HOMOLOGY THEORIES, SPECTRA

John Rognes

February 17th 1999

Lecture 8. Homology theories and spectra

8.1. Homology theories.

Let GrAb be the category of ℤ-graded abelian groups, let CW_{∗} be the category of pointed CW complexes and cellular maps, and let CW_{f∗} be its full category of finite pointed CW-complexes.

A (reduced) homology theory on CW_{f∗} is a functor

Ẽ_{∗}: CW_{f∗} ⟶ GrAb

and a natural equivalence σ: Ẽ_{∗}(X) ≅ Ẽ_{∗+1}(ΣX) satisfying (1) and (2):

(1) Ẽ_{∗} is a homotopy functor, i.e., given homotopic maps f, g: X → Y the induced homomorphisms f_{∗}, g_{∗}: Ẽ_{∗}(X) → Ẽ_{∗}(Y) are equal.

(2) Ẽ_{∗} is exact, i.e., given a CW complex X and a subcomplex A let i: A → X be the inclusion and j: X → X/A the projection to the quotient complex. Then the diagram

Ẽ_{∗}(A) ⟶^{i_{∗}} Ẽ_{∗}(X) ⟶^{j_{∗}} Ẽ_{∗}(X/A)

is exact.

We can relate this definition to the Eilenberg–Steenrod axioms as follows: In the cofiber sequence X → X ∪ CA → ΣA the projection c: X ∪ CA → X/A is a homotopy equivalence, so there is a natural connecting map

∂: Ẽ_{n}(X/A) ≅ Ẽ_{n}(X ∪ CA) → Ẽ_{n}(ΣA) ≅ Ẽ_{n−1}(A)

and a long exact sequence

⋯ → Ẽ_{n}(A) ⟶^{i} Ẽ_{n}(X) ⟶^{j} Ẽ_{n}(X/A) ⟶^{∂} Ẽ_{n−1}(A) → ….

Excision is automatic, since for U ⊆ A ⊆ X (with the closure of U in the interior of A), there is a homeomorphism (X \ U)/(A \ U) ≅ X/A inducing the excision isomorphism.

The homology theory extends to CW_{∗} by

Ẽ_{∗}(X) := colim_{X_{γ}⊆X} Ẽ_{∗}(X_{γ})

Typeset by AMS-TEX

where X_{γ} runs over the category of finite subcomplexes of X. The resulting functor Ẽ_{∗}: CW_{∗} → GrAb satisfies J.W. Milnor’s wedge axiom:

(3) The natural map

⨁_{α} Ẽ_{∗}(X_{α}) ⟶^{≅} Ẽ_{∗}(⋁_{α} X_{α})

is an isomorphism for any family of spaces X_{α}.

For finite families this follows from exactness, but is an additional hypothesis for infinite families. Conversely, for any functor X: C → CW_{∗} with C a small and filtering category, we have a cofiber sequence

Σ ⋁_{f : α→β∈C} X_{α} ⟶ Σ ⋁_{γ∈C} X_{γ} ⟶ Σ colim_{γ∈C} X_{γ} .

Here X_{α} = X(α) for α ∈ C, etc. and applying Ẽ_{n+1} and the suspension isomorphism yields an exact sequence

⨁_{f : α→β∈C} Ẽ_{n}(X_{α}) ⟶ ⨁_{γ∈C} Ẽ_{n}(X_{γ}) ⟶ Ẽ_{n}(colim_{γ∈C} X_{γ})

which induces an isomorphism

colim_{γ∈C} Ẽ_{n}(X_{γ}) ⟶^{≅} Ẽ_{n}(colim_{γ∈C} X_{γ}) .

So conversely, any Ẽ_{∗}: CW_{∗} → GrAb satisfying (1), (2) and (3) is naturally equivalent to one extended from CW_{f∗} in the above manner.

Extend to T (the category of pointed topological spaces) by

Ẽ_{∗}(X) := Ẽ_{∗}(| Sing(X)|)

where Sing(X) is the simplicial set of singular simplices in X, and | | denotes geometric realization. There is a natural weak equivalence | Sing(X)| → X for any space X, which is a homotopy equivalence if and only if X has the homotopy type of a CW complex.

The resulting Ẽ_{∗}: T → GrAb satisfies the weak equivalence axiom:

(4) If f: X → Y is a weak equivalence, then f_{∗}: Ẽ_{∗}(X) → Ẽ_{∗}(Y) is an isomorphism.

And conversely, if Ẽ_{∗}: T → GrAb satisfies (1), (2), (3) and (4), then it is naturally equivalent to one extended from CW_{∗} as above.

We extend homology theories to pairs (X, A) of spaces by

Ẽ_{∗}(X, A) := Ẽ_{∗}(X/A)

for A ⊆ X a subcomplex in CW_{f∗} or in CW_{∗}, and by

Ẽ_{∗}(X, A) := Ẽ_{∗}(| Sing(X)|/| Sing(A)|)

for A ⊆ X any subspace in T . Then Ẽ_{∗}(X, A) ≅ Ẽ_{∗}(X/A) if the inclusion A ⊆ X is a closed cofibration.

We also have unreduced homology theories, setting E_{∗}(X) := Ẽ_{∗}(X_{+}) for X ∈ U (unbased topological spaces), where X ↦ X_{+} is the functor U → T adding a disjoint basepoint. For A ⊆ X in U we let E_{∗}(X, A) := Ẽ_{∗}(X_{+}, A_{+}). So E_{∗}(X, A) ≅ Ẽ_{∗}(X, A) for A ⊆ X in T .

8.2. Cohomology theories.

A (reduced) cohomology theory on CW_{f∗} is a functor

Ẽ^{∗}: CW_{f∗}^{op} ⟶ GrAb

and a natural equivalence σ: Ẽ^{∗}(X) ≅ Ẽ^{∗+1}(ΣX), satisfying (1*) Ẽ^{∗} is a homotopy functor, and (2*) Ẽ^{∗} is exact, i.e., for i: A ⊆ X and j: X → X/A the induced homomorphisms

form an exact sequence

Ẽ^{∗}(A) ⟵^{i^{∗}} Ẽ^{∗}(X) ⟵^{j^{∗}} Ẽ^{∗}(X/A) .

Such a functor admits an extension to CW_{∗}^{op} satisfying Milnor’s wedge axiom: (3*) The natural map

Ẽ^{∗}(⋁_{α} X_{α}) ⟶^{≅} ∏_{α} Ẽ^{∗}(X_{α})

is an isomorphism.

The extension will be the so-called representable cohomology theory Ẽ^{∗}. Filtering X by its skeleta X^{(k)} the cofiber sequence

Σ ⋁_{k} X^{(k)} ⟶ Σ ⋁_{k} X^{(k)} ⟶ ΣX

leads to the long exact sequence

⋯ → Ẽ^{∗}(X) ⟶ ∏_{k} Ẽ^{∗}(X^{(k)}) ⟶ ∏_{k} Ẽ^{∗}(X^{(k)}) → …

and the Milnor exact sequence:

0 → lim^{1}_{k} Ẽ^{n−1}(X^{(k)}) ⟶ Ẽ^{n}(X) ⟶ lim_{k} Ẽ^{n}(X^{(k)}) → 0

Here lim^{1} is the right derived functor of lim.

We can extend to T^{op} by

Ẽ^{∗}(X) := Ẽ^{∗}(| Sing(X)|)

as before. The resulting theory satisfies the weak equivalence axiom:

(4*) If f: X → Y is a weak equivalence, then f^{∗}: Ẽ^{∗}(Y) → Ẽ^{∗}(X) is an isomorphism.

Again, any theory satisfying (1*), (2*), (3*) and (4*) arises from a cohomology theory on finite CW complexes by this extension procedure.

8.3. Spectra.

Given a pointed space Z, let π^{Z}: CW_{f∗}^{op} → Ens_{∗} (the category of pointed sets) be the functor X ↦ [X, Z] (pointed homotopy classes of maps). If Z is a double loop space, π^{Z} naturally takes values in abelian groups. A functor F: CW_{f∗}^{op} → Ens_{∗} naturally equivalent to π^{Z} is said to be representable, and the space Z then represents the functor F.

Let Ẽ^{∗} be a cohomology theory and suppose each X ↦ Ẽ^{n}(X) is represented by a space E_{n}, for n ∈ ℤ. So Ẽ^{n}(X) ≅ [X, E_{n}]. The suspension isomorphism gives a chain of isomorphisms

[X, E_{n}] ≅ Ẽ^{n}(X) ≅ Ẽ^{n+1}(ΣX) ≅ [ΣX, E_{n+1}] ≅ [X, ΩE_{n+1}]

which are natural in finite pointed CW complexes X. Thus there is a weak homotopy equivalence E_{n} ≃_{w} ΩE_{n+1}. G.W. Whitehead and E.L. Lima were thus lead to study sequences of spaces E_{n} for n ∈ ℤ equipped with maps E_{n} → ΩE_{n+1} or ΩE_{n+1} → E_{n}, with suitable properties. The first possibility defines a (pre)spectrum in the algebro-topological sense. The second possibility seems less fruitful.

A prespectrum E = {E_{n}, e_{n}}_{n∈ℤ} is a sequence of pointed spaces E_{n} and pointed maps e_{n}: ΣE_{n} → E_{n+1}, for all n ∈ ℤ. We call E_{n} the nth space of E, call e_{n} the nth structure map, and write ẽ_{n}: E_{n} → ΩE_{n+1} for the nth adjoint structure map.

Let us say that E is a Σ-spectrum if each E_{n} is a CW complex, and each map e_{n} embeds ΣE_{n} as a subcomplex of E_{n+1}. (J.F. Adams only considers such spectra.)

A prespectrum E is called an Ω-spectrum if each adjoint structure map ẽ_{n} is a homotopy equivalence. (J.P. May insists that each ẽ_{n} must be a homeomorphism.)

Example: The sphere spectrum S has nth space S^{n} = S^{1} ∧ ⋯ ∧ S^{1} (with n circle factors), and nth structure map s_{n}: ΣS^{n} = S^{n+1} the identity map. The adjoint structure map s̃_{n}: S^{n} → ΩΣS^{n} is (2n − 1)-connected by Freudenthal’s suspension theorem, but is not a homotopy equivalence.

To a prespectrum E we associate a homology theory Ẽ_{∗}: CW_{f∗} → GrAb and a cohomology theory Ẽ^{∗}: CW_{f∗}^{op} → GrAb, by the following formulas:

Ẽ_{n}(X) := colim_{k} π_{n+k}(E_{k} ∧ X)
Ẽ^{n}(X) := colim_{k}[Σ^{k}X, E_{n+k}]

The colimits are taken with respect to the composite maps

π_{n+k}(E_{k} ∧ X) ⟶^{σ} π_{n+k+1}(ΣE_{k} ∧ X) ⟶^{e_{k}} π_{n+k+1}(E_{k+1} ∧ X)

and

[Σ^{k}X, E_{n+k}] ⟶^{σ} [Σ^{k+1}X, ΣE_{n+k}] ⟶^{e_{n+k}} [Σ^{k+1}X, E_{n+k+1}] .

If E is an Ω-spectrum, then all the composite maps in the colimit defining Ẽ^{n}(X) are isomorphisms, so Ẽ^{n}(X) ≅ [X, E_{n}] for all n ∈ ℤ.

It remains to verify that these formulas define (co)homology theories.

For homology, consider a cofiber sequence A → X → X ∪ CA and the induced diagram

Ẽ_{n}(A) ⟶^{i_{∗}} Ẽ_{n}(X) ⟶^{j_{∗}} Ẽ_{n}(X ∪ CA) .

Since ji is null-homotopic, the composite j_{∗}i_{∗} = 0. Conversely, consider a class

[f] ∈ colim_{k} π_{n+k}(E_{k} ∧ X) = Ẽ_{n}(X)

represented by a map f: S^{n+k} → E_{k} ∧ X, and suppose that j_{∗}[f] = [jf] = 0 in Ẽ_{n}(X ∪ CA). By possibly choosing a larger k, we may assume that jf: S^{n+k} → E_{k} ∧ X ∪ CA is null-homotopic. Choosing a null-homotopy amounts to choosing an extension g: D^{n+k+1} → E_{k} ∧ X ∪ CA of jf. Passing to quotients, we get a map h: S^{n+k+1} → E_{k} ∧ ΣA. Identify the target with ΣE_{k} ∧ A, and compose with e_{k} ∧ 1 to get a map e = (e_{k} ∧ 1)h: S^{n+k+1} → E_{k+1} ∧ A. Then the class

[e] ∈ colim_{ℓ} π_{n+ℓ}(E_{ℓ} ∧ A) = Ẽ_{n}(A)

(with ℓ = k + 1) satisfies i_{∗}([e]) = [f]. This proves exactness at Ẽ_{n}(X). It is clear that X ↦ Ẽ_{n}(X) is a homotopy functor, so Ẽ_{∗} is indeed a homology theory.

To show that Ẽ^{∗} is a cohomology theory, note that for every k the sequence

[Σ^{k}A, E_{n+k}] ⟵ [Σ^{k}X, E_{n+k}] ⟵ [Σ^{k}X ∪ CA, E_{n+k}]

is exact, thus also the colimit over k. So Ẽ^{∗} is a cohomology theory on CW_{∗} (not just on CW_{f∗}), and extends to T as described in the previous section.

Example: S̃_{n}(X) = π_{n}^{S}(X) is the stable homotopy of X, and S̃^{n}(X) = π_{S}^{n}(X) is the stable cohomotopy of X.

8.4. Brown’s representability theorem.

Conversely, all (co)homology theories on CW_{f∗} arise in this way.

Theorem (E.H. Brown, J.F. Adams). Let Ẽ^{∗}: CW_{f∗}^{op} → GrAb be a cohomology theory. There is a spectrum E = {E_{n}, e_{n}}_{n} and universal elements ι_{n} ∈ Ẽ^{n}(E_{n}) inducing a natural equivalence

ι: Ẽ^{n}(X) ≅ Ẽ^{n}(X)

for all X ∈ CW_{f∗}. We may assume that E is both a Σ-spectrum and an Ωspectrum, i.e., that each E_{n} is a CW complex, each structure map e_{n} embeds ΣE_{n} as a subcomplex of E_{n+1}, and each adjoint structure map ẽ_{n}: E_{n} → ΩE_{n+1} is a homotopy equivalence.

The natural equivalence ι takes a class [f] ∈ Ẽ^{n}(X) with f: Σ^{k}X → E_{n+k} to the image of f^{∗}(ι_{n+k}) ∈ Ẽ^{n+k}(Σ^{k}X) ≅ Ẽ^{n}(X) by way of the k-fold iterated suspension isomorphism.

E.H. Brown proved a version of this theorem representing suitably exact homotopy functors defined on CW_{∗} and taking values in pointed sets. J.F. Adams proved a second version, for functors on CW_{f∗} taking values in abelian groups. Our cohomology theories are integer graded versions of the latter kind of functor.

The proof of Brown’s theorem is by inductive construction, building each CW complex E_{n} by skeleta so as to satisfy a version of the theorem for finite pointed CW complexes X of dimensions ≤ d at the dth stage of the induction. Adams’ argument is a little more involved.

Given Ẽ^{∗} on CW_{f∗},the resulting cohomology theory Ẽ^{∗} on CW_{∗} is the representable extension of Ẽ^{∗} over CW_{f∗} ⊂ CW_{∗}.

8.5. Spanier–Whitehead duality.

Using Spanier–Whitehead duality, one may likewise prove:

Theorem. Let Ẽ_{∗}: CW_{f∗} → GrAb be a homology theory. There is a spectrum E = {E_{n}, e_{n}}_{n} and a natural equivalence

Ẽ_{n}(X) ≅ Ẽ_{n}(X)

for all X ∈ CW_{f∗}. We may assume that E is both a Σ-spectrum and an Ω-spectrum.

Given these two representation theorems we will no longer distinguish notationally between Ẽ^{∗} and the represented theory Ẽ^{∗}, by writing Ẽ^{∗} for both. Likewise for homology.