NOTES ON COHOMOLOGY THEORIES, BROWN REPRESENTABILITY AND SPECTRA

ANDREW BAKER [17/01/2002]

1. Basic ideas of homotopy theory Let AbGp be the category of abelian groups. Let Top be the category of topological spaces or a ‘reasonable’ full subcategory, and let

Top∗ be the associated category of based spaces. We will also denote by h-Top and h-Top∗ the associated homotopy categories in which we factor the morphisms of Top and Top∗ by the homotopy relation (for Top∗ we have to use base-point preserving homotopies). There is a forgetful functor Top∗ Ã Top which sends (X, x0) to X; there is also a functor

()+ : Top à Top∗; X+ = X q {∗} which adjoins a disjoint base-point ∗ to a space. This allows us to always work with based spaces from now on. We will also usually write X for a based space (X, x0) unless we want to make the base-point explicit. If f : X −→ Y is a map in Top the (unreduced) mapping cone of f is

Cf = ((X × I) q Y ) / (X × {1}) .

When f = idX : X −→ X we write CX = CidX . Then

Cf = (CX qY ) / (x ∼ f(x): x ∈ X) .

Notice that CX is contractible, i.e., homotopic to a point. If f :(X, x0) −→ (Y, y0) is a map of based spaces then we have the (reduced) mapping cone of f Cf = ((X × I) q Y ) / (X × {1} ∪ {x0} × I) .

Notice that Y ⊆ Cf and X ⊆ CX . In the based situation, CX is contractible through a base-point preserving homotopy.

f

X Y Cf

Figure 1.1

The mapping cone construction is clearly functorial in the triple (X, Y, f) in the sense that a commutative diagram

f X −−−−→ Y     (1.1a) y y

f 0 X0 −−−−→ Y 0 1 2 ANDREW BAKER extends to a commutative diagram

f iY X −−−−→ Y −−−−→ Cf       (1.1b) y y y

0 0 f 0 iY 0 X −−−−→ Y −−−−→ Cf 0 The cone construction also has an important universal property.

Proposition 1.1. A map f is null-homotopic if and only if f extends to a map f˜:CX −→ Y .

Proposition 1.2. If f : X −→ Y is a cofibration, then there is a homotopy equivalence Cf ' Y/fX.

So Cf provides a substitute for the naive quotient in the homotopy categories h-Top and h-Top∗. For later use we introduce the idea of the smash product of two based spaces (X, x0) and (Y, y0), X ∧ Y = X × Y/ (X × {y0} ∪ {x0} × Y ) . The (reduced) suspension of X is 1 1 ¡ 1 ¢ ΣX = S ∧ X = (S × X)/ {1} × X ∪ S × {x0} . We can iterate this to obtain the m-th iterated suspension ΣmX = Σ(Σm−1X). It can be shown that ΣmX is homeomorphic to Sm ∧ X. For based spaces (X, x0) and (Y, y0) let [X,Y ] denote the set of based homotopy classes of based maps X −→ Y . As a special case we have the homotopy groups n πn(X, x0) = πn(X) = [S ,X](n > 0).

We set πn(X) = 0 if n < 0. π0(X) is the set of path components of X. Clearly each πn(X) is a functor h-Top∗ Ã Sets, but in fact it takes values in some subcategories.

Proposition 1.3. For any based space (X, x0), πn(X) is a group for every n > 1 and is abelian if n > 2. If X is group-like then πn(X) is a group for n = 0 and abelian for n > 1.

In this statement, group-like means that it gives a group object in h-Top∗. A based map f : X −→ Y is called a weak equivalence if the induced homomorphism f : π (X) −→ π (Y ) is an isomorphism for all n > 0. ∗ n n W Finally, if {Xλ}λ∈Λ is some collection of spaces, their wedge λ∈Λ Xλ is their one-point union with open sets being those which intersect all the subspaces Xλ in open sets.

2. CW complexes A based space X is a CW complex if it has a filtration by subspaces

X0 ⊆ X1 ⊆ · · · ⊆ Xn ⊆ Xn+1 ⊆ · · · for which S • X = n>0 Xn, W 0 • X0 = S , λ∈Λ0 λ W . • for each n, there is a map f : Sn −→ X for which X = C . n λ∈Λn λ n n+1 gn Theorem 2.1 (Whitehead Theorem). For CW complexes X and Y , a map f : X −→ Y is a weak equivalence if and only if it is a homotopy equivalence. Theorem 2.2 (CW Approximation Theorem). For a space X there is a CW complex X0 and a weak equivalence g : X0 −→ X; furthermore, if g0 : X00 −→ X is a second such map, then there is a homotopy equivalence h: X0 −→ X00. COHOMOLOGY THEORIES, BROWN REPRESENTABILITY AND SPECTRA 3

Let CW denote the full subcategory of Top∗ consisting of CW complexes, and h-CW its −1 associated homotopy category. Let h-Top∗[WE ] denote the localization of h-Top∗ with respect to the weak equivalences (this is a topological version of the algebraic derived category).

−1 Corollary 2.3. The inclusion of h-CW into h-Top∗[WE ] is an equivalence of categories.

3. Cohomology theories A sequence of contravariant functors n K : Top∗ Ã AbGp (n ∈ Z) is a (reduced) cohomology theory if it has the following properties. For a maps f : X −→ Y we usually write f ∗ = Knf for the induced homomorphisms. (A) If two maps f, g : X −→ Y in Top are homotopic then f ∗ = g∗ : Kn(Y ) −→ Kn(X)(n ∈ Z). (B) For the singleton based space ∗ = {∗}, Kn(∗) = 0 (n ∈ Z). (C) If f : X −→ Y is a based map, there is a long exact sequence

i∗ ∗ i∗ δ n Y n f n δ n+1 Y 0 ··· −−−−→K (Cf ) −−−−→K (Y ) −−−−→K (X) −−−−→K (Cf ) −−−−→· · · and in this is functorial in the sense that given the situation of (1.1) gives rise to a commutative diagram

i∗ 0∗ i∗ δ n Y 0 n 0 f n 0 δ n+1 Y 0 ··· −−−−→K (Cf 0 ) −−−−→K (Y ) −−−−→K (X ) −−−−→K (Cf 0 ) −−−−→· · ·         y y y y i∗ ∗ i∗ δ n Y n f n δ n+1 Y ··· −−−−→ K (Cf ) −−−−→ K (Y ) −−−−→ K (X) −−−−→ K (Cf ) −−−−→· · · Property (C) is often replaced by the equivalent Mayer-Vietoris property: (MV) If X = U ∪ V = U o ∪ V o (where U o,V o denote the interiors of U, V in X), there is a long exact sequence which is natural in the triple (X, U, V )

f ∗⊕g∗ i∗ −i∗ · · · −→ Kn(X) −−−−→Kn(U) ⊕ Kn(V ) −−−−→KU V n(U ∩ V ) −→Kδ n+1(X) −→ · · · Notice that (A) says that a cohomology theory factors through a sequence of functors n 0 K : h-Top∗ Ã AbGp on the homotopy category. Also, if two based spaces X and X are based-homotopic then Kn(X) =∼ Kn(X0) for all n ∈ Z. A natural transformation of cohomology theories Φ: K∗ −→ L∗ is a sequence of natural n n transformations Φ = {Φn : K −→ L }n∈Z which induces commutative diagrams linking the long exact sequences of (C):

i∗ ∗ i∗ δ n Y n f n δ n+1 Y ··· −−−−→K (Cf ) −−−−→K (Y ) −−−−→K (X) −−−−→K (Cf ) −−−−→· · ·         Φny Φny Φny Φn+1y i∗ ∗ i∗ δ n Y n f n δ n+1 Y ··· −−−−→ L (Cf ) −−−−→ L (Y ) −−−−→ L (X) −−−−→ L (Cf ) −−−−→· · · If S0 = {1, −1} is the two-point space based at 1, the graded group K∗ = K∗(S0) forms the coefficients of the theory. It is sometimes useful to consider a cohomology theory for which the following Weak Homotopy Equivalence axiom holds: ∼ (WHE) Every weak equivalence f : X −→ Y induces isomorphisms f ∗ : Kn(Y ) −→K= n(X) for all n ∈ Z. 4 ANDREW BAKER

A cohomology theory K∗ satisfying (WHE) is determined up to natural isomorphism by it values on h-CW. Indeed, given a cohomology theory defined on h-CW, there is an essentially unique extension to a cohomology theory defined on h-Top∗ which satisfies (WHE).

Example 3.1. For a map f : X −→ Y , we may consider the inclusion i = iY : Y −→ Cf and its mapping cone CiY (see Figure 3.1). Then the long exact sequence of (C) becomes ∗ ∗ i ∗ i δ n Cf n i n δ n+1 Cf ··· −−−−→K (Ci) −−−−→K (Cf ) −−−−→K (Y ) −−−−→K (Ci) −−−−→· · ·

It is fairly clear that Ci is homotopic to CX /X which is homeomorphic to the suspension of X, ΣX. The long exact sequence becomes

∗ δ n n i n δ n+1 ··· −−−−→K (ΣX) −−−−→K (Cf ) −−−−→K (Y ) −−−−→K (ΣX) −−−−→· · · Comparing this with the long exact sequence for f, we find that there is a commutative diagram

∗ δ n n i n δ n+1 ··· −−−−→ K (ΣX) −−−−→K (Cf ) −−−−→K (Y ) −−−−→K (ΣX) −−−−→· · ·         y y y y ∗ i∗ ∗ f n−1 δ n Y n f n δ ··· −−−−→K (X) −−−−→K (Cf ) −−−−→K (Y ) −−−−→ K (X) −−−−→· · · and from this we see that Kn(X) =∼ Kn+1(ΣX). It can be shown that ΣmX is homeomorphic to Sm ∧ X, so Kn(X) =∼ Kn+m(ΣmX). In particular, ΣmS0 is homeomorphic to Sm so we have Kn(S0) =∼ Kn+m(Sm).

f i

X Y Cf Ci

Figure 3.1

4. Ordinary cohomology and Eilenberg-MacLane spaces The best known examples of cohomology theories are the ordinary cohomology theories. The word unique in the next result really means unique up to natural isomorphism of functors. For basesd

Theorem 4.1. Let A be an abelian group. (i) There is a unique cohomology theory H∗(; A) for which ( A if n = 0, Hn(S0; A) = 0 otherwise. (ii) For each n, there is a space K(A, n) and a natural isomorphism of functors Hn(X; A) =∼ [X, K(A, n)]. Hence, ( A if k = n, πk(K(A, n)) = 0 otherwise. For n < 0, K(A, n) is contractible. COHOMOLOGY THEORIES, BROWN REPRESENTABILITY AND SPECTRA 5

(iii) For each homomorphism of abelian groups ϕ: A −→ B there is an associated map ϕ: K(A, n) −→ K(B, n) and a diagram of natural transformations ∼ Hn(; A) −−−−→= [ , K(A, n)]     ϕ˜y yϕ∗ ∼ Hn(; B) −−−−→= [ , K(B, n)] where ϕ˜ is the change of coefficients natural transformation. The existence of the Eilenberg-MacLane spaces K(A, n) is an important indicator of a general phenomenon. We note one other important property of these spaces. We know that Hn(K(A, n); A) =∼ Hn+1(ΣK(A, n); A), hence there is a unique element [αn] ∈ [ΣK(A, n), K(A, n + 1)] corresponding to the identity map in [K(A, n), K(A, n)]. By a standard adjunction result, such a map is equivalent to a map αn 0 S1 αn : K(A, n) −→ K(A, n + 1) . It can be shown that 0 =∼ S1 αn∗ : π∗(K(A, n)) −→ π∗(K(A, n + 1) ), which suggests that K(A, n) and K(A, n + 1)S1 are essentially the same space up to homotopy. If the K(A, n) are constructed as cellular complexes this is indeed the case. It is usual to write ΩX = XS1 for the space of based maps S1 −→ X; then we always have

Top∗(ΣX,Y ) = Top∗(X, ΩY ), h-Top∗(ΣX,Y ) = h-Top∗(X, ΩY ) when these spaces are reasonably nice.

5. Brown representability and spectra A cohomology theory K∗ is said to satisfy the Wedge Axiom if for any wedge of spaces W λ∈Λ Xλ, _ Y n n (WA) K ( Xλ) = K (Xλ). λ∈Λ λ∈Λ Theorem 5.1 (Brown representability). If the cohomology theory K∗ defined on CW satisfies the Wedge Axiom, then there are spaces Kn (n ∈ Z) and natural isomorphisms of functors n K () −→ [ ,Kn] as well as homeomorphisms αn : Kn −→ ΩKn+1 which induce the isomorphisms n ∼ ∼ n+1 K (X) = [X,Kn] −→ [X, ΩKn+1] −→ [ΣX,Kn+1] = K (ΣX).

∗ The sequence of spaces K = {Kn}n∈Z forms a spectrum and we say that K represents K . A sequence of maps {hn : Kn −→ Ln}n∈Z is called a map of spectra if there are commutative diagrams 0 αn Kn −−−−→ ΩKn+1     hny yΩhn+1

0 βn Ln −−−−→ ΩLn+1 0 where βn : Ln −→ ΩLn+1 is the above structure homeomorphism for {Ln}n∈Z. We usually write h: K = {Kn}n∈Z −→ L = {Ln}n∈Z for such a map of spectra. It is possible to define a good category of spectra Spectra and an associated homotopy cat- egory h-Spectra. However, the technical details required to obtain many important properties 6 ANDREW BAKER of Spectra (as opposed to h-Spectra) are formidable and some of these have only been worked out recently. We also have Theorem 5.2. Associated to a natural transformation of cohomology theories Φ: K∗ −→ L∗ defined on CW and represented by spectra K and L is a map of spectra ϕ: K −→ L for which n n each Φn : K −→ L is induced by ϕn :[ ,Kn] −→ [ ,Ln]. Notice that this result does not have any homotopy uniqueness associated to the choice of ϕ.

6. Multiplicative cohomology theories and ring spectra If A is a commutative ring then H∗(; A) takes values in the category of graded commutative ∗ rings, and in fact of graded commutative A-algebras. For a space X,H (X+; A) is a unital A-algebra. This structure corresponds to pairings K(A, m) ∧ K(A, n) −→ K(A, m + n) with various properties and these make the Eilenberg-MacLane spectrum K(A) = {K(A, n)}n∈Z into a commutative ring object in h-Spectra. In general, a cohomology theory K∗ which takes values in the category of commutative graded rings will have a similar product K ∧K −→ K in h-Spectra but this need not give a ring object. If it does then K is a ring spectrum realizing the product on K∗; an even stronger requirement is that such a ring structure exists as a map K ∧ K −→ K in Spectra, and this may exist for one representative of the homotopy class of K but not for others! In fact, to even formulate this question requires the existence of a coherent smash product bifunctor ( ) ∧ ( ) on Spectra before passage to homotopy and this was only constructed in the past decade. It is possible to realize such a structure for K(A) provided the correct model is built in Spectra, however there are also examples known where this cannot be done.