Rational orthogonal calculus

We show that one can use model categories to construct rational orthogonal calculus. That is, given a continuous functor from vector spaces to based spaces one can construct a tower of approximations to this functor depending only on the rational homology type of the input functor, whose layers are given by rational spectra with an action of O(n). By work of Greenlees and Shipley, we see that these layers are classified by torsion H∗(BSO(n))[O(n)/SO(n)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{H}}}^*({{\mathrm{B}}}SO(n))[O(n)/SO(n)]$$\end{document}-modules.


Introduction
Orthogonal calculus constructs a Taylor tower for functors from vector spaces to spaces. The nth layer of this tower is determined by a spectrum with O(n) action. Orthogonal calculus has a strong geometric flavour, for example it was essential to the results of [1] which shows how the rational homology of a manifold determines the rational homology of its space of embeddings into a Euclidean space. Working rationally is also central to work of Reis and Weiss [15]. Thus it is natural to ask if one can construct a rationalised version of orthogonal calculus where the tower of a functor F depends only on the (objectwise) rational homology type of F. In this paper we apply the work of [3] to construct suitable model categories that capture the notion of rational orthogonal calculus. In particular, we show that the layers of the rational tower are classified by rational spectra with an action of O(n). By the work of Greenlees and Shipley [9], we see that these layers are classified by torsion H * (B SO(n); Q)[O(n)/SO(n)]-modules. Thus we have a strong technical foundation for rational orthogonal calculus and a simpler, algebraic characterisation of the layers which should reduce the amount of effort required in future calculations. This paper also gives a nice demonstration of how Pontryagin classes are at the heart of orthogonal calculus, as the graded ring H * (B SO(n)) is polynomial on the Pontryagin (and Euler) classes.
The main difficulty in the work is setting up the model structures by careful use of Bousfield localisations. There are some subtleties involved as we are mixing left and right Bousfield localisations. The source of the difficulty is that in general left localisations require left properness of the model structure, but do not preserve right properness, whilst the reverse situation generally holds for right localisation. For example, the model category of H Q-local spaces in the sense of Bousfield [4] is created via a left localisation of the standard model structure on spaces and is not right proper.
Organisation We recap the basic notions of orthogonal calculus in Sect. 2: polynomial and homogeneous functors, the classification theorem and the tower, and the derivative of a functor. Section 3 has the main result: the rational version of the classification of n-homogeneous functors, Theorem 3.3.
The rest of the paper is where we give the details of the proof of the main theorem. In Sect. 4 we give a brief recap of the notion of Bousfield localisation and review the construction of the model categories used in orthogonal calculus.
In Sects. 5 and 6 we establish model structures for rational n-polynomial and rational n-homogeneous functors. We extend the classification results of Weiss in Sect. 7 to the rational setting and finish our description of rational n-homogeneous functors.

Orthogonal calculus
We give a brief overview of orthogonal calculus, introducing the relevant categories and definitions. The primary reference is [17].

Continuous functors
Let L be the category of finite dimensional real inner product spaces and isometries. To ensure this category is skeletally small, we assume that these vector spaces are all subspaces of some universe, R ∞ . Note that for U ⊆ V , L(U, where V −U denotes the orthogonal complement of U in V . We define a new category J 0 : it has the same objects as L but its morphism spaces are given by J 0 (U, V ) = L(U, V ) + .
Orthogonal calculus studies continuous functors from L to (based or unbased) topological spaces. We restrict ourselves to the based version and consider Top-enriched functors F : J 0 → Top where Top denotes the category of based topological spaces. Thus F is a functor with the property that the induced map of spaces is continuous (and is associative, unital and compatible with composition of maps of vector spaces). We denote the category of such functors J 0 Top. Whenever we talk of functors from vector spaces to spaces we mean an object of J 0 Top.

Polynomial and homogeneous functors
Definition 2.1 An object F ∈ J 0 Top is said to be n-polynomial if for each V ∈ J 0 the inclusion map is a weak homotopy equivalence.
This definition captures the idea of the value of F at V being recoverable from the value of F at vector spaces of higher dimension (and the maps between these values).
By [17,Proposition 4.2] we can give an equivalent definition of n-polynomial using a sequence of vector bundles γ n over L. Let and define Sγ n (U, V ) be the unit sphere in γ n (U, V ).
A functor F is n-polynomial if and only if η F : F → τ n F is an objectwise weak homotopy equivalence, where η is induced by the projections Sγ n (V, W ) + → J 0 (V, W ).
Using τ n we can construct the n-polynomial approximation to an element of J 0 Top.

Definition 2.3
For F ∈ J 0 Top let T n F be the functor The map F → T n F induced by η F is the n-polynomial approximation to F. This map has the desired property that if f : F → G is a map in J 0 Top with G an n-polynomial functor, then f factors over F → T n F (up to homotopy) in a unique way (up to homotopy).
Since this is not clear from the definitions, we give [17, Proposition 5.1].

Proposition 2.4
If F is (n − 1)-polynomial then F is n-polynomial.
Hence we have a map (unique up to homotopy) from T n E → T n−1 E for any E ∈ J 0 Top.

Definition 2.5 A functor F ∈ J o
Top is said to be n-homogeneous if F is n-polynomial and T n−1 F is objectwise weakly equivalent to a point. For E ∈ J 0 Top we let D n E denote the homotopy fibre of the map T n E → T n−1 E. We call D n E the nhomogeneous approximation to E.
The functor D n E is n-homogeneous as T n and T n−1 commute with homotopy fibres (and finite homotopy limits in general). In particular T n−1 T n E = T n−1 E.

The tower and the classification
Now we are in a place to be more definite about orthogonal calculus. Given a functor F ∈ J 0 Top, orthogonal calculus constructs the n-polynomial approximations T n F and the n-homogenous approximations D n F. These can be arranged in a tower of fibrations (analogous to the Postnikov tower).
For each n 0 there is a fibration sequence D n F → T n F → T n−1 F, which can be arranged as below.
In the above, S R n ⊗V is the one-point compactification of R n ⊗ V . This has O(n)action induced from the regular representation of O(n) on R n . The smash product is equipped with the diagonal action of O(n), and h O(n) denotes homotopy orbits.

Motivation
Calculations in the orthogonal calculus can often be difficult. By considering only the rational information of a functor, these calculations can often be simplified, such as in [15] or [1]. One implementation of this idea is to alter the constructions so that if f : F → G in J 0 Top induces a levelwise isomorphism on rational homology, then the tower of F and G should agree (up to weak homotopy equivalence).
It follows that we need to construct a rational n-polynomial replacement functor T Q n . This should have the property that given any functor F ∈ J 0 Top, T Q n F is the closest functor that is both rational and n-polynomial. We construct such a functor in Sect. 5. Hence we have a rational n-homogeneous functor D Q n F (the homotopy fibre of T Q n F → T Q n−1 F). We study rational n-homogeneous functors in detail in Sect. 6. A strong piece of evidence that this implementation is the correct one is that our rational n-homogeneous functors are classified in terms of rational spectra with an action of O(n). To that end, we introduce the algebraic model for rational spectra with an action of O(n) from [9] so that we can state the rationalised version of Weiss's classification theorem.

The algebraic model
The algebraic category is based on the group cohomology of SO(n). The following calculation is well-known. The elements p i have degree 4i and represent the Pontryagin classes, the element e has degree k and represents the Euler class.
From now on we shall omit the Q from our notation for cohomology. We will also use the notation W = W O(2) SO(2) = O(2)/SO (2). We construct a rational differential graded algebra H * (B SO(n)) [W ]. It has basis given by symbols w i · r , where r ∈ H * (B SO(n)) and i is either 0 or 1. The multiplication is given by the formulae below, where r, s ∈ H * (B SO(n)) and w(r ) is the image of r under the ring isomorphism w, where w( p i ) = p i and w(e) = −e. The unit is w 0 · 1 and all differentials are zero. We need to add one final condition to get the correct (model) category, see [9] for details. This category has a model structure where the weak equivalences are the homology isomorphisms of underlying chain complexes and the cofibrations are the monomorphisms. This is a substantial simplification of the classification in terms of spectra with an O(n)-action. For example, the first derivative of an object of J 0 Top is uniquely classified (up to homotopy) by a chain complex N , with an action of W . Furthermore, the second derivative of an object of J 0 Top is uniquely classified (up to homotopy) by an chain complex M, which has an action of Q[e] (e has degree 2) and an action of W , where w * (e · m) = −e · (w * m).

The main theorem
The Quillen equivalence of Greenlees and Shipley is a composite of a number of adjunctions, so we leave the details to the reference. However, if E is an O(n)-spectrum and M ∈ tors-H * (B SO(n))[W ]-mod is the corresponding object (under the series of Quillen equivalences), then the homology of M is determined by the relation In the above, H SO(n) * (X ) denotes homology of the Borel construction applied to the spectrum X (that is, the homology of the SO(n)-homotopy orbits of X ) and A n is the adjoint representation of SO(n) at the identity. Thus A n is the n(n −1)/2-dimensional vector space of skew symmetric matrices with SO(n) acting by conjugation. The cap product induces the action of H * (B SO(n)). The spectrum E had an O(n)-action and we have taken SO(n)-orbits (or fixed points), hence W = O(n)/SO(n) acts on the homology and homotopy groups above.
The Quillen equivalence is also equipped with an Adams spectral sequence [9, Theorem 9.1] allowing one to perform calculations easily. In the following, r is the rank of the maximal torus of O(n), so this is n/2 for even n and (n − 1)/2 when n is odd.

Theorem 3.4 Let X and Y be free rational O(n)-spectra. There is a natural Adams spectral sequence
It is a finite spectral sequence concentrated in rows 0 to r and is strongly convergent for all X and Y .
We give some examples to illustrate Theorem 3.3. Many more calculations in rational orthogonal calculus can be found in Reis and Weiss [15].

Example 3.5 Consider the functor which sends a vector space
This functor can also be described as J n (0, V ), see Sect. 7. We want to find the rational nth-derivative of this functor. To do so, we use the Quillen equivalences of Propositions 7.8 and 7.9. Letting L indicate derived functors, we have the following diagram of objects.
If we then work rationally, the nth-derivative is O(n) + ∧ H Q. Furthermore, we can identify the algebraic model for this spectrum in tors-H * (B SO(n))[W ]-mod. Taking homotopy groups we obtain with the Pontryagin (and Euler) classes acting as zero for degree reasons. If X is a chain complex having this homology, then X n(n−1)/2 Q[W ] by a simple formality argument. Hence the rational nth-derivative of V → S nV is given by n(n−1)/2 Q[W ].

Example 3.6
The previous example is the case U = 0 of the representable functor V → J n (U, V ) as defined in Sect. 7. This functor from vector spaces to spaces is like a shift desuspension of V → J n (0, V ) = S nV . Indeed, the corresponding object to J n (U, −) in the homotopy category of n-homogeneous functors is the functor V → O(n) + ∧ J n (U, V ), which is the shift desuspension of J n (0, −). As the nhomogeneous model structure is stable, we want to find the algebraic model for the The O(n)-spectrum S nR corresponds to the object Q[n] in the algebraic model: the sign representation of O(n) in degree n with zero differential. Similarly S nU for U a vector space of dimension u corresponds to Q ⊗u [nu]. So for even u, this is just Q in degree nu and for odd u it is Q in degree nu.
It follows that the algebraic model for , which in turn corresponds to the sphere spectrum (with trivial action) in O(n) Sp. So rationally the derivative is H Q. The algebraic model for this spectrum has homology given by H where A n is the adjoint representation of SO(n) at the identity. If we ignore the action of W , this is essentially a suspension of H * (B SO(n)) with the Pontryagin classes acting by the cap product. Hence the rational nth-derivative is often considered as the orthogonal calculus equivalent of the identity functor. It has first derivative the sphere spectrum and second derivative the desuspension of the by sphere spectrum (each with trivial action) by [17,Example 2.7]. The remaining are rationally trivial by [2,Theorem 4]. It follows that the algebraic model for derivatives of B O are Q in degree 0, Q in degree −1 and 0 for higher derivatives.

Model categories for orthogonal calculus
To prove the rational classification theorem we use the language of model categories and Quillen functors. In this section we recall the construction of the n-polynomial and n-homogeneous model structures from [3]. We start with some basic model category notions.

Cellular, topological and proper model categories
Just as one has a simplicial model category, there is the notion of a topological model category. See [13,Definition 5.12] for more details. induced by Hom M (i, id) and Hom M (i, p). We say that M is a topological model category if M(i * , p * ) is a Serre fibration of spaces whenever i is a cofibration and p is a fibration and further that M(i * , p * ) is a weak equivalence if (in addition) either i or p is a weak equivalence.

Definition 4.2
Let M be a model category and let the following be a commutative square in M: M is called left proper if, whenever f is a weak equivalence, i a cofibration and the square is a pushout, then g is also a weak equivalence. M is called right proper if, whenever g is a weak equivalence, j a fibration, and the square is a pullback, then f is also a weak equivalence.  Relevant examples of cellular model categories include: simplicial sets, topological spaces, sequential spectra, symmetric spectra and orthogonal spectra (with a group acting). These are all proper and (with the exception of simplicial sets) topological.

Bousfield localisations
With the exception of Sect. 3 we will be using topological model categories. These categories have the advantage that we can use the enrichment in topological spaces to define the weak equivalences of left or right localisations. This avoids the more complicated terminology of homotopy function complexes, see [10,Section 17]. The essential point is that for cofibrant X and fibrant Y in a topological model category M we have Where Hom M (−, −) denotes the enrichment, ⊗ is the tensoring, and [−, −] M is maps in the homotopy category of M. For general X and Y in M we define where c denotes cofibrant replacement and f denotes fibrant replacement in M.
We now summarise Hirschhorn's results on left and right Bousfield localisations, see [10,Sections 4 and 5]. The techniques of Bousfield localisation of model categories allow one to construct a new model structure from a given model category with a larger class of weak equivalences. This was used in [3] to construct the n-polynomial and n-homogeneous model structures on J 0 Top from the objectwise model structure. We begin with the necessary terminology. We say that Z is S-fibrant if it is S-local and fibrant in M.
We have an almost dual set of definitions for right localisations. The K -equivalences we define below are sometimes called K -coequivalences or K -colocal equivalences. An object C is said to be K -colocal if for any K -equivalence f : X → Y in S, there is a weak homotopy equivalence of spaces We say that C is K -cofibrant if it is K -colocal and cofibrant in M.
Notice that every weak equivalence of M is both a K -equivalence and an Sequivalence for any set of objects K and any set of maps S.

Application to orthogonal calculus
The model structures used in [3] are defined using Bousfield localisations of (more standard) model structures. We give the relevant results and indicate where we have Quillen equivalences.
We begin with the objectwise or projective model structure on continuous functors from J 0 to Top. We let sk L denote a skeleton of the category of finite dimensional vector spaces and isometries (it is also a skeleton of J 0 ).

Proposition 4.8
There is an objectwise model structure on J 0 Top where a fibration is an objectwise Serre fibration of based spaces and a weak equivalence is an objectwise weak homotopy equivalence. This model category is cellular and proper with generating cofibrations and acyclic cofibrations given by Proof See [13, Theorem 6.5].
We now alter the objectwise model structure on J 0 Top to obtain a model category whose homotopy category is the category of n-polynomial objects up to homotopy. For details see [3,Section 6]. In particular that reference shows that the model structure is right proper, despite it being a left Bousfield localisation. We construct the n-homogeneous model structure from the n-polynomial model structure using a right Bousfield localisation. Definition 4. 10 We define n-homog-J 0 Top, the n-homogeneous model structure on J 0 Top to be the right Bousfield localisation of n-poly-J 0 Top at the set of objects The following result summarises [3, Proposition 6.9]. The weak equivalences of the n-homogeneous model structure may also be described as those maps f such that D n F is an objectwise weak equivalence.

Rational polynomial functors
In this section we construct a category of rational n-polynomial functors, that is, a model category whose fibrant objects are both (objectwise) rational and n-polynomial. We begin by recapping the construction of a model category of rational spaces. Remark 5.2 A simply connected space that is H Q-local is usually called a rational space. In this paper we do not make any assumption on the connectivity of our spaces.

Rational spaces
Working with H Q-local spaces (as opposed to rational spaces) has two main advantages. Firstly we can consider non-nilpotent (and non-simply connected) spaces such as B O(n) without having to use the category of spaces over some classifying space B K as in [15]. Secondly, the existence of well-behaved model structures allows us to phrase the classification of rational n-homogeneous functors in terms of Quillen equivalences and make use of the existing work on model categories for orthogonal calculus. We note Walter [16] takes the other approach and studies homotopy functor calculus of rational spaces in the setting of Goodwillie calculus. We extend this model structure on spaces to J 0 Top in the expected manner.
These sets define a cellular and left proper model structure by the same argument as for the objectwise model structure. The statements about the fibrant objects and weak equivalences are immediate.
It is clear that this model structure is the localisation of the objectwise model structure at the set of morphisms J 0 ∧ J Q Top .

Double localisations
We will construct a model category of rational n-polynomial functors as a double localisation of the objectwise model structure on J 0 Top. One localisation is at the set S n , which makes the fibrant objects n-polynomial. The other is localisation at the set J 0 ∧ J Q Top , which makes the fibrant objects H Q-local.
We give a couple of lemmas examining the structure of double localisations. The first is essentially the language of Bousfield lattices adapted to more general localisations. In the following we let f denote fibrant replacement in C, f S denote fibrant replacement in L S C and f T denote fibrant replacement in L T C.

Lemma 5.4 Let C be a left proper and cellular topological model category and S and T be sets of maps in
is a weak homotopy equivalence of spaces. For any X and Y , we have By symmetry, the fibrant objects of L S (L T C) are exactly the fibrant objects of L S∪T C. It follows that these two model categories have the same weak equivalences (and the same cofibrations as C), so they are equal.
The weak equivalences are the S ∪ T -local equivalences by definition. Any S-local equivalence is a weak equivalence in L T (L S C) as this is a Bousfield localisation. Similarly every T -local equivalence is a weak equivalence in L T (L S C).
It is important to note that we do not claim that the fibrant replacement functors f S and f T commute. In general, this will be false (consider localising spectra at H Q and MZ/ p). We will add an assumption about the interaction of the fibrant replacement functors. This assumption will hold in the case of rational n-polynomial functors.

The weak equivalences are characterised as T -equivalences between S-localisations. Moreover, f is an S ∪ T -local equivalence if and only if f S f is a T -local equivalence.
Proof The description of fibrant replacements is routine, so we turn to the statement about weak equivalences. Let f : C → C be an S ∪ T -local equivalence. Then we have a commutative diagram of maps in C as below.
The lower horizontal map is a S ∪ T -local equivalence between S ∪ T -local objects and hence is a weak equivalence in C. It follows that f S f is a T -local equivalence. The converse is immediate. Proof Recall J Q Top , the set of generating acyclic cofibrations for the rational model structure on based spaces. We extended these to J 0 Top as the set

Rational polynomial functors
We define the rational n-polynomial model structure as To complete the proof, we apply Lemmas 5.4 and 5.5. Thus we must show that if F is objectwise H Q-local then so is T n F. Let F be objectwise H Q-local. We claim that for any V ∈ J 0 , the space is H Q-local, so that τ n F is objectwise H Q-local. That is, we want to show that It is easily checked that this smash product is a Quillen bifunctor with respect to the H Q-local model structures on J 0 Top and Top. Hence the claim follows.
The functor T n F is the homotopy colimit over k of τ k n F and this homotopy colimit is constructed objectwise. Since homotopy colimits preserve H Q-local spaces, we see that T n F is H Q-local.
For more details on the interactions between localisations and Quillen bifunctors, see [8].
Corollary 5.7 Let F ∈ J 0 Top. Then the homotopy fibre of is an n-homogeneous functor that is objectwise H Q-local. We denote this D Q n F.
We want to show that such functors are classified by rational spectra with an action of O(n). We do so by proving that there is a Quillen equivalence between a model structure for rational n-homogeneous functors and a model structure for rational spectra with an action of O(n).
Remark 5.8 A formal consequence of the definitions is that for F ∈ J 0 Top, T Q n F = T n L H Q F is the closest functor to F that is both n-polynomial and objectwise H Qlocal. But it should be noted that for F ∈ J 0 Top, the functor f H Q T n F is weakly equivalent to F in n-poly-J 0 Top Q , but it is not necessarily n-polynomial.

Rational homogeneous functors
In this section we will left Bousfield localise the n-homogeneous model structure on J 0 Top to ensure that the fibrant objects are also objectwise L H Q -local. This requires some care to ensure both that the n-homogeneous model structure admits a left Bousfield localisation and that a localisation with the correct properties exists.
In order that we can perform left Bousfield localisations on a model structure we need to know that it is left proper and cellular. The following lemma builds upon Proposition 4.11 and shows that these properties hold for n-homog-J 0 Top. The key fact is that this model structure is stable: the (objectwise) suspension is an equivalence on the homotopy category. This follows from the fact that any homogeneous functor has a de-looping, see also [3, Corollary 10.2].

Lemma 6.1
The pushout of an n-homogeneous weak equivalence along a cofibration (of the objectwise model structure) is an n-homogeneous weak equivalence. In particular, the n-homogeneous model structure on J 0 Top is left proper (and hence is proper). Moreover this model structure is cellular.
Proof These statements follow from the proofs of [6, Proposition 5.8 and Theorem 5.9]. In each case we do not need the original category (n-poly-J 0 Top) to be stable, only that its right localisation (n-homog-J 0 Top) is stable. Now we need to specify a set of maps at which to localise n-homog-J 0 Top. A reasonable first guess would be J 0 ∧ J Q Top as we used this set to make the model structure of rational n-polynomial functors. But this set would cause us some technical problems as the functor J 0 (U, −) is (most likely) not cofibrant in n-homog-J 0 Top. Instead the cofibrant objects are built (in a well-defined sense) from the objects J n (U, −) see [10, Definition 5.1.4 and Theorem 5.1.5]. Thus we will left Bousfield localise n-homog-J 0 Top at the set If we had chosen the set J 0 ∧ J Q Top then we would need to cofibrantly replace the codomain and domain in the definition of the localised weak equivalences. This would make it much harder to understand the fibrant objects. As it is, we will not see until Proposition 7.9 that this model structure has all the properties we desire.

Theorem 6.2
There is a rational n-homogenous model structure on J 0 Top, denoted n-homog-J 0 Top Q . This model structure is cellular and proper. The fibrant objects are the n-homogeneous functors F such that Ind n 0 F is objectwise H Q-local.
Proof We define the model category n-homog-J 0 Top Q to be the left Bousfield localisation of n-homog-J 0 Top at the set of maps So we have shown that the fibrations of n-homog-J 0 Top Q are precisely the fibrations of n-homog-J 0 Top such that Ind n 0 f is an objectwise rational fibration. It follows that the fibrant objects are the n-polynomial functors F such that Ind n 0 F is objectwise H Q-local.

The intermediate categories
Recall the vector bundles γ n over L: Define J n (U, V ) to be the Thom space of γ n (U, V ). There is a composition map which induces a composition on the corresponding Thom spaces. Thus J n has the structure of a category enriched over based topological spaces. Furthermore, the standard action of the group O(n) on R n induces an action on the vector bundles γ n (U, V ) that is compatible with the composition maps above. Hence J n is a category enriched over based topological spaces with an action of O(n).
Since we have encountered spaces with O(n)-action, we briefly recap that notion. We say that a map f : The category O(n) (J n Top) was denoted O(n)E n in [3]. There is an adjunction between this category and spectra with an action of O(n), see [3, Section 8] for details.

Definition 7.2
There is a morphism of enriched categories α n : J n → J 1 which sends V to R n ⊗ V and acts on mapping spaces by This induces a functor α * n : O(n) (J n Top) → O(n) Sp. It is defined as (α * n E)(V ) = E(R n ⊗ V ), but the action on the O(n)-space E(R n ⊗ V ) is altered to use both the pre-existing action and also the action induced by O(n) acting on R n ⊗ V . This has a left adjoint called J n ∧ J 1 (−).
We also want to compare O(n) (J n Top) and J 0 Top. The inclusion {0} = R 0 → R n induces a map of vector bundles γ 0 → γ n and a map of enriched categories i n : J 0 → J n . This creates an adjoint pair between O(n) (J n Top) and J 0 Top.
The cofibrant objects have no fixed points (except the base point), the weak equivalences are the underlying weak homotopy equivalences.
Using this model structure on O(n)-spaces, we can equip O(n) (J n Top) and O(n) Sp with levelwise model structures similar to that for J 0 Top. These model structures are proper and cellular. We want to make them into stable model categories. For that, we need a new class of weak equivalences. The idea is to generalise the notion of π * -isomorphisms of spectra. For full details, see [3,Section 7].
For V a vector space, R n ⊗ V has the O(n)-action induced by the standard representation of O(n) on R n . Let S nV be the one-point compactification of R n ⊗ V , then Just as with the definition of the stable homotopy groups of a spectrum, we can use these maps (and n-fold suspension) to construct the nπ * -homotopy groups of E.
is a weak homotopy equivalence. These model structures are stable, cellular and proper.
Proof These model structures can be constructed as left Bousfield localisations. We localise the levelwise model structures at the following sets of maps, the first is for O(n) (J n Top) and the second is for O(n) Sp.
We now need to rationalise O(n) (J n Top) and O(n) Sp. Each is left proper and cellular, so we may use [10] to perform a left Bousfield localisation of each of the categories. Recall the set J Q Top of generating acyclic cofibrations for the rational model structure on based spaces. We use this set to rationalise spectra with an O(n)-action. It is easily checked that the fibrant objects of O(n) (J n Top) Q are the objectwise H Q-local objects whose adjoints of suspensions maps are weak homotopy equivalences. We are now ready to prove our main result, Theorem 3.3. on nonrationalised model categories is a Quillen equivalence by [3,Section 8]. So we can apply [11,Proposition 2.3] to see that the localised adjunction is a Quillen equivalence.
If we apply the left derived functor of Res n 0 /O(n) to the maps in Q n we obtain the set J n ∧ J Q Top = {J n (U, −) ∧ j | j ∈ J Q Top U ∈ sk J n } which we used to localise the n-homogeneous model structure. Hence we obtain the following result by the same proof as for Proposition 7.8. Again we use [3,Section 8] to see that (Res n 0 /O(n), Ind n 0 ε * ) is a Quillen equivalence between the non-rationalised model categories. Ind n 0 ε * Thus we have now shown that n-homog-J 0 Top Q has the correct homotopy category, namely the homotopy category of rational spectra with an action of O(n). We can identify the derived composite of the above adjunctions and learn more about the cofibrant-fibrant objects of n-homog-J 0 Top Q . . Let F be a cofibrant and fibrant object of n-homog-J 0 Top Q . Then F is an objectwise H Q-local n-homogeneous functor.
Proof Combining the two propositions above at the level of homotopy categories gives the equivalence. That the equivalence agrees with Weiss's classification theorem follows from the proof of [3, Theorem 10.1]. Let F be a cofibrant and fibrant object of n-homog-J 0 Top Q . Then F is in particular n-homogeneous, so it defines a spectrum F with O(n)-action. We know that Ind n 0 F is objectwise H Q-local, hence the spectrum F is objectwise H Q-local by the Quillen equivalences above. Now consider the n-homogeneous functor defined by F : This functor is objectwise H Q-local and it is objectwise weakly equivalent to F, thus F itself is objectwise H Q-local.
Thus we have shown that n-homog-J 0 Top Q is a model for the homotopy theory of n-homogeneous functors in J 0 Top that are objectwise H Q-local. Secondly we have shown that such functors are determined, up to homotopy, by rational spectra with an action of O(n).