A Postnikov decomposition for a simply connected CW-complex X is a commutative diagram
such that pn∗ : πr(X) → πr(Pn(X)) is an isomorphism for r ≤ n and πr(Pn(X)) = 0 for r > n. Let Fn be the homotopy fiber of qn. Then the exact sequence
πr+1(PnX) →qn∗ πr+1(Pn−1X) → πr(Fn) → πr(PnX) →qn∗ πr(Pn−1X)
shows that Fn is an Eilenberg–MacLane space K(πnX, n). Constructing Pn+1(X) inductively from Pn(X) requires knowing the nth k-invariant, which is a map of the form kn : Pn(X) → Yn. The space Pn+1(X) is then the homotopy fiber of kn. Thus there is a homotopy fibration sequence
K(πn+1X, n+1) → Pn+1(X) → Pn(X) → Yn
This means that K(πn+1X, n+1) is homotopy equivalent to the loop space ΩYn. Consequently,
πr(Yn) ≅ πr−1(ΩYn) ≅ πr−1(K(πn+1X, n+1) = πn+1X, r = n+2,
= 0, otherwise.
and we see that Yn is a K(πn+1X, n+2). Thus the nth k-invariant is a map kn : Pn(X) → K(πn+1X, n+2)
Note that it induces the zero map on all homotopy groups, but is not necessarily homotopic to the constant map. The original space X is weakly homotopy equivalent to the inverse limit of the Pn(X).
Applying the paradigm of Eckmann–Hilton duality, we arrive at the homology decomposition principle from the Postnikov decomposition principle by changing:
- the direction of all arrows
- π∗ to H∗
- loops Ω to suspensions S
- fibrations to cofibrations and fibers to cofibers
- Eilenberg–MacLane spaces K(G, n) to Moore spaces M(G, n)
- inverse limits to direct limits
A homology decomposition (or Moore space decomposition) for a simply connected CW-complex X is a commutative diagram
such that jn∗ : Hr(X≤n) → Hr(X) is an isomorphism for r ≤ n and Hr(X≤n) = 0 for
r > n. Let Cn be the homotopy cofiber of in. Then the exact sequence
Hr(X≤n−1) →in∗ Hr(X≤n) → Hr(Cn) →in∗ Hr−1(X≤n−1) → Hr−1(X≤n)
shows that Cn is a Moore space M(HnX, n). Constructing X≤n+1 inductively from X≤n requires knowing the nth k-invariant, which is a map of the form kn : Yn → X≤n.
The space X≤n+1 is then the homotopy cofiber of kn. Thus there is a homotopy cofibration sequence
Yn →kn X≤n →in+1 X≤n+1 → M(Hn+1X, n+1)
This means that M(Hn+1X, n+1) is homotopy equivalent to the suspension SYn. Consequently,
H˜r(Yn) ≅ H∼r+1(SYn) ≅ Hr+1(M(Hn+1X, n+1)) = Hn+1X, r = n,
= 0, otherwise
and we see that Yn is an M(Hn+1X, n). Thus the nth k-invariant is a map kn : M(Hn+1X, n) → X≤n
It induces the zero map on all reduced homology groups, which is a nontrivial statement to make in degree n:
kn∗ : Hn(M(Hn+1X, n)) ∼= Hn+1(X) → Hn(X) ∼= Hn(X≤n)
The original space X is homotopy equivalent to the direct limit of the X≤n. The Eckmann–Hilton duality paradigm, while being a very valuable organizational principle, does have its natural limitations. Postnikov approximations possess rather good functorial properties: Let pn(X) : X → Pn(X) be a stage-n Postnikov approximation for X, that is, pn(X)∗ : πr(X) → πr(Pn(X)) is an isomorphism for r ≤ n and πr(Pn(X)) = 0 for r > n. If Z is a space with πr(Z) = 0 for r > n, then any map g : X → Z factors up to homotopy uniquely through Pn(X). In particular, if f : X → Y is any map and pn(Y) : Y → Pn(Y) is a stage-n Postnikov approximation for Y, then, taking Z = Pn(Y) and g = pn(Y) ◦ f, there exists, uniquely up to homotopy, a map pn(f) : Pn(X) → Pn(Y) such that
homotopy commutes. Let X = S2 ∪2 e3 be a Moore space M(Z/2,2) and let Y = X ∨ S3. If X≤2 and Y≤2 denote stage-2 Moore approximations for X and Y, respectively, then X≤2 = X and Y≤2 = X. We claim that whatever maps i : X≤2 → X and j : Y≤2 → Y such that i∗ : Hr(X≤2) → Hr(X) and j∗ : Hr(Y≤2) → Hr(Y) are isomorphisms for r ≤ 2 one takes, there is always a map f : X → Y that cannot be compressed into the stage-2 Moore approximations, i.e. there is no map f≤2 : X≤2 → Y≤2 such that
commutes up to homotopy. We shall employ the universal coefficient exact sequence for homotopy groups with coefficients. If G is an abelian group and M(G, n) a Moore space, then there is a short exact sequence
0 → Ext(G, πn+1Y) →ι [M(G, n), Y] →η Hom(G, πnY) → 0,
where Y is any space and [−,−] denotes pointed homotopy classes of maps. The map η is given by taking the induced homomorphism on πn and using the Hurewicz isomorphism. This universal coefficient sequence is natural in both variables. Hence, the following diagram commutes:
Here we will briefly write E2(−) = Ext(Z/2,−) so that E2(G) = G/2G, and EY (−) = Ext(−, π3Y). By the Hurewicz theorem, π2(X) ∼= H2(X) ∼= Z/2, π2(Y) ∼= H2(Y) ∼= Z/2, and π2(i) : π2(X≤2) → π2(X), as well as π2(j) : π2(Y≤2) → π2(Y), are isomorphisms, hence the identity. If a homomorphism φ : A → B of abelian groups is onto, then E2(φ) : E2(A) = A/2A → B/2B = E2(B) remains onto. By the Hurewicz theorem, Hur : π3(Y) → H3(Y) = Z is onto. Consequently, the induced map E2(Hur) : E2(π3Y) → E2(H3Y) = E2(Z) = Z/2 is onto. Let ξ ∈ E2(H3Y) be the generator. Choose a preimage x ∈ E2(π3Y), E2(Hur)(x) = ξ and set [f] = ι(x) ∈ [X,Y]. Suppose there existed a homotopy class [f≤2] ∈ [X≤2, Y≤2] such that
j∗[f≤2] = i∗[f].
η≤2[f≤2] = π2(j)∗η≤2[f≤2] = ηj∗[f≤2] = ηi∗[f] = π2(i)∗η[f] = π2(i)∗ηι(x) = 0.
Thus there is an element ε ∈ E2(π3Y≤2) such that ι≤2(ε) = [f≤2]. From ιE2π3(j)(ε) = j∗ι≤2(ε) = j∗[f≤2] = i∗[f] = i∗ι(x) = ιEY π2(i)(x)
we conclude that E2π3(j)(ε) = x since ι is injective. By naturality of the Hurewicz map, the square
commutes and induces a commutative diagram upon application of E2(−):
It follows that
ξ = E2(Hur)(x) = E2(Hur)E2π3(j)(ε) = E2H3(j)E2(Hur)(ε) = 0,
a contradiction. Therefore, no compression [f≤2] of [f] exists.
Given a cellular map, it is not always possible to adjust the extra structure on the source and on the target of the map so that the map preserves the structures. Thus the category theoretic setup automatically, and in a natural way, singles out those continuous maps that can be compressed into homologically truncated spaces.