Homotopically Truncated Spaces.

The Eckmann–Hilton dual of the Postnikov decomposition of a space is the homology decomposition (or Moore space decomposition) of a space.

A Postnikov decomposition for a simply connected CW-complex X is a commutative diagram

such that p_n∗ : π_r(X) → π_r(P_n(X)) is an isomorphism for r ≤ n and π_r(P_n(X)) = 0 for r > n. Let F_n be the homotopy fiber of q_n. Then the exact sequence

π_r+1(P_nX) →^q_n∗ π_r+1(P_n−1X) → π_r(F_n) → π_r(P_nX) →^q_n∗ π_r(P_n−1X)

shows that F_n is an Eilenberg–MacLane space K(π_nX, n). Constructing P_n+1(X) inductively from P_n(X) requires knowing the nth k-invariant, which is a map of the form k_n : P_n(X) → Y_n. The space P_n+1(X) is then the homotopy fiber of k_n. Thus there is a homotopy fibration sequence

K(π_n+1X, n+1) → P_n+1(X) → P_n(X) → Y_n

This means that K(π_n+1X, n+1) is homotopy equivalent to the loop space ΩY_n. Consequently,

π_r(Y_n) ≅ π_r−1(ΩY_n) ≅ π_r−1(K(π_n+1X, n+1) = π_n+1X, r = n+2,

= 0, otherwise.

and we see that Y_n is a K(π_n+1X, n+2). Thus the nth k-invariant is a map k_n : P_n(X) → K(π_n+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 P_n(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 j_n∗ : H_r(X_≤n) → H_r(X) is an isomorphism for r ≤ n and H_r(X_≤n) = 0 for

r > n. Let C_n be the homotopy cofiber of i_n. Then the exact sequence

H_r(X_≤n−1) →^i_n∗ H_r(X_≤n) → H_r(C_n) →^i_n∗ H_r−1(X_≤n−1) → H_r−1(X_≤n)

shows that C_n is a Moore space M(H_nX, n). Constructing X_≤n+1 inductively from X_≤n requires knowing the nth k-invariant, which is a map of the form k_n : Y_n → X_≤n.

The space X_≤n+1 is then the homotopy cofiber of k_n. Thus there is a homotopy cofibration sequence

Y_n →^k_n X_≤n →^i_n+1 X_≤n+1 → M(H_n+1X, n+1)

This means that M(H_n+1X, n+1) is homotopy equivalent to the suspension SY_n. Consequently,

H^˜_r(Y_n) ≅ H^∼_r+1(SY_n) ≅ H_r+1(M(H_n+1X, n+1)) = H_n+1X, r = n,

= 0, otherwise

and we see that Y_n is an M(H_n+1X, n). Thus the nth k-invariant is a map k_n : M(H_n+1X, n) → X_≤n

It induces the zero map on all reduced homology groups, which is a nontrivial statement to make in degree n:

k_n∗ : H_n(M(H_n+1X, n)) ∼= H_n+1(X) → H_n(X) ∼= H_n(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 p_n(X) : X → P_n(X) be a stage-n Postnikov approximation for X, that is, p_n(X)_∗ : π_r(X) → π_r(P_n(X)) is an isomorphism for r ≤ n and π_r(P_n(X)) = 0 for r > n. If Z is a space with π_r(Z) = 0 for r > n, then any map g : X → Z factors up to homotopy uniquely through P_n(X). In particular, if f : X → Y is any map and p_n(Y) : Y → P_n(Y) is a stage-n Postnikov approximation for Y, then, taking Z = P_n(Y) and g = p_n(Y) ◦ f, there exists, uniquely up to homotopy, a map p_n(f) : P_n(X) → P_n(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_∗ : H_r(X_≤2) → H_r(X) and j_∗ : H_r(Y_≤2) → H_r(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, π_n+1Y) →^ι [M(G, n), Y] →^η Hom(G, π_nY) → 0,

where Y is any space and [−,−] denotes pointed homotopy classes of maps. The map η is given by taking the induced homomorphism on π_n 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 E^Y (−) = 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].

Then

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) = ιE^Y 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.

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