EQUIVARIANT STABLE HOMOTOPY THEORY

Contents

Lecture 1 1

1. G-spaces 1

2. The equivariant stable category 2

Lecture 2 4

Lecture 3 7

3. Plan of the seminar 9

lecture 4 10

4. G-CW complexes 10

5. The Whitehead Theorem 12

Lecture 5 14

6. The Wirtm¨uller isomorphism 14

Lecture 6 16

7. The Adams-isomorphism 16

8. Suspension Spectra 20

9. The Freudenthal Suspension Theorem 22

Lecture 1

The aim of this first lecture is firstly to give a quick definition of the equivariant stable category for a finite groupG, and secondly to agree on what we want to study in greater detail.

1. G-spaces

By a G-space we mean a pointed topological space with a continous action of G fixing the base point. A G-map f: X → Y of G-spaces is a (continous) map respecting the group-actions. The following example introduces some notation.

Example 1.1.

(i) We consider every pointed space as a G-space with trivial action of G, that is, g·x=x for all x∈X and g ∈G.

(ii) If X and Y are G-spaces, then X × Y becomes a G-space by defining g ·(x, y) = (g ·x, g·y). In particular, if I denotes the unit interval, then I×X is a G-space with g·(t, x) = (t, g·x).

(iii) If X andY areG-spaces, then so isX∧Y. In particularG₊∧X ∼=W

g∈GX is a G-space.

1

(iv) IfX and Y areG-spaces, then we can form the space map(X, Y) of pointed maps from X to Y. This space becomes a G-space by defining g · f as the function with (g·f)(x) =g ·(f(g⁻¹·x)). In particular map(G₊, X)∼= Q

g∈GX is a G-space.

(v) IfX is aG-space, thenX^∧G =V

g∈GX has a uniqueG-space structure such that the projection map(G₊, X) = Q

x∈GX →V

g∈GX =X^∧G is a G-map.

(vi) If H is a subgroup of G, then every G-space X is also considered as an H-space.

Given G-maps f₀, f₁: X → Y, a G-homotopy F from f₀ to f₁ consists of an G- map F: I₊∧X → Y with F(i, x) = f_i(x) for i = 0,1. If H is a subgroup of G we write [X, Y]_H for the set of equivalence classes of H-maps from X to Y with respect to the equivalence relation given by H-homotopy. In particular we denote by [X, Y]_Gthe set of equivalence classes ofG-maps fromX toY with respect to the equivalence relation given by G-homotopy.

2. The equivariant stable category

Definition 2.1. The category Sp^G of G-spectra is defined as follows.

(i) A G-spectrum X is a sequence (X₀, X₁, X₂, . . .) of G-spaces together with structure G-maps σ_n^X: (S¹)^∧G∧X_n→X_n+1.

(ii) A map f: X → Y of G-spectra consists of maps fn: Xn → Yn satisfying that σ_n^Y ◦fn =fn+1◦σ^X_n for all n.

Example 2.2.

(i) The sphereG-spectrumShas Sn= (Sⁿ)^∧G and structure maps given by the obvious homeomorphisms.

(ii) Let X be a G-spectrum and let K be a G-space. There is a G-spectrum map(K, X) with map(K, X)_n= map(K, X_n).

(iii) Let K be a G-space and let X be a G-spectrum. There is a G-spectrum K ∧X with (K ∧X)_n = K ∧X_n. The spectrum K ∧S is traditionally denoted Σ^∞_GK.

Let H be a subgroup of G. Given an H map α: S^m∧(Sⁿ)^∧G → X_n+k we can produce the H-map

S^m∧(Sⁿ⁺¹)^∧G ∼= (S¹)^∧G∧S^m∧(Sⁿ)^∧G−−−→^id∧α (S¹)^∧G∧X_n+k ^σ

X

−−−→n+k X_n+1+k. Since this is compatible with H-homotopy we obtain a function

[S^m∧(Sⁿ)^∧G, X_n+k]_H →[S^m∧(Sⁿ⁺¹)^∧G, X_n+1+k]_H

Definition 2.3. A mapf: X →Y ofG-spectra is aweak equivalenceif the function colim

n [S^m∧(Sⁿ)^∧G, X_n+k]_H →colim

n [S^m∧(Sⁿ)^∧G, Y_n+k]_H is bijective for every subgroup H of G and allm, k ≥0.

Remark 2.4. This definition is taken from the paper “Enriched Functors and Mo- tivic Stable Homotopy Theory” of Dundas, R¨ondigs and Østvær, with the only difference that we work with topological spaces as opposed to simplicial sets.

Remark 2.5. The set colim

n [S^m,(Sⁿ)^∧G, X_n]_H is the underlying set of an abelian group usually denoted π^H_m(X). This is them-th equivariant stable homotopy groups of the H-fixed points of X. We shall later see that it also is possible to define negative homotopy groups of G-spectra.

Theorem 2.6. There exists a category hoSp^G, the stable equivariant category, and a functor γ: Sp^G → hoSp^G with the property that for every functor F: Sp^G → C satisfying that F(f) is an isomorphism for every weak equivalence f of G-spectra there exists a unique functor F: hoSp^G → C with F =F ◦γ.

Note that the category hoSp^G is unique up to isomorphism.

Remark 2.7. In the case G={e}we recover the familiar category of spectra.

Lecture 2

A G-spectrum X as defined last week consists of a sequence (X₀, X₁, . . .) of G- spaces together with G-maps σ_n^X: (S¹)^∧G ∧X_n → X_n+1. A map f: X → Y of G-spectra is a weak equivalence if the function

colim

n [S^m∧(Sⁿ)^∧G, X_n+k]_H →colim

n [S^m∧(Sⁿ)^∧G, Y_n+k]_H

is bijective for every subgroup H of G and all m, k ≥0. We ended the last lecture by claiming that there exists a localizationγ:Sp^G →hoSp^G with the property any functor fromSp^G taking weak equivalences to isomorphisms factors throughγ. The category hoSp^G, the equivariant stable category, is our object of study.

The statement of the following theorem means that hoSp^G(X, Y) is an abelian group for allG-spectraX andY, and that composition in hoSp^G is compatible with the abelian group structure.

Theorem 2.8. The category hoSp^G has an enrichement in the category of abelian groups.

As a precusor of what is to come we mention that hoSp^G(S,S) is isomorphic to the unerlying abelian group of the Brunside ring, that is, the group-completion of the set of isomorphism classes of finiteG-sets with addition given by disjoint union.

We will also give a hint on how to obtain the ring-structure on hoSp^G(S,S).

The category Sp^G of G-spectra admits many variations in the sense that there exist many categories with weak equivalences having the same homotopy category.

We describe some right away. Let A denote a finite G-set. A (G, A)-spectrum Y consists of a sequence (Y₀, Y₁, Y₂, . . .) of G-spaces together with G-maps

σ_n^Y : (S¹)^∧A∧Yn→Yn+1.

Thus a (G, G)-spectrum is exactly the same as a G-spectrum. As before we can form the colimit colim

n [(Sⁿ)^∧A, Yn+k]H for every subgroup H of G and every k ∈Z, and we can use this colimit to define weak equivalences of (G, A)-spectra. As for G-spectra there is a resulting homotopy category hoSp^(G,A).

Proposition 2.9. IfAis a finite freeG-set, then there is an equivalence of categories between hoSp^G and hoSp^(G,A).

Proof. A choise of representatives of the G-orbits of A gives a G-equivariant iso- morphism A ∼= G×G\A, where G acts trivially on G\A. There is an induced homeomorphism (S¹)^∧A ∼= ((S¹)^∧G)^∧G\A. Given a (G, A)-spectrumY we construct a G-spectrum X with X_|G\A|n+k = (S^k)^∧G ∧Y_n for k = 0,1, . . . ,|G\A| −1. The structure mapsσ_|G\A|n+k^X are obvious homeomorphisms fork = 0, . . . ,|G\A| −2 and σ|G\A|n+|G\A|−1^X is given by the composition

(S¹)^∧G∧X|G\A|n+|G\A|−1 ∼= (S¹)^∧A∧Y_n →Y_n+1 =X_|G\A|(n+1).

Conversely, given a G-spectrum X, there is an associated (G, A)-spectrumY with Y_n =Xn|G\A|. In order to check that this gives rise to an equivalence of homotopy

categories we note that |G\A|N is cofinal in N.

The above equivalence of categories is preferred in the sense that up to isomor- phism it is independant of the choises made in the proof.

Above we only considered free G-sets, but when we come to the socalled Adams isomorphism we shall need the categories hoSp^(G,A) for A of the form G/N for a normal subgroup N of G.

Remark 2.10. The above preferred eqivalence of categories is the source of several kinds of trouble. There is an elegant way to avoid by choosing a different model for the equivariant stable category. (There are many good choises, but I tend to prefer the ones obtained as functor categories. Simplicial functors or orthogonal spectra are two such.)

One problem arises when we want to form smash-products ofG-spectraX andY. There is a (G, GqG)-spectrumX∧Y with (X∧Y)_n =X_n∧Y_n, and this gives rise to a symmetric monoidal pairing ∧on the homotopy category hoSp^G. In particular we can talk about monoids in the stable equivariant category. However there are reasons to desire a more restrictive concept of monoids. For example if we want to work with free resolutions and similar concepts known from homological algebra. Most of these technical problems are solved by now but it took a lot of work before elegant solutions were found. The search for such categories of spectra led to interaction with representation theory, theoretical physics and homological algebra through the theory of higher coherences.

Before stating some fundamental results relating the equivariant stable category for G to the equivariant stable category for subgroups of G we need the notation introduced in the next example.

Example 2.11.

(i) If H →G is a homomorphism of groups, then everyG-spectrum X can be considered as an (H, G)-spectrum res^H_GX by neglect of structure. Thus the is a functor res^H_G: hoSp^G →hoSp^H.

(ii) IfH is a subgroup ofGandY is an (H, G)-spectrum, there is aG-spectrum G₊∧_H Y with (G₊∧_H Y)_n = G₊∧_H Y_n, where the subsctipt H indicates that we have divided G₊∧Y_n by the equivalence relation (a, hy) ∼(ah, y) for a ∈ G, h ∈ H and y ∈ Y_n. The action of G on G₊ ∧_H Y_n is defined by g · (a, y) := (ga, y). The structure map of G₊ ∧_H Y is given by the composition

(S¹)^∧G∧(G₊∧_H Y_n) ∼= G₊∧_H ((S¹)^∧G∧Y_n)

→ G₊∧Y_(n+1).

(The above homeomorphism takes (s, g, y) to (g, g⁻¹s, y).) Thus there is a functor hoSp^H →hoSp^G, Y 7→G+∧Y.

(iii) If H is a subgroup ofG andY is a (H, G)-spectrum, there is a G-spectrum map_H(G₊, Y) with map_H(G₊, Y)_n = map_H(G₊, Y_n). The action of G on map_H(G₊, Y_n) is given by (g ·f)(a) = f(ag). Let us note that there is a G-map (S¹)^∧G∧map_H(G₊, Y_n) →map_H(G₊,(S¹)^∧G∧Y_n) taking (s, f) to the map g 7→ gs∧f(g). The structure map for map_H(G₊, Y) is given by

the compositon

(S¹)^∧G∧map_H(G₊, Y_n) → map_H(G₊,(S¹)^∧G∧Y_n)

→ map_H(G₊, Y_(n+1)).

Thus there is a functor Y 7→map_H(G₊, Y) from hoSp^H to hoSp^G. Let H be a subgroup of G and letY be an (H, G)-spectrum. There is a map

i: G₊∧_H Y →map_H(G₊, Y) of G-spectra induced by the injective maps

G₊∧_H Y_n →map_H(G₊, Y_n)

taking (a, y) ∈ G₊∧Y_n to the function f: G₊ → Y_n taking b ∈ G₊ to hy if there exists h∈H with b=ha⁻¹ and taking b ∈G+ to the basepoint of Yn otherwise.

Theorem 2.12. The map i is a weak equivalence of G-spectra.

The above theorem can be used to obtain an isomorphism of the form hoSp^H(res^H_GX, Y)∼= hoSp^G(X,map_H(G, Y))∼= hoSp^G(X, G+∧H Y)

for a G-spectrum X and H-spectrum Y. Lewis, May and Steinberger call this the generalized Wirtm¨uller isomorphism. In view of the much simpler isomorphism

hoSp^H(Y,res^H_GX)∼= hoSp^G(G₊∧_H Y, X)

this shows that Y 7→G₊∧_H Y is both a left- and a right adjoint functor.

Lecture 3

Let H be a subgroup of Gand let Y be an (H, G)-spectrum. Thus Y consists of H-spacesY₀, Y₁, . . . and H-maps (S¹)^∧G∧Y_n→Y_n+1. We ended the last lecture by observing that there is a map

i: G+∧H Y →map_H(G+, Y)

ofG-spectra and stating as a theorem that iis a weak equivalence ofG-spectra. For good historical reasons this result is refered to as the Wirhtm¨uller isomorphism. We pause to remark on a structural consequence of this result.

There is an isomorphism G/H₊ ∧ Y ∼= G₊ ∧_H Y induced by the map taking (g, y) to (g, g⁻¹y). If K ⊆ H ⊆ G are inclusions of subgroups of G, and X is a G-spectrum, then there is a morphism G/H₊∧X →G/K₊∧X in hoSp^G given by the composition

G/H₊∧X ∼=G₊∧_HX →map_H(G₊, X)→map_K(G₊, X)←^'−G₊∧_KX ∼=G/K₊∧X.

Elaborating on this the above theorem can also be used to obtain a morphism B₊∧X →A₊∧X in hoSp^G for every G-map A→B of finite (unpointed)G-sets.

On the other hand, we have the morphismA₊∧X →B₊∧Xinduced by theG-map A+ →B+. The above maps give rise to Mackey-Functors. Given G-spectra X and Y and a finite G-set A, we let MA = M^A = hoSp^G(A+∧X, Y). Given a G-map A →B as above we obtain homomorphisms M^f: M^B →M^A and M_f: M_A →M_B induced by the above morphisms of G-spectra. This is a Mackey-Functor in the following sense:

(i) We have M_A=M^A for every finite G-set A.

(ii) If A = A₁ ∪ A₂ is a disjoint union of G-sets, then the homomorphism M^A → M^A1 ×M^A2 induced by the inclusions A₁ ⊆ A and A₁ ⊆ A is an isomorphism.

(iii) The assignment f 7→ M^f is a contravariant functor from finite G-sets to abelian groups.

(iv) The assignmentf 7→Mf is a covariant functor from finite G-sets to abelian groups.

(v) If

A₀ −−−→^f A₁

g





y ^g

0



 y

B₀ ^f

0

−−−→ B₁

is a pull-back diagram of finite G-sets, then M_f ◦M^g =M^g0 ◦M_f⁰.

Mackey-functors also enter in the study of representations of finite groups. The homological algebra of Mackey functors, introduced by Dress, is highly related to equivariant stable homotopy theory. Taking X = Y = S we obtain a Mackey- functor Ω with Ω_A equal to the Burnside ring of A, that is, the ring obtained by group-completing the semi-ring of isomorphism classes ofG-sets overAwith addition given by disjoint union and with multiplication given by fibred product over A.

The above theorem has a companion theorem, originally due to Adams, concerning surjectionsf: G→J of groups instead of subgroups ofG. Let N denote the kernel of the surjection f: G→J. The diagonal embedding induces a J-homeomorphism (S¹)^∧J ∼= ((S¹)^∧G)^N. Thus to aG-spectrumX we can associate aJ-spectrum Φ^NX with (Φ^NX)_n= (X_n)^N. The structure map is given by the composition

(S¹)^∧J∧X_n^N ∼= ((S¹)^∧G)^N ∧X_n^N = ((S¹)^∧G∧X_n)^N →X_n+1^N .

This is the geometric fixed point spectrum of X. There is another construction of fixed points that goes as follows: LetV denote theG-inner product spaceR[G]. Then S^V ∼= (S¹)^∧G. Note that the one-point compactification has the property that it takes sums to smash-products. More precisely, there is a homeomorphismS^V1⊕V2 → S^V1 ∧S^V2 taking (v₁, v₂) ∈ V₁ ×V₂ = V₁⊕V₂ ⊆ S^V1⊕V2 to v₁ ∧v₂ ∈ S^V1 ∧S^V2 and taking ∞to∞ ∧ ∞. This map is clearly continous and bijective, so by compactness it is a homeomorphism. Now any subspace W ⊆V has an orthogonal complement V −W. In particular S^V ∼=S^W ∧S^V−W. We define XnW = map(S^n(V−W), Xn) for W ⊆ V and n ≥ 1. In particular X_nV ∼= X_n. This is a (G, S^W)-spectrum in the sense that there is a structure map S^W ∧X_nW →X_(n+1)W given by the composition S^W ∧X_nW →map(S^n(V−W), S^W ∧X_n)→map(S^(n+1)(V−W), S^V ∧X_n)→X_(n+1)W, where the first map takes (w, f) tox7→f(x)∧w, the second map is smash-product with the identity on S^V−W, and the last map is induced by the structure map ofX.

The fixed point spectrum X^N is the J-spectrum with (X_nV^N)^N as n-th space and with structure map given by the composition

(S¹)^∧J ∧(X_nV^N)^N ∼=S^VN ∧(X_nV^N)^N ∼= (S^VN ∧X_nV^N)^N →X_(n+1)V^N N.

In order for this construction to carry homotopical meaning we need X to be an

“Ω-spectrum” or “fibrant”. We will come back to this.

Remark 2.13. It is not difficult extend the above construction to a prespectrum in the sense of Peter May et al, that is, to a collection of G-spaces X_U for all G- representations U in a complete G-universe together with structure maps S^U−U0 ∧ X_U⁰ → X_U. Hint: start by letting X_W1⊕···⊕W_n = map(S^{(V−W1)⊕···⊕(V−Wn)}, X_n) for irreducible subrepresentations W₁, . . . , W_n of V. Up to homeomorphism this is in- dependent of the order of W₁, . . . , W_n. Continue by choosing one representative for each irreducible representation.

Consider a (G, J)-spectrumZ, that is, a sequenceZ0, Z1, . . . ofG-spaces together with maps (S¹)^∧J ∧Zn → Zn+1. Using inner product spaces we obtain a homeo- morphism of G-spaces of the form (S¹)^∧G∼=S^V ∼=S^VN∧S^V−VN ∼= (S¹)^∧J∧S^W for W =V −V^N. There is a G-spectrumi_∗Z with (i_∗Z)_n =Z_n∧S^nW. The structure map (S¹)^∧G∧(i∗Z)_n→(i∗Z)_n+1 is given by the composition

(S¹)^∧G∧Z_n∧S^nW ∼= (S¹)^∧J ∧S^W ∧Z_n∧S^nW

∼= (S¹)^∧J ∧Zn∧S^(n+1)W

→ Z_n+1∧S^(n+1)W.

The generalized Adams isomorphism gives a description of the J-spectrum (i∗Z)^N under the assumption that the normal subgroup N acts trivially on the spaces Zn. LetN\Z denote theJ-spectrum given by the sequenceN\Z₀, N\Z₁, . . . ofJ-spaces together with the structure maps

(S¹)^∧J ∧N\Zn∼= ((S¹)^∧G)^N ∧N\Zn ∼=N\(((S¹)^∧G)^N ∧Zn)→N\Zn+1. Theorem 2.14 (generalized Adams isomorphism). If N acts trivially on the spaces Z_n, then there exists a weak equivalence trf : N\Z →(i∗Z)^N of J-spectra.

The above isomorphism theorem can also be formulated as a kind of adjunction.

Let Z be an N-free (G, J)-spectrum and let Y be a J-spectrum. Then there is a natural isomorphism

hoSp^J(Y, N\Z)→hoSp^J(Y,(i_∗Z)^N)∼= hoSp^G(res^G_JY, i_∗Z).

There is a second Adams isomorphism, of the form

hoSp^J(N\Z, Y)∼= hoSp^G(i∗Z,res^G_JY),

also for Z an N-free (G, J)-spectrum. A technical remark: I would like the N- freeness to be coupled with a statement about cofibrancy.

3. Plan of the seminar

I suggest that we take as our first goal to prove the above theorems. They are all contained in the first 100 pages of the book of Lewis, May and Steinberger.

However we can try to stay more elementary. First we could digress a little on G-spaces and G-CW complexes. Next we could prove the Wirhtm¨uller- and the Adams-isomorphism for suspension spectra following Adams. After that we can turn to the existence of the equivariant stable category and the proof of the general versions of the Adams- and Wirhtm¨uller isomorphisms. When we are there we can choose between either algebraic of geometric aspects of equivariant stable homotopy theory, for example Mackey functors or bordism theory. I expect us to need one to two session on CW-complexes and two or three sessions on the paper of Adams.

There are many aspects of equivariant stable homotopy theory that I have not at all touched yet, such as:

• The Segal conjecture.

• Homotopy orbits and homotopy fixed points.

• The Segal-tom Dieck splitting and geometric fixed points.

• Highly structured smash-product- and function G-spectra.

• Equivariant bordism and Thom spectra.

• Equivariant K-theory.

• Equivariant homology and cohomology theories.

The approved plan is to start from the top of the list and to make a fresh start when we get to highly structured smashproducts and function spectra.

lecture 4 4. G-CW complexes

In this lecture we will introduce G-CW complexes and outline some rudimentary facts about them. We start with the definition of G-CW complexes. Then we prove the G-versions of the Whitehead- and Freudental theorems. (Adams 2.7 and 3.3, tom Dieck 2.6 and 2.10).

At this point I unfortunately have to back up on notation. In the last lectures we were mostly interrested in pointedG-spaces, and we agreed thatG-space should mean pointed G-space. In this section we also need to work with unpointed spaces, and thus we mean UNPOINTED G-space when we say G-space here.

We extend the category of pointed spaces by considering the category of pairs (X, A) of spaces with A a subspace of X. (When we insist on A being a one- point space we obtain the category of pointed spaces.) Given a pust-out diagram of UNPOINTED G-spaces of the form

A −−−→ X



 y



 y B −−−→ Y,

whereA→B is a closed cofibration, we say thatY is obtained fromX by attaching B along A. (Recall that A → B is a cofibration if there exists a retract of the map A× [0,1]∪A×{0} B × {0} → B × [0,1]. It is closed if the images of closed subsets of A are colsed in B. In the above situation it is a formal consequence that X → Y is a closed cofibration.) In the special situation where A is of the form A = `

j∈JG/H_j ×Sⁿ⁻¹, where B is of the form B =`

j∈JG/H_j ×Dⁿ and where A→B is the standard inclusion we say thatY is obtained by attachingn-cells toX.

A relative G-CW-complex is a pair (X, A) of G-spaces together with an increasing sequence

A=X−1 ⊆X₀ ⊆X₁ ⊆. . . of subspaces of X satisfying

(i) X is the union of the spaces X_n and a subset of X is closed if and only its intersection with each of the spaces X_n is closed.

(ii) For every n≥0 the space X_n is obtained from Xn−1 by attaching n-cells.

If (X,∅) is a relative G-CW-complex, then we say that X is a G-CW-complex.

Example 4.1.

(i) If X and Y are G-CW-complexes, then X ×Y is a G-CW-complex with (X×Y)_n =S

k+l=nX_k×Y_land with cells given by decomposition of products of cells for X and Y.

(ii) If (X, A) is a relativeG-CW-complex, then (X/A, A/A) is a relativeG-CW- complex with the same cells.

Example 4.2. For every finite G-set A the underlying G-space of the pointed G- space (S¹)^∧A is a G-CW-complex. More generally, if V is a finite-dimensional real

inner-product space with an action of Gpreserving the inner product, then the one- point compactification S^V =V ∪ {∞} is a a finite G-CW-complex. In order to see this we choose a complete fan Σ on V with the property that C ∈ Σ and gC =C implies that gv = v for every v ∈ C. (The following construction shows that any fan can be subdivided into a fan with this property: Suppose that gC = C and that gx 6= x for some x ∈ C. The hyperplane in V consisting of points with the same distance to x and gx cuts C into two cones C₁ and C₂ with gC₁ =C₂.) For each g ∈G we obtain a new fan gΣ consisting of the cones gC for C ∈Σ. The fan Σ^G =∩g∈GgΣ with cones of the form∩g∈GC_g, whereC_g ∈gΣ has the property that g permutes the cones and that C ∈ Σ^G and gC = C implies that gv= v for every v ∈C. IfV_k denotes the union of cones in Σ^G of dimension less that or equal tok, then we have an increasing sequence

{∅} ⊆V0 ⊆V1· · · ⊆V

of subspaces of V, where V_k is obtained from Vk−1 by attaching k-cones, that is, there is a pust-out diagram of the form

`

C∈Σ^G,dim(C)=k∂C −−−→ Vk−1



 y



 y

`

C∈Σ^G,dim(C)=kC −−−→ V_k.

Taking one-point compactification we obtain push-out diagrams of the form

`

j∈JG/Hj ×S^k−1 `

C∈Σ^G,dim(C)=kS^k−1 −−−→ S^Vk−1



 y



 y

`

j∈JG/Hj ×D^k `

C∈Σ^G,dim(C)=kD^k −−−→ S^Vk.

The dimension dim(X, A) of a relative G-CW-complex (X, A) is the function on the set of subgroups of G with dim(X, A)(H) equal to the supremum of the set {n: X_n−1^H 6=X_n^H}. If A=∅is the empty space we write dim(X) = dim(X,∅).

Let ν denote a function from the set of subgroups of G to the natural numbers.

A pair (Y, B) ofG-spaces is ν-connected if for every relative G-CW-complex (X, A) with dim(X, A)≤ν every map f: (X, A)→ (Y, B) is homotpoic relA to a map of X into B. Here homotopic rel A means that the homotopy is constant on A, that is, the homotopy F: X×[0,1]→Y satisfies

(i) F(x,0) =f(x) for all x∈X,

(ii) F(a, t) = f(a) for all a∈A and all t∈I and (iii) F(x,1)∈A for all x∈X.

Exercise 4.3. Suppose that ν is constant on conjugacy classes of subgroups of G.

Show that a pair (Y, B) is isν-connected if and only if for every n and H ≤Gwith n ≤ν(H), any map (G/H×Dⁿ, G/H×Sⁿ⁻¹)→(Y, B) is homotopic relG/H×Sⁿ⁻¹ to a map of G/H×Dⁿ into B. Deduce from this that the pair (Y^H, B^H) is ν(H)- connected. (Hint: tom Dieck Proposition 1.5 on page 97)

5. The Whitehead Theorem

Lemma 5.1. Let (X, A)a relativeG-CW-complex of dimensiondim(X, A)≤ν and let (Y, B)be a ν-connected pair of G-spaces. Given any map h: X× {0} ∪A×{0}A× [0,1] → Y with h(a,1) ∈ B for all a ∈ A, there exists a map H: X ×[0,1] → Y satisfying

(i) H(x,0) =h(x,0) for all x∈X,

(ii) H(a, t) = h(a,2t) for all a∈A and all t∈[0,¹₂] and (iii) H(a, t) = H(a,1) for all a∈A and all t ∈[¹₂,1]

(iv) H(x,1)∈B for all x∈X.

Proof. Since A ⊆ X is a cofibration there exists H₁: X ×[0,1] → Y such that H₁(x,0) =h(x,0) all x ∈ X and H₁(a, t) = h(a, t) for all a ∈ A and all t ∈ [0,1].

Since (Y, B) isν-connected and dim(X, A)≤νthere existsH₂: X×[0,1]→Y with (i) H₂(x,0) =H₁(x,1) all x∈X,

(ii) H₂(a, t) = H₂(a,0) all a∈A and all t∈[0,1], (iii) H₂(x,1)∈B all x∈X.

The map H defined by H(x, t) =H₁(x,2t) for t∈[0,¹₂] and H(x, t) =H₂(x,2t−1)

for t∈[¹₂,1] has the desired property.

Note that iff, g:X →Y are homotopic, then there exists a homotopyH: X×I → Y betwen f and g with H(x, t) =H(x,1) for t∈[¹₂,1].

Proposition 5.2. Let (Y, B) be a ν-connected pair and let X be a G-CW-complex.

The induced map[X, B]_G →[X, Y]_GofG-homotopy classes is surjective ifdim(X)≤ ν and it is bijective if dim(X)≤ν−1.

Proof. Apply the above lemma to (X,∅) in order to get surjectivity and apply it to (X×I, X×∂I) in order to get injectivity.

Recall that the mapping zylinder Z_h of a G-map h: B →Y is the space Z_h = (B×I)∪(B×{1})Y = (B ×I)qY /∼,

where (b,1)∼ h(b). Note that B ∼= B× {0} ⊆ Z_h is a closed cofibration and that there is a homotopy equivalence Z_h → Y. A map h: B → Y is ν-connected if the pair (Z_h, B) is ν-connected.

Proposition 5.3. Let f: Y → Z be ν-connected and let X be a G-CW-complex.

The function f∗: [X, Y]G →[X, Z]G is surjective if dim(X) ≤ν, and it is bijective if dim(X)≤ν−1.

In the following formulation the above proposition is a G-equivariant version of the Whitehead theorem.

Corollary 5.4. Every map f of G-CW-complexes which isν-connected for every ν is a G-homotopy equivalence.

Proof. This is pure category theory. The map f∗: [X, Y]_G → [X, Z]_G is bijective for all G-CW-complexes X. (Here we use that the above proposition also holds for ν = ∞.) In particular there exists g: Z → Y with f∗(g) = [f ◦g] = [id_Z]. Now

f∗: [Y, Y]G → [Y, Z]G is bijective and f∗(g∗(f)) = [f ◦g◦f] = [f] = f∗(idY), and

thus g∗(f) = [g◦f] = [idY]

Lecture 5

6. The Wirtm¨uller isomorphism

In this section we follow tom Dieck and Wirtm¨uller’s proof of the so-called Wirt- m¨uller isomorphism, stated in lecture two as Theorem 2.11.

Given a subgroup H of G and an (H, G)-spectrum Y we considered the G- spectrum map

i:G₊∧_H Y →map_H(G₊, Y), (g, y)7→ g₁ 7→

(g₁gy if g₁g ∈H

∗ otherwise.

!

The claim of the theorem is that i is a weak equivalence. In the special case, where Y_n = (S^n−m)^∧G∧Y_m and the structure maps forY are the obvious hoemeomrphisms, this claim follows from the following lemma. In particular i is a weak equivalence when Y =S∧Y0 is a suspension spectrum.

Lemma 6.1. For every k∈Z and every n, j ∈N with n+k−m≥j+ 2 the map [S^j∧(Sⁿ)^∧G, G₊∧_H Y_n+k]⁰_G →[S^j ∧(Sⁿ)^∧G,map_H(G₊, Y_n+k)]⁰_G

is bijective.

Proof. In order to simplify notation we writeB forY_n+k. LetK be a subgroup ofG.

LetK be a subgroup ofGand let D=K\G/H. Considered as a map ofK-spaces, the map j has the following decomposition

G₊∧_H B −−−→ map_H(G₊, B)

W

KgH∈DKgH₊∧_H B −−−→ Q

KgH∈Dmap_H(Hg⁻¹K₊, B).

If g⁻¹Kg⊆H, then the maps

Fix(g⁻¹Kg, B)→KgH+∧H B, b 7→g∧b and

KgH₊∧_H B →Fix(g⁻¹Kg, B), kgh∧b =gh∧b =g∧hb7→hb

are inverse homeomorphisms. If g⁻¹Kg is not contained in H, then we can choose k ∈K with g⁻¹kg /∈H. Then for everyb ∈B we have

g⁻¹kb /∈H ⇒e∧b g⁻¹kg∧b ⇒g∧bkg∧b, and thus (KgH₊∧_H B)^K =∗ if g⁻¹Kg is not contained in H.

On the other hand there is a homeomorphism

map_H(Hg⁻¹K₊, B)^K →Fix(g⁻¹Kg∩H, B), f 7→f(g⁻¹), whose inverse takes a fixed point y to the map hg⁻¹k 7→hy.

If we let D(1) ={KgH ∈D:g⁻¹Kg⊆H} and D(2) =D\D(1), then i has the

form W

KgH∈D(1)Fix(g⁻¹Kg, B)

i



 y Q

KgH∈D(1)Fix(g⁻¹Kg, B)×Q

KgH∈D(2)Fix(g⁻¹Kg∩H, B).

Let V =R[G]^n+k−m and recall that B =Yn+k= (S^n+k−m)^∧G∧Y0 ∼=S^V ∧Y0. Since (S^V)^K is a sphere of dimension

dim((S^V)^K) = dim(V^K) = (n+k−m)|G/K|,

the value of the connectivity of i at the subgroup K is at least the minimumν(K) of

{(n+k−m)|G/(g⁻¹Kg∩H)| −1 = dim Fix(g⁻¹Kg∩H, V)−1 : KgH ∈D(2)}

and

{2(n+k−m)|G/g⁻¹Kg| −1 = 2 dim Fix(g⁻¹Kg, V)−1 :KgH ∈D(1)}.

In particular

dim(S^j ∧Fix(K, S^V)) =j + (n+k−m)|G/K| ≤ν(K)−1

for every subgroup K of G. The statement of the lemma is now an application of

Proposition 5.3 proceeding the Whitehead theorem.

We have now in particular proven the Wirhtm¨uller isomorphism theorem for sus- pensionG-spectra. The passage to an arbitraryG-spectrumY is rather formal. First we writ Y as a homotopy colimit of spectra Yhmi with Yhmi_n+m = (Sⁿ)^∧G ∧Y_m. Now we only need to know that bothY 7→G+∧HY and Y 7→map_H(G+, Y) respect the passage to directed homotopy colimits. I will return to this next week, where I will also give a description of an inverse map to i.

Lecture 6

7. The Adams-isomorphism

Before we go on we need to repair the fixed-point construction, and to fix a small problem in our definition of weak equivalences.

For now let V be an inner-product real G-representation. Given a (G, S^V)- spectrum Z, we define a new (G, S^V)-spectrumQZ by

(QZ)_m = (QZ)(S^mV) = hocolim

n map(S^nV, Z(S^(m+n)V)).

Definition 7.1. A map X → Y of (G, S^V)-spectra is a weak equivalence if the induced map QX(S^mV)→QY(S^mV) is a weak equivalence for every m≥0.

This almost amounts to our old definition, in that it is equivalent to colim

n [S^k∧S^nV, X(S^(m+n)V)]_H →colim

n [S^k∧S^nV, Y(S^(m+n)V)]_H being a bijection for every m, k ≥0 and every H ≤G.

There is a map Z → QZ of (G, S^V)-spectra, and by inspection on the definition this is a weak equivalence. This is somewhat confusing. It is a good exercise to work this out. (A reference discussing this is a paper on general symmetric spectra of Hovey.) Let me warn that, although it is not explicit from the notation, the construction Z 7→QZ depends on V.

Given W ⊆ V we let (i^∗Z)(S^W) = Z(S^W) := map(S^V−W, Z(S^V)), and similarly (i^∗Z)(S^nW) = Z(S^nW) := map(S^n(V−W), Z(S^nV)). This is a (G, S^W)-spectrum i^∗Z.

Given H ⊆ G, the fixed point spectrum of Z is the spectrum (N H/H, S^VH)- spectrum Q(Z)^H with Q(Z)^H(S^mVH) = (Q(Z)(S^mVH))^H.

Last time we proved that if Z is a G-spectrum, then there is a weak equivalence G₊∧_H Z →map_H(G₊, Z) for every subgroup H ofG. In view of the isomorphisms

G₊∧_H Z →G/H₊∧Z, (g, z)7→(g, gz) and

map_H(G₊, Z)→map(G/H₊, Z), f 7→(gh 7→gf(g⁻¹)), we have actually proved:

Theorem 7.2. For every G-spectrum Z and every finite G-set A, the inclusion A₊∧Z →map(A₊, Z)

of a wedge of copies of Z into a product of copies of Z is a weak equivalence.

Definition 7.3. We say that a (G, S^W)-spectrum isbounded below if there exists a constant c such thatZ(S^nW) is (ndim(W)−c)-connected for all n. (Here dim(W) is a function on the conjugacy classes of subgroups of G.)

From now on V shall denote the representationV =R[G]. The following theorem is a consequence of the Freudenthal suspension theorem. I choose to postpone the proof.

Theorem 7.4 (Adams Proposition 5.5). Let N be a normal subgroup of G, let Z be a bounded below (G, S^VN)-spectrum and let X be a finite pointedG-CW-complex.

If N acts freely on X, then the map colim

n [X∧S^m∧S^nVN, Z(S^(n+k)VN)]_H → colim

n [X∧S^m∧S^nVN ∧S^n(V−VN), Z(S^(n+k)VN)∧S^n(V−VN)]_H is bijective for every subgroup H of G and every m, k ≥0.

Now we are almost able to formulate the Adams-isomorphism. For every bounded below (G/N, S^VN)-spectrumY and every finiteN-freeG-setAwe have the following sequence of weak equivalences for Z = map(A+, j^∗Y):

(QZ(S^mVN))^N

∼=

−→ hocolim

n map(S^nVN, Z(S^(m+n)VN))^N

−→' hocolim

n map(S^nVN ∧S^{(m+n)(V−VN)}, Z(S^(m+n)VN)∧S^{(m+n)(V−VN)})^N

∼=

−→ hocolim

n map(S^m(V−VN),map(S^nV, Z(S^(m+n)VN)∧S^{(m+n)(V−VN)}))^N

−→' map(S^m(V−VN),hocolim

n map(S^nV, Z(S^(m+n)VN)∧S^{(m+n)(V−VN)}))^N

= map(S^m(V−VN), Q(i∗Z)(S^mV))^N

= Q(i∗Z)^N(S^mVN)

Letj denote the projectionj: G→G/N. IfA is aN-finite free G-set andY is a (G/N, S^VN)-spectrum, then we have the following zig-zag chain of weak equivalences between (A₊∧j^∗Y)/N and Q(i∗(A₊∧j^∗Y))^N:

(A₊∧j^∗Y)/N ∼= A/N₊∧Y

−→' Q(A/N₊∧Y)

−→' Q(map(A/N₊, Y))

∼= Q(map(A+, j^∗Y))^N

−→' Q(i∗map(A+, j^∗Y))^N

∼= Q(map(A₊, i∗j^∗Y))^N

←'− Q(A₊∧i∗j^∗Y)^N

∼= Q(i∗(A₊∧j^∗Y))^N.

Here the first arrow is a weak equivalence by the above general remark, the second and third arrows are weak equivalences by the Wirthm¨uller isomorphism, and the last arrow is the above sequence of weak equivalences.

Proposition 7.5. Up to weak equivalence the functorsZ 7→Z/N andZ 7→(Q(i∗Z))^N commute with homotopy colimits ofN-free (G, S^VN)-spectra bounded below by a com- mon bound. In particular they commute with geometric realizations of such.

Proof. The statement about the functorZ 7→Z/N is a consequence of this functor being a left Quillen functor. The statement about Z 7→(Q(i∗Z))^N is a connectivty estimate using a particular simplicial model of the homotopy colimit. (Compare wedge-sums with restricted products by an estimate using connectivity.) (Still only

on my sheets A and B!)

If we can produce a natural (in A₊∧j^∗Y) weak equivalence (A₊∧j^∗Y)/N → Q(i_∗(A₊∧j^∗Y))^N, then we can conclude:

Theorem 7.6 (The Adams isomorphism). For every N-free and bounded below (G, S^VN)-spectrum Z, there is a natural weak equivalence Z/N →(Q(i_∗Z))^N. Proof. Every N-free and bounded below (G, S^VN)-spectrum Z can be obtained as a retract of a homotopy colimit of spectra of the form G/H+ ∧j^∗Y. Actually it suffices to consider Y of the form ΣⁿS⁰ and ΣⁿI forn ∈Z. Thus for an N-free (G, S^VN)-spectrum Z we want to construct a map Z/N → Q(i∗Z)^N combining to a commutative diagram with the above. We work withG×G- sets. Let

G−→^ι1 G×G−^p→¹ G and G−→^ι2 G×G−^p→² G

denote the canonical inclusions and projections and letG×Gact onG+ by the rule (g₁, g₂)·a=g₂ag⁻¹₁ . Given a G-space K, there is a G×G-homeomorphism

G₊∧p^∗₁K →G₊∧p^∗₂K, (a, k)7→(a, ak)

(test: (g1, g2)·(a, k) = (g2ag⁻¹₁ , g1k)7→(g2ag₁⁻¹, g2ak) = (g1, g2)·(a, ak)). Let

∆_N ∈map(G₊,map(G₊, p^∗₂Q₀(SV))^e×N)^G×G denote the composition

G₊→G/N₊ → map(G/N₊, S⁰)

→ map(G/N₊, p^∗₂Q₀(S^V))

∼= map(G+, p^∗₂Q₀(SV))^e×N.

LetE be aG×G-space satisfying thatE^H =∅ifH∩(N×N)6=eand thatE^H ' ∗ if H∩(N ×N) =e.

Lemma 7.7. The map

map(E+,map(G+, Q0(G+∧p^∗₂S^V)^e×N))→

map(E₊,map(G₊,map(G₊, p^∗₂Q₀(SV)^e×N))) is a weak equivalence of G×G-spaces.

Proof. It suffices to show that ifH∩(N×N) = e, then the induced map of H-fixed points is a weak equivalence. Let DG= {(g, g) : g ∈ G} ⊆ G×G. The projection

G×G→ G, (g₁, g₂) 7→g₂g₁⁻¹ defines a bijection (G×G)/DG∼= G of G×G-sets.

Given a G×G-space U, there are homeomorphisms

map(G+, U)^H ∼= map((G×G)/DG, U)^H

∼= Y

(G×G)/H

map(H/(H∩DG), U)^H

∼= Y

(G×G)/H

U^H∩DG

Because of this it sufices to note that since H∩(N ×N) = ethe map (Q₀(G₊∧p^∗₂SV)^e×N)^H∩DG →(map(G₊, p^∗₂Q₀(SV)^e×N))^H∩DG

is a weak equivalence. This is an application of the Wirthm¨uller isomorphism for

the group (e×N)×(H∩DG).

Using the above lemma we can choose an element τ⁰ mapping to an element in the path-component of the map on E₊ with value ∆_N on every non-base point. By adjunction the map τ⁰ corrresponds to a G×G-map

τ⁰⁰: E₊∧G₊ →Q₀(G₊∧p^∗₂SV)^e×N. Given a (G, S^VN)-spectrum Z we let τ_m denote the composition

p^∗₂Z(S^mVN)∧E₊∧G₊ ∼= p^∗₁Z(S^mVN)∧E₊∧G₊

id×τ⁰⁰

−−−→ p^∗₁Z(S^mVN)∧Q₀(G₊∧p^∗₂SV)^e×N

→ Q0(p^∗₁Z(S^mVN)∧G+∧p^∗₂S^V)^e×N

∼= Q0(p^∗₂Z(S^mVN)∧G+∧p^∗₂S^V)^e×N

→ Q0(p^∗₂Z(S^mVN)∧p^∗₂S^V)^e×N

= p^∗₂(Q(Z)^N(S^mVN))

→ p^∗₂(Q(i_∗Z))^N(S^mVN)

Dividing out the action ofi₁(G) = G×e ⊆G×Gwe obtain aG=G×G/(G×e)-map Z(S^mVN)∧E₊^δ ∼= (p^∗₂Z(S^mVN)∧E₊∧G₊)/(G×e)→

(p^∗₂(Q(i∗Z))^N(S^mVN))/(G×e) =Q(i∗Z))^N(S^mVN).

Heree E^δ denotes the space E with G-action through the diagonal. Dividing out the action of N we obtain a natural map

tm: Z(S^mVN ∧E+)/N →(Q(i∗Z))^N(S^mVN).

These maps assemble to a map t: (Z ∧E₊)/N →Q(i∗Z)^N of (G/N, S^VN)-spectra.

However the map (Z∧E₊)/N →Z/N induced by E₊ →S⁰ is a weak equivalence.

We now check that for Z of the form A₊ ∧Y, where G acts trivially on Y, the

map t is a weak equivalence. We do this by checking that the following diagram is commutative:

(A₊∧E₊)/N ∧Y −−−→^t Q(i_∗(A₊∧j^∗Y))^N



 y



 y

map(A₊, Q(j^∗Y))^N −−−→ map(A₊, Q(i∗j^∗Y))^N

In order to check this we turn back to G×G-spaces and note that the outher and the lower squares in the following diagram are commutative:

p^∗₁(A₊∧Y_m)∧E₊∧G₊ ^id∧τ

00

−−−→ p^∗₁(A₊∧Y_m)∧Q₀(G₊∧SV)^e×N



 y



 y

Q₀(p^∗₂A∧Y_m∧G₊∧p^∗₂SV) −−−→^∼= Q₀(p^∗₁A∧Y_m∧G₊∧p^∗₂SV)



 y



 y

map(p^∗₂A₊, Q₀(G₊∧Y_m∧p^∗₂SV))^e×N map(G₊, Q₀(p^∗₁A₊∧Y_m∧p^∗₂SV))^e×N



 y



 y

map(G₊∧p^∗₂A₊, Q₀(Y_m∧p^∗₂SV))^e×N −−−→^∼= map(G₊∧p^∗₁A₊, Q₀(∧Y_m∧p^∗₂SV))^e×N To see that the transfer t⁰ for A+ ∧Y ∧G+ has anything to do with the transfer for A₊∧Y we choose orbit representatives forAin order to obtain an isomorphism (A₊ ∧G₊)/N ∼= A/N₊∧G₊. Upon restricting the action to i₂(G) ⊆ G×G, the maps in the lower square in the above diagram all become weak equivalences, and thus the upper square commutes up to weak equivalence after restricting the action to i₂(G). From this point it is easy to obtain the commutativity of the diagram to be checked. Since every (G, S^VN)-spectrum is weakly equivalent to a retract of a cell-spectrum this together with the statement about homotopy colimits suffices to finish the proof of the Adams Isomorhism Theorem.

8. Suspension Spectra

In this section we shall work out a description of hoSp^G(S∧X,S∧Y) forG-spaces X and Y.

Lemma 8.1. Let H be a subgroup of G, let V =R[G] and let Z be a G-spectrum.

There is a natural isomorphism

hoSp^G(S, Z)∼= colim

n [S^nV, Z(S^nV)]⁰_G.

Proof. Let us construct natural maps both ways. Given a G-map S^nV → Z(S^nV) we obtain aG-mapS^mV ∼=S^(m−n)V ∧S^nV →S^(m−n)V ∧Z(S^nV)→Z(S^mV) for every m ≥n. If Z(n) and S(n) denote theG-spectra given by the sequences

Z(n) = (∗, . . . ,∗, Zn, Zn+1, . . .) S(n) = (∗, . . . ,∗, S^nV, S^(n+1)V, . . .),

then we have produced a map S(n) → Z(n). However, the obvious maps Z(n) → Z and S(n) → S are weak equivalences. Thus we have produced an element in hoSp^G(S, Z). This construction gives a map

colim

n [S^nV, Z(S^nV)]⁰_G→hoSp^G(S, Z).

Conversely, the functor

Sp^G → Ab, Z 7→colim

n [S^nV, Z(S^nV)]⁰_G

takes weak equivalence to isomorphisms, so there is an induced functor hoSp^G → Ab, Z 7→colim

n [S^nV, Z(S^nV)]⁰_G. In particular, we have a homomorphism

hoSp^G(S, Z) → Ab(colim

n [S^nV, S^nV]⁰_G,colim

n [S^nV, Z(S^nV)]⁰_G)

→ Ab(colim

n [G₊∧S^nV, S^nV]⁰_G,colim

n [S^nV, Z(S^nV)]⁰_G)

∼= Ab(Z,colim

n [S^nV, Z(S^nV)]⁰_G)

ev1

−−→ colim

n [S^nV, Z(S^nV)]⁰_G.

We leave it as an exercise to check that we have constructed inverse isomorphisms.

Theorem 8.2. If X is a G-CW-complex and Z is a G-spectrum, then there is a natural isomorphism

hoSp^G(S∧X, Z)∼= colim

n [S^nG∧X, Z(S^nV)]_G.

Proof. The functors Y 7→ Y ∧X and Z 7→map(X, Z) form a Quillen adjoint pair.

Therefore they pass to an isomorphism

hoSp^G(Y ∧X, Z)∼= hoSp^G(Y,map(X, Z)).

Using the above result we obtain isomorphisms

hoSp^G(S∧X, Z) ∼= hoSp^G(S,map(X, Z))

∼= colim

n [S^nV,map(X, Z(S^nV))]⁰_G

∼= colim

n [S^nV ∧X, Z(S^nV)]⁰_G

As a special case of the above theorem, we can take Z =S∧Y for a G-space Y and obtain an isomorhism

hoSp^G(S∧X,S∧Y)∼= colim

n [S^nG∧X, S^nV ∧Y]⁰_G.