚ}

we have a new cofibre sequence for any spectra X and Y.

0 → [X, ∨_{i}𝕊 ∧ Y]^{H} →^{f_{∗}} [X, ∨_{j}𝕊 ∧ Y]^{H} → [X, 𝕊_{ℚ} ∧ Y]^{H} → 0

but if X is H-small we obtain an isomorphic cofibre sequence

0 → ⊕_{i}[X, Y]^{H} →^{f_{∗}} ⊕_{i}[X, Y]^{H} → [X, Y]^{H} ⊗ ℚ → 0

which is the same as

0 → [X, Y]^{H} ⊗ R →^{f_{∗}} [X, Y]^{H} ⊗ F → [X, Y]^{H} ⊗ ℚ → 0.

That is, [X, 𝕊_{ℚ} ∧ Y]^{H} ≅ [X, Y]^{H} ⊗ ℚ for H-small X.

Any construction of a rational sphere spectrum such that π_{∗}^{H}(X ∧ 𝕊_{ℚ}) ≅ π_{∗}^{H}(X) ⊗ ℚ for all subgroups H of G will give the same rational equivalences as in our construction. From this it follows that the model category we have defined is independent of the construction of the rational sphere spectrum. The reader may feel that localising with respect to the rational sphere spectrum is different to their normal version of rationalisation. The lemma below deals with this.

Lemma 4.3 For a map f: X → Y the following are equivalent:

(i). f_{∗}^{H}: H_{∗}(X^{H}; ℚ) → H_{∗}(Y^{H}; ℚ) is an isomorphism for all H.

(ii). f_{∗}^{H}: π_{∗}(X^{H}) ⊗ ℚ → π_{∗}(Y^{H}) ⊗ ℚ is an isomorphism for all H.

(iii). f ∧ 1: X ∧ 𝕊_{ℚ} → Y ∧ 𝕊_{ℚ} is a π_{∗}-isomorphism.

(iv). f: X → Y is a π_{∗}^{ℚ}-isomorphism.

Proof The previous proposition shows that the last three statements are the same. The first two are well known to be equivalent, explicitly we use Remark 4.4 to express homotopy groups in terms of maps in the homotopy category of prespectra. Then note that for orthogonal spectra of the form Σ^{∞}G/K_{+}, the homotopy groups are simply the stable homotopy groups of G/K_{+} and when rationalised this is the same as rational homology. The statement then follows from the fact that spectra of the form Σ^{∞}G/K_{+} are a localising subcategory for orthogonal spectra [SS03b, Example 3.4(i)].

Remark 4.4 For H any subgroup of G, and any integer q,

[Σ^{q}𝕊 ∧ G/H_{+}, X]^{G} = π_{q}^{H}(X).

This follows from [MM02, III, Theorem 4.16] which expresses maps of orthogonal spectra in terms of their underlying prespectra and [Ada74, III, Proposition 2.8] which relates these maps in the homotopy category to homotopy groups. In particular, A(H) = [𝕊, 𝕊]^{H} ≅ [𝕊 ∧ G/H_{+}, 𝕊]^{G}.

Now we turn our attention to proving some useful results on the structure of GI𝒮 _{ℚ} and its smash product. We write f̂_{ℚ} for fibrant replacement in this category and maps in the homotopy category between spectra X and Y will be written [X, Y]^{ℚ}. We restate [MM02, III, Theorem 3.5] in terms or rational homotopy for clarity.

Proposition 4.5 (i). A map of G-prespectra is a π_{∗}^{ℚ}-isomorphism if and only if its suspension is a π_{∗}^{ℚ}-isomorphism.

(ii). The homotopy groups of a wedge of prespectra are the direct sums of the homotopy groups of the wedge summands, hence a wedge of π_{∗}^{ℚ}-isomorphisms of G-prespectra is a π_{∗}^{ℚ}-isomorphism.

(iii). If i: A → X is an h-cofibration and a π_{∗}^{ℚ}-isomorphism of G-prespectra and f: A → Y is any map of G-prespectra, then the cobase change j: Y → X ∨_{A} Y is a π_{∗}^{ℚ}-isomorphism.

(iv). If i and i′ are h-cofibrations and the vertical arrows are π_{∗}^{ℚ}-isomorphisms is the diagram below, then the induced map of pushouts is a π_{∗}^{ℚ}-isomorphism.

[figure: X; A; i; Y; X′; A′; i′; Y′]

(v). If X is the colimit of a sequence of h-cofibrations X_{n} → X_{n+1}, each of which is a π_{∗}^{ℚ}-isomorphism, then the map from the initial term X_{0} into X is a π_{∗}^{ℚ}- isomorphism.

(vi). For any map f: X → Y of G-prespectra and any H ⊂ G, there are natural long exact sequences

\xymatrix{… \ar[r] & π_{q}^{H}(F f) ⊗ ℚ \ar[r] & π_{q}^{H}(X) ⊗ ℚ \ar[r] & π_{q}^{H}(Y) ⊗ ℚ \ar[r] & π_{q−1}^{H}(F f) ⊗ ℚ \ar[r] & …, \\ … \ar[r] & π_{q}^{H}(X) ⊗ ℚ \ar[r] & π_{q}^{H}(Y) ⊗ ℚ \ar[r] & π_{q}^{H}(Cf) ⊗ ℚ \ar[r] & π_{q−1}^{H}(X) ⊗ ℚ \ar[r] & …,}

and the natural map ν: F f → ΩCf is a π_{∗}-isomorphism.

Proof This follows from [MM02, III, Theorem 3.5], Lemma 4.3, the fact that ℚ is flat (that is, − ⊗ ℚ preserves exact sequences) and that smashing with 𝕊_{ℚ} is a left adjoint.

Theorem 4.6 For any X and Y, [X, Y]^{ℚ} is a rational vector space and when X is G-small, there is a natural isomorphism

[X, Y]^{ℚ} ≅ [X, Y]^{G} ⊗ ℚ.

Proof For each integer n we have a self-map of f̂𝕊 which represents multiplication by n at the model category level, applying (−)∧X we obtain a self-map of f̂𝕊∧X. Since this map is an isomorphism of rational homotopy groups it is a weak equivalence of GI𝒮 _{ℚ} and so in the homotopy category of GI𝒮 _{ℚ} we have an isomorphism n: X → X hence [X, Y]^{ℚ} is a rational vector space.

By definition of the fibrant replacement in GI𝒮 _{ℚ} we have a π_{∗}-isomorphism X ∧ 𝕊_{ℚ} → f̂_{ℚ}X ∧ 𝕊_{ℚ} and by Proposition 3.4 (since cofibrant replacement is the same in both categories) we have [X, Y]^{ℚ} ≅ [X, f̂_{ℚ}Y]^{G}. Now we have the following series of natural isomorphisms:

[X, Y]^{G} ⊗ ℚ ≅ [X, Y ∧ 𝕊_{ℚ}]^{G}  by Remark 4.2,
≅ [X, f̂_{ℚ}Y ∧ 𝕊_{ℚ}]^{G} by the comments above,
≅ [X, f̂_{ℚ}Y]^{G} ⊗ ℚ by Remark 4.2,
≅ [X, Y]^{ℚ} ⊗ ℚ  by Proposition 3.4,
≅ [X, Y]^{ℚ}

Note that in GI𝒮 _{ℚ}, the map Y → Y ∧ 𝕊_{ℚ} is a weak equivalence.

Corollary 4.7 If the spectrum X is H-small in GI𝒮 then it is H-small in GI𝒮 _{ℚ}. In particular, the generators of GI𝒮 _{ℚ} are all G-small, or equivalently, 𝕊 is H-small in GI𝒮 _{ℚ} for all H.

Corollary 4.8 Fibrant objects in GI𝒮 _{ℚ} have rational homotopy groups. In general, maps in to an 𝕊_{ℚ}-local object in the homotopy category of GI𝒮 form a rational vector space.

5 The categories F𝒮 and F̃𝒮

Definition 5.1 A collection of subgroups of G, ℱ, is called a family if it is closed under conjugation and taking subgroups. The complement of this set in the set of all subgroups of G is a cofamily, the cofamily associated to the family ℱ will be denoted ℱ′.

We have the universal ℱ -space Eℱ. This is a G-CW complex with all cells of orbit type G/H, defined by the universal property: Eℱ^{H} is a contractible space for H ∈ ℱ and Eℱ^{H} = ∅ for H ∉ ℱ. We then have the fancily named isotropy separation cofibering that is, we define a map ε: Eℱ_{+} → S^{0} by using the contraction Eℱ = Eℱ^{e} → ∗ and then adding a disjoint point to both. The cofibre of ε, C_{ε}, will be called the universal ℱ′-space and will be written Eℱ̃. Applying the basic fact: C_{f^{H}} ≅ (C_{f})^{H}, we can then see by a simple calculation that Eℱ̃^{H} is a contractible space for H ∈ ℱ and Eℱ̃^{H} = S^{0} for H ∉ ℱ.

Remark 5.2 It is worth noting that since Σ^{∞} is a left Quillen functor and commutes with (−) ∧ A for any G-space A, it preserves cofibre sequences, that is, Σ^{∞}C_{f} ≅ C_{Σ^{∞}f}.

We ask the reader to recall the material of [MM02, IV, Section 6], where for a family of subgroups ℱ, the G-space Eℱ_{+} is used to localise GI𝒮 , we use the notation F𝒮 for L_{Eℱ} GI𝒮 . They also use the corresponding space Eℱ̃associated to the cofamily ℱ′ to construct F̃𝒮 := L_{Eℱ̃}GI𝒮 .

Theorem 5.3 The category GI𝒮 with weak equivalences the Eℱ_{+}-equivalences, cofibrations the q-cofibrations and fibrations as necessary is a cofibrantly generated, proper, monoidal model category satisfying the monoid axiom with Eℱ_{+}-fibrant objects the fibrant Eℱ_{+}-local objects. We write F𝒮 for GI𝒮 with this model structure.

Theorem 5.4 The category GI𝒮 with weak equivalences the Eℱ̃-equivalences, cofibrations the q-cofibrations and fibrations as necessary is a cofibrantly generated, proper, monoidal model category satisfying the monoid axiom with Eℱ̃-fibrant objects the fibrant Eℱ̃-local objects. We write F̃𝒮 for GI𝒮 with this model structure.

Proposition 5.5 The following conditions on a map f: X → Y are equivalent.

(i). f is an Eℱ_{+}-equivalence.

(ii). f_{∗}: π_{∗}^{H}(X) → π_{∗}^{H}(Y) is an isomorphism for all H ∈ ℱ.

A map satisfying the second condition is called an ℱ -equivalence. This definition is important due to the following result which is [MM02, IV, Theorem 6.5 and Theorem 6.9].

Theorem 5.6 The category GI𝒮 has an ℱ -model structure with weak equivalences the ℱ -equivalences, cofibrations the ℱ -cofibrations and fibrations as defined by the lifting property. This is a compactly generated proper model structure and the identity functor gives the left adjoint of a Quillen equivalence from the ℱ -model structure on GI𝒮 to F𝒮 .

For q-cofibrations we start with a set, I, of G-space maps of the form G/H_{+} ∧ A → G/H_{+} ∧ B and then take F I to be the generating cofibrations. For the ℱ - cofibrations we only consider such maps with H ∈ ℱ, and then apply F to obtain the generating ℱ -cofibrations. The same process is also applied to the set J which constructs the generating trival cofibrations to construct the generating acyclic ℱ - cofibrations. We let [X, Y]^{ℱ} denote the set of maps between spectra X and Y in the homotopy category of F𝒮 (or GI𝒮 with the F-model structure) and similarly we use [X, Y]^{ℱ′} for F̃𝒮 .

Theorem 5.7 For any orthogonal spectra X and Y we have a natural isomorphism

[X, Y]^{ℱ} ≅ [X ∧ Eℱ_{+}, Y ∧ Eℱ_{+}]^{G}

and thus we have an equivalence of categories between HoF̃𝒮 and the full subcategory of objects X ∧ Eℱ_{+} in HoGI𝒮 . The map ρ: X → F (Eℱ_{+}, X) induced by Eℱ_{+} → S^{0} gives an Eℱ_{+}-equivalence from X to an object that is (up to π_{∗}- isomorphism) Eℱ_{+}-local.

This is [MM02, IV, Theorem 6.11] and [MM02, IV, Theorem 6.14] below is the equivalent for Eℱ̃. These results are extremely useful, they tell us exactly how the homotopy categories of the localised model categories behave.

Theorem 5.8 For any orthogonal spectra X and Y we have a natural isomorphism

[X, Y]^{ℱ′} ≅ [X ∧ Eℱ̃, Y ∧ Eℱ̃]^{G}

and thus we have an equivalence of categories between HoF̃𝒮 and the full subcategory of objects X ∧ Eℱ̃in HoGI𝒮 .

6 The Splitting

Definition 6.1 Recall the definition of the product model category: given model categories M_{1} and M_{2} we can put a model category structure on M_{1} × M_{2}. A map (f_{1}, f_{2}) is a cofibration, weak equivalence or fibration if and only if f_{1} is so in M_{1} and f_{2} is so in M_{2}.

Remark 6.2 Note that if M_{1} and M_{2} satisfy any of the following: left properness, right properness, the pushout product axiom, the monoid axiom or cofibrant generation then so does M_{1} × M_{2}.

Proposition 6.3 Given E, F cofibrant orthogonal G-spectra or cofibrant G-spaces, and a model category M such that L_{E}M and L_{F} M exist (such as GI𝒮 or some localisation of it), then we have a strong monoidal Quillen adjunction

Δ : M ⇄ L_{E}M × L_{F} M : ×

(we follow the usual convention of placing the left adjoint on top).

Proof Take a map f: X → Y in M, then Δ(f) = (f, f): (X, X) → (Y, Y). For a map (a, b): (A, B) → (C, D), ×(a, b) is given by a × b: A × B → C × D. That these are an adjoint pair is easy to see since all of the isomorphisms and equalities below are natural.

L_{E}M × L_{F} M((X, X), (A, B)) = L_{E}M(X, A) × L_{F} M(X, B)
= M(X, A) × M(X, B)
≅ M(X, A × B)

If f is a (acyclic) cofibration of M then it is clear that (f, f) is a (acyclic) cofibration of L_{E}M × L_{F} M, thus Δ is a left Quillen functor. All that remains is to show that this is a strong monoidal adjunction, which amounts to noting that the unit of M is cofibrant and that Δ is strong monoidal [SS03a, Definition 3.6] which is obvious from the definitions.

Proposition 6.4 Take a map f: E → F, then a map g: X → Y which is an Eequivalence and a C_{f}-equivalence is also an F -equivalence.

Proof Consider the map of cofibre sequences

\xymatrix{E ∧ X \ar[r] \ar[d]^{1_{E}∧f} & F ∧ X \ar[r] \ar[d]^{1_{F} ∧f} & C_{f} ∧ X \ar[d]^{1_{C_{f}} ∧f} \\ E ∧ Y \ar[r] & F ∧ Y \ar[r] & C_{f} ∧ Y}

This gives a map of long exact sequences of homotopy groups [MM02, III, Theorem 3.5(vi)] and we then apply the five lemma.

In fact, the above proof shows that for any cofibre sequence of cofibrant objects, E → F → G, one only has to check that two out of three of the maps g ∧ 1_{E}, g ∧ 1_{F} and g ∧ 1_{G} are weak equivalences to conclude that all three are weak equivalences. Also note that if E and F are cofibrant, then so is C_{f}.

Corollary 6.5 Take a family of subgroups ℱ, then a map f: X → Y is a π_{∗}- isomorphism if and only if f is an Eℱ_{+}-equivalence and an Eℱ̃-equivalence.

Proof We have the cofibre sequence Eℱ_{+} → 𝕊 → Eℱ̃and an 𝕊-equivalence is a π_{∗}-isomorphism.

Lemma 6.6 For any family ℱ, π_{∗}^{H}(Eℱ_{+} ∧ Eℱ̃) = 0.

Proof This is true at the space level by the calculation below and then we apply the left Quillen functor Σ^{∞} and Ken Brown’s lemma [Hov99, 1.1.12] to see that this also holds at the level of spectra.

\begin{aligned}π_{∗}^{H}(Eℱ_{+} ∧ Eℱ̃) & = & π_{∗}((Eℱ_{+} ∧ Eℱ̃)^{H}) \\ & = & π_{∗}(Eℱ_{+}^{H} ∧ Eℱ̃^{H} ) & ≅ & \{ \begin{matrix}π_{∗}(S^{0} ∧ ∗) & H ∈ ℱ \\ π_{∗}(∗ ∧ S^{0}) & H ∉ ℱ \end{matrix} \end{aligned}

Lemma 6.7 Let E and F be cofibrant orthogonal spectra such that E ∧F has trivial homotopy groups. Then the following are equivalent

(i). L_{E}X_{F} → ∗ is a π_{∗}-isomorphism

(ii). X_{F} → ∗ is an E-equivalence

(iii). X_{F} ∧ E → ∗ is a π_{∗}-isomorphism

where X_{F} is any F -local object, and L_{E} is E-local replacement (fibrant replacement in L_{E}GI𝒮 ).

Proof The last two are equivalent by definition, the first is equivalent since X_{F} → ∗ is an E-equivalence if and only if L_{E}X_{F} → ∗ is an E-equivalence (X_{F} → L_{E}X_{F} is always an E-equivalence) and an E-equivalence between E-local objects is a π_{∗}- isomorphism.

We now turn to a case where the adjunction of the Proposition 6.3 is a Quillen equivalence. We must use the rationalised model category of orthogonal spectra, so that we can apply the following rational equivariant information. We also use the rationalised categories F𝒮 _{ℚ} and F̃𝒮 _{ℚ}, which are constructed in the same way from F𝒮 and F̃𝒮 (respectively) as GI𝒮 _{ℚ} is from GI𝒮 We define A(G), Burnside ring of G, to be [𝕊, 𝕊]^{G}. By our understanding of GI𝒮 _{ℚ} we know that A(G) ⊗ ℚ := [𝕊, 𝕊]^{ℚ} so [a], an element of the Burnside ring can, be represented by a self map a of f̂_{ℚ}𝕊 and hence for any spectrum X we have 1 ∧ a, a self-map of X ∧ f̂_{ℚ}𝕊. We would now like to use this information to let A(G) ⊗ ℚ act on GI𝒮 _{ℚ}, by defining a spectrum aX for any element of the Burnside ring [a] and any spectrum X. Note that the rational homotopy group of any spectrum is an A(G) ⊗ ℚ module since X → X ∧ f̂_{ℚ}𝕊 induces an isomorphism of rational homotopy groups for any spectrum X.

Definition 6.8 For a sequence of maps

\xymatrix{X_{0} \ar[r]_{f_{0}} & X_{1} \ar[r]_{f_{1}} & X_{2} \ar[r]_{f_{2}} & X_{3} \ar[r]_{f_{4}} & X_{4} \ar[r]_{f_{5}} & …}

we define the homotopy sequential colimit or telescope to be the colimit of the top row of the following diagram.

[figure: X_{0}; g_{0}; h_{0}; =; X′_{1}; g_{1}; h_{1}; ∼; X′_{2}; g_{2}; h_{2}; ∼; X′_{3}; g_{3}; h_{3}; ∼; X′_{4}; g_{4}; h_{4}; ∼; …; X_{0}; f_{0}; X_{1}; f_{1}; X_{2}; f_{2}; X_{3}; f_{4}; X_{4}; f_{5}; …]

Where h_{i+1} ∘ g_{i} is a factorisation of f_{i} ∘ h_{i} into a q-cofibration followed by an acylic fibration. We denote this construction by HoColim_{i} X_{i}.

Proposition 6.9 For a diagram as below (in any of GI𝒮 , F𝒮 , F̃𝒮 or their rationalisations), with each f_{i} and g_{i} a h-cofibration, the induced map of colimits is a weak equivalence.

[figure: X_{0}; g_{0}; h_{0}; ∼; X_{1}; g_{1}; h_{1}; ∼; X_{2}; g_{2}; h_{2}; ∼; X_{3}; g_{3}; h_{3}; ∼; X_{4}; g_{4}; h_{4}; ∼; …; Y_{0}; f_{0}; Y_{1}; f_{1}; Y_{2}; f_{2}; Y_{3}; f_{4}; Y_{4}; f_{5}; …]

Proof We begin by extending the diagram to include the cofibres of the h_{i}, we have l_{i}: Y_{i}+→ C_{h_{i}}, which is a h-cofibration and k_{i}: C_{h_{i}}+→ C_{h_{i+1}} which we now prove to be a h-cofibration and a weak equivalence. It is a h-cofibration by inspecting the relevant diagram and by our strong form of properness in these categories we see that each C_{h_{i}} is weakly equivalent to an acyclic spectrum and thus k_{i} must be a weak equivalence. We now apply the variant of [MM02, III, Theorem 3.5(v)] that applies to our category to see that Colim_{i}(C_{h_{i}}) is acyclic. Apply the standard yoga of colimits and left adjoints (recall that CX_{i} = X_{i} ∧ I) to see that Colim_{i}(C_{h_{i}}) ≅ C_{Colim_{i}(h_{i})} so both are acyclic and thus we can conclude that Colim_{i}(h_{i}) is a weak equivalence.

Corollary 6.10 From this it follows that the choice of factorisation in the definition of a telescope is unimportant, as different choices will be weakly equivalent.

Proposition 6.11 If X_{0} is cofibrant then HoColim_{i} X_{i} is cofibrant. If either of X_{0} or Y is cofibrant then there is a weak equivalence

HoColim_{i}(Y ∧ X_{i}) → Y ∧ HoColim_{i}(X_{i}).

Proof This follow by the standard manipulations of the definitions.

Definition 6.12 Let e be a self-map of f̂_{ℚ}𝕊 such that [e] ∈ A(G) ⊗ ℚ is an idempotent and thus e ∘ e ≃ e. We define eX to be the homotopy colimit of

X ∧ f̂_{ℚ}𝕊 ⟶^{1∧e} X ∧ f̂_{ℚ}𝕊 ⟶^{1∧e} X ∧ f̂_{ℚ}𝕊 ⟶^{1∧e} X ∧ f̂_{ℚ}𝕊 ⟶^{1∧e} …,

the map 𝕊 → f̂_{ℚ}𝕊 then provides a map X → eX.

Proposition 6.13 Let e be an idempotent of the rational Burnside ring and let X be any orthogonal spectrum then we have the following commutation relation.

π_{∗}^{H}(eX) ⊗ ℚ ≅ eπ_{∗}^{H}(X) ⊗ ℚ

In general, for a sequential colimit diagram X_{0} ⟶^{f_{0}} X_{1} ⟶^{f_{1}} X_{2} ⟶^{f_{2}} X_{3} ⟶^{f_{3}} we understand the homotopy groups of the homotopy colimit.

π_{∗}^{H}(HoColim_{i} X_{i}) ≅ Colim_{i} π_{∗}^{H}(X_{i})

Proof Note that for homotopy colimit of an idempotent the algebraic colimit of the obvious sequential diagram is the same as the image of the idempotent. This result is simply the statement that the generators 𝕊 ∧ G/H_{+} are G-small, we give an argument below. which shows that the suspension spectrum of a compact (in the standard topological sense) G-trivial space Z is compact. For any sequential diagram of h-cofibrations we have the following sequence of natural isomorphisms

[Σ^{∞}Z, Colim_{i} X_{i}]^{G} ≅ [Z, Ω^{∞} Colim_{i} X_{i}]^{G𝒯}
≅ [Z, Colim_{i} X_{i}(0)]^{G𝒯}  colimits are defined levelwise,
≅ [Z, (Colim_{i} X_{i}(0))^{G}]^{𝒯} taking G-fixed points,
≅ [Z, Colim_{i}(X_{i}(0)^{G})]^{𝒯} since the maps are inclusions,
≅ Colim_{i}[Z, X_{i}(0)^{G}]^{𝒯}  by the compactness of Z,
≅ Colim_{i}[Z, X_{i}(0)]^{G𝒯}
≅ Colim_{i}[Σ^{∞}Z, X_{i}]^{G}.

We remind the reader that h-cofibrations of spectra are levelwise h-cofibrations of G-spaces ([MMSS01, page 457]) and thus are levelwise closed inclusions ([Hov99, page 50-52]). Hence can we utilise the fact that colimits of inclusions commute with taking G-fixed points ([MM02, III, Lemma 1.6]). The commutation relation we use above for spaces is on [MMSS01, page 458] and the fact that homotopy for spaces is given in terms of maps out of the standard cylinder object Z ∧ I_{+} which is compact when Z is.

Proposition 6.14 The map X → X ×X → eX ×(1−e)X is a rational equivalence for any X and any idempotent e of the Burnside ring.

Proof They certainly have isomorphic rational homotopy groups and examination of the maps involved shows that the result is true.

We will need [Gre98, Proposition 3.4], stated below as a definition, for the coming proposition.

Definition 6.15 Define Γ(G) to be the set of conjugacy classes of subgroups with topology induced from the Hausdorff metric on the set of closed subgroups of G. The support of an map f ∈ [𝕊, 𝕊]^{G}, S(f) is defined to be {(K) ∈ Γ(G) | α(f)(K) ≠ 0} where α: [𝕊, 𝕊]^{G} → C(Γ(G), ℚ), α(f)(H) is the degree of the H-fixed point map.

Proposition 6.16 Let X be a cofibrant orthogonal spectrum, and take [e], an idempotent of the rational Burnside ring of G with support S(e). Then for any (K) ∈ S(e), Φ^{K}(eX) is weakly equivalent to Φ^{K}(X), otherwise Φ^{K}(eX) is acyclic.

Proof We perform much of the proof in the category of rational orthogonal Kspectra, that is, we apply the forgetful functor ι^{∗} to all the spectra involved, (ι is the inclusion of L in G). This functor ι^{∗} preserves fibrations, q-cofibrations, weak equivalences and more, [MM02, V, Lemma 2.2] so this presents us with no difficulty.

In general when write Φ^{H}(X) for a G-spectrum X and H a subgroup of G, we mean Φ^{H}(ι^{∗}X). Also note that ι^{∗}(eX) is weakly equivalent to ι^{∗}(e)ι^{∗}(X) as rational K-spectra.

We now apply Proposition 6.11 to obtain a weak equivalence (of K-spectra) eX → e𝕊 ∧ X between cofibrant objects and hence (by Ken Brown’s lemma) we have a weak equivalence of non-equivariant spectra Φ^{K}(eX) → Φ^{K}(e𝕊∧X). Using [MM02, V, Section 4] we have a further weak equivalence

Φ^{K}(e𝕊) ∧ Φ^{K}(X) → Φ^{K}(e𝕊 ∧ X)

and we know that Φ^{K} commutes with colimits of h-cofibrations so that Φ^{K}(e𝕊) ≃ Φ^{K}(e)Φ^{K}𝕊.

Since 𝕊 is the suspension spectrum of S^{0} which is G-fixed, Φ^{K}𝕊 ≅ 𝕊, now we only have to understand the map Φ^{K}(e). The construction of tom Dieck’s isomorphism

A(G) ⊗ ℚ ≅ C(FG/G, ℚ)

takes a map of spectra f to the map H ↦ degΦ^{H}(f), where degree has the usual algebraic topology definition in terms of homology, which coincides with homotopy (since we are working rationally and stably). Thus

π_{∗}(Φ^{K}(e)𝕊) ≅ Φ^{K}(e)_{∗}π_{∗}(𝕊) = degΦ^{K}(e)π_{∗}(𝕊)

which is either zero or π_{∗}(𝕊) according to whether or not K ∈ S(e). This last statement follows from [Gre98, Lemma 4.1 and Lemma 4.3] which imply that degΦ^{K}(e) = 1 if and only if K ∈ ℱ. When K ∉ S(e) we are smashing Φ^{K}(X) with an acyclic object, hence (since Φ^{K} preserves q-cofibrations and X is cofibrant) Φ^{K}(eX) is an acyclic non-equivariant orthogonal spectrum. For the other case we have a weak equivalence of non-equivariant spectra 𝕊 → Φ^{K}(e)𝕊 (since Φ^{K}(e) is a π_{∗}- isomorphism) and so we have our result.

Proposition 6.17 Let f: X → Y be a map of cofibrant G-equivariant orthogonal spectra, then (using ι^{∗}: GI𝒮 → H𝒮 that we can apply Φ^{H}) Φ^{H}f is a weak equivalence of non-equivariant orthogonal spectra if and only if f is a weak equivalence of G-equivariant orthogonal spectra.

Proof This statement above appears to be well known and is one of the reasons why geometric fixed point functors are important. We apologise for the long winded statement above but we feel that the clarity is essential here.

The ‘if’ part is immediate by Ken Brown’s lemma applied to Φ^{H}. The converse begins with [MM02, V, Proposition 4.17] to see that the hypothesis is equivalent to the maps

(R(X ∧ Eℱ̃_{H}))^{H} ⟶ (R(Y ∧ Eℱ̃_{H}))^{H}

being weak equivalences of non-equivariant spectra for all H. The family ℱ_{H} is the collection of subgroups of H which do not contain H: it is the family of proper subgroups of H. We use R to denote fibrant replacement in the category of Gequivariant spectra.

The result now follows by a series of natural isomorphisms

[Σ^{n}𝕊, (R(Y ∧ Eℱ̃_{H}))^{H}] ≅ [Σ^{n}𝕊 ∧ G/H_{+}, R(Y ∧ Eℱ̃_{H})]^{G}
≅ [Σ^{n}𝕊 ∧ G/H_{+}, Y ∧ Eℱ̃_{H}]^{G}
≅ [Σ^{n}𝕊, Y ∧ Eℱ̃_{H}]^{H}
≅ [Σ^{n}𝕊, Y]^{H}

where the last isomorphism holds by noting that S^{0} → Eℱ̃_{H} is a π_{∗}^{H}-isomorphism, and these are preserved by smashing with cofibrant objects (this is contained in the proof that this operation preserves weak equivalences).

Corollary 6.18 The above proposition also holds in the rationalised case: f: X → Y is a rational equivalence of G-spectra if and only if Φ^{H}f is a rational equivalence of non-equivariant spectra for all subgroups H of G.

Proof This follows by the fact that Φ^{H}𝕊_{ℚ} = 𝕊_{ℚ} for all H and the usual properties of Φ^{H} [MM02, V, Section 4].

Corollary 6.19 Let ℱ be an idempotent family of subgroups of G, with corresponding cofamily ℱ′. Let e_{ℱ} and e_{ℱ′} = 1 − e_{ℱ} be the idempotents associated to these subspaces. Then the above results show that Eℱ_{+} → e_{ℱ} S^{0} given by the composition of Eℱ_{+} → S^{0} and S^{0} → e_{ℱ} S^{0} is a rational π_{∗}-isomorphism of orthogonal G-spectra. Equally the map S^{0} → Eℱ̃ gives a rational equivalence e_{ℱ′} S^{0} → e_{ℱ′} Eℱ̃ and we also have a rational equivalence Eℱ̃→ e_{ℱ′} Eℱ̃, so we can conclude that Eℱ̃and e_{ℱ′} S^{0} are weakly equivalent.

Proposition 6.20 For an idempotent family ℱ and idempotents as above we have the following collection of rational equivalences for any orthogonal G-spectrum X.

\begin{aligned}e_{ℱ} X & ≃ & e_{ℱ} 𝕊 ∧ X & ≃ & Eℱ_{+} ∧ X \\ e_{ℱ′} X & ≃ & e_{ℱ′} 𝕊 ∧ X & ≃ & Eℱ̃∧ X \end{aligned}

Theorem 6.21 For ℱ an idempotent family of subgroups of G, the strong monoidal Quillen adjunction of Proposition 6.3 as it applies to GI𝒮 _{ℚ} is a Quillen equivalence.

Proof We use the characterisation of Quillen equivalences of [HSS00, Lemma 4.1.7]. So letting K denote fibrant replacement in L_{Eℱ_{+}} GI𝒮 × L_{Eℱ̃}GI𝒮 we must show that

(i). × detects and preserves weak equivalences between fibrant objects,

(ii). X → ×ΔX → ×KΔX is a π_{∗}^{ℚ}-isomorphism for all cofibrant X in GI𝒮 .

Take A ∈ F𝒮 and B ∈ F̃𝒮 then by the definition of the product model category, (A, B) is fibrant in L_{Eℱ_{+}} GI𝒮 _{ℚ} × L_{Eℱ̃}GI𝒮 _{ℚ} if and only if A is Eℱ_{+}-local and B is Eℱ̃-local. So take (f, g): (A, B) → (C, D) a map in L_{Eℱ_{+}} GI𝒮 _{ℚ} × L_{Eℱ̃}GI𝒮 _{ℚ} between fibrant objects. Now recall that an Eℱ_{+}-equivalence between Eℱ_{+}-local objects is a π_{∗}-isomorphism and similarly for Eℱ̃. We complete the proof of the first part: (f, g) is a weak equivalence if and only if f is an Eℱ_{+}-equivalence and g is an Eℱ̃-equivalence. This is so if and only if f and g are π_{∗}^{ℚ}-isomorphisms and this occurs exactly when f × g is a π_{∗}^{ℚ}-isomorphism.

The second part is marginally more difficult: fix X a cofibrant object of GI𝒮 , and label its fibrant replacement in L_{Eℱ_{+}} GI𝒮 _{ℚ} by X_{ℱ} and by X_{ℱ′} in L_{Eℱ̃}GI𝒮 _{ℚ}. We must show that

X → X × X → X_{ℱ} × X_{ℱ′}

is a π_{∗}^{ℚ}-isomorphism. We do so by proving that it is both an Eℱ_{+}-equivalence and an Eℱ̃-equivalence and then applying Corollary 6.5. We apply several transformations translate to this question before we are in a place to answer it, repeatedly using the two-out-of-three property of weak equivalences. X → X_{ℱ} × X_{ℱ′} is an Eℱ_{+}- equivalence if and only if X_{ℱ} × X_{ℱ′} → X_{ℱ} is an Eℱ_{+}-equivalence. Now since X_{ℱ} ∨ X_{ℱ′} → X_{ℱ} × X_{ℱ′} is a π_{∗}-isomorphism, X_{ℱ} × X_{ℱ′} → X_{ℱ} is an Eℱ_{+}- equivalence if and only if X_{ℱ} ∨ X_{ℱ′} → X_{ℱ} is an Eℱ_{+}-equivalence.

Now we must prove that X_{ℱ} ∨X_{ℱ′} ∧Eℱ_{+} → X_{ℱ} ∧Eℱ_{+} is a a π_{∗}^{ℚ}-isomorphism. We know by Lemma 6.7 and Lemma 6.6 that X_{ℱ′} ∧ Eℱ_{+} has trivial homotopy groups and Eℱ_{+} is cofibrant and so preserves π_{∗}^{ℚ}-isomorphisms, thus X_{ℱ} ∧ Eℱ_{+} → X_{ℱ} ∧ Eℱ_{+} is a π_{∗}^{ℚ}-isomorphism. Thus we have our weak equivalence X_{ℱ} ∨X_{ℱ′} ∧Eℱ_{+} → X_{ℱ} ∧ Eℱ_{+}.

We can apply a similar argument to reduce the claim that X_{ℱ} ∨ X_{ℱ′} → X_{ℱ′} is an Eℱ̃-equivalence to showing that X_{ℱ} ∧ Eℱ̃has trivial homotopy groups. Since X_{ℱ} is Eℱ_{+}-local we have a π_{∗}-isomorphism X_{ℱ} → F (Eℱ_{+}, X_{ℱ} ) and we can replace F (Eℱ_{+}, X_{ℱ} ) ∧ Eℱ̃by the rationally equivalent spectrum e_{ℱ′} F (Eℱ_{+}, X_{ℱ} ). So we have now reduced the problem to showing that

π_{∗}^{H}(e_{ℱ′} F (Eℱ_{+}, X_{ℱ} )) = e_{ℱ′} π_{∗}^{H}(F (Eℱ_{+}, X_{ℱ} )) = 0

for all subgroups H. Using the description of homotopy groups as maps in the homotopy category and the standard adjunctions we must show that for all integers n and all subgroups H

(e_{ℱ′} )_{∗}[Σ^{n}Eℱ_{+}, X_{ℱ} ]_{ℚ}^{H} ≅ (e_{ℱ′} )_{∗}[e_{ℱ} Σ^{n}𝕊, X_{ℱ} ]_{ℚ}^{H} = 0.

Since X_{ℱ} is an 𝕊_{ℚ}-local object (and hence also an Ω-G spectrum) we have the following series of isomorphisms:

[e_{ℱ} Σ^{n}𝕊, X_{ℱ} ]_{ℚ}^{H} ≅ [e_{ℱ} Σ^{n}𝕊, X_{ℱ} ]^{H} ≅ [e_{ℱ} Σ^{n}𝕊, X_{ℱ} ]_{l}^{H}.

Now we apply [MM02, III, Theorem 2.7(v)] since e_{ℱ} Σ^{n}𝕊 is a colimit of cofibrant objects and hence they are non-degenerately based by [MM02, III, Theorem 2.4]. Thus we obtain the lim^{1} exact sequence of pointed sets.

∗ → lim^{1}[Σ^{n+1}𝕊, X_{ℱ} ]_{l}^{H} → [e_{ℱ} Σ^{n}𝕊, X_{ℱ} ]_{l}^{H} → lim[Σ^{n}𝕊, X_{ℱ} ]_{l}^{H} → ∗

Since e_{ℱ}^{∗} (that is, e_{ℱ} acting on the first factor) is an idempotent on sets of maps in the rational homotopy category and [Σ^{n+1}𝕊, X_{ℱ} ]_{l}^{H} ≅ [Σ^{n+1}𝕊, X_{ℱ} ]_{ℚ}^{H} the tower it creates satisfies the Mittag-Leffler condition ([Wei94, Definition 3.5.6]) hence the lim^{1} term is zero.

So we see that

[e_{ℱ} Σ^{n}𝕊, X_{ℱ} ]_{l}^{H} ≅ e_{ℱ}^{∗} [Σ^{n}𝕊, X_{ℱ} ]_{l}^{H}

(a limit of idempotent maps is equivalent to taking the image). This is zero by the action of the Burnside ring on sets of maps in the homotopy category and the fact that e_{ℱ′} ∘ e_{ℱ} is trivial in the homotopy category. In particular,

e_{ℱ}^{∗} [f] = [f] ∧ [e_{ℱ} ] and (e_{ℱ′} )_{∗}[f] = [f] ∧ [e_{ℱ′} ]

hence e_{ℱ}^{∗} (e_{ℱ′} )_{∗} = (e_{ℱ′} )_{∗}e_{ℱ}^{∗} = 0.

In the above proof we could have used a more symmetrical argument to prove statement (ii), that is, we could have argued via idempotents to show that X → X_{ℱ} ×X_{ℱ′} is an Eℱ_{+}-equivalence. However, this is true for any family ℱ, it is when only showing that the above map is an Eℱ̃-equivalence that we need to use the idempotent argument and hence must use the rationalised category.

Index

Burnside ring, 13

cofamily, 10

tom Dieck’s isomorphism, 1

E-acyclic, 4 E-equivalence, 4 E-local, 4 E-localisation, 4 E-model structure, 4

ℱ -cofibration, 11 ℱ -equivalence, 11 ℱ -model structure, 11 F𝒮 , 10 F̃𝒮 , 10 family, 10 FG, 1

GI𝒮 , 2 GI𝒮 _{ℚ}, 7 Γ(G), 15

h-cofibration, 5 H-small, 7 homotopy sequential colimit, 13

idempotent family, 1 isotropy separation cofibering, 10

localisation functor, 4

monoid axiom, 5

product model category, 11 properness, 6 pushout product axiom, 5

rational equivalence, 7 rational fibration, 7 rational sphere spectrum, 3

support, 15

telescope, see homotopy sequential colimit

universal ℱ -space, 10 universal ℱ′-space, 10

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