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Simple Postnikov stages are classified by k-invariants

Statement

Assume AC. Let $n \geq 2$ , let $B$ be a connected simple CW $(n - 1)$ -type, and let $A$ be an abelian group. Marked simple Postnikov extensions of $B$ by $A$ --that is, ordinary Hurewicz fibration stages with fibers of CW homotopy type $K (A, n)$ ,

$F ≃ K (A, n) ⟶ E \overset{q}{\to} B,$

with trivial monodromy and fixed identifications of the base and fiber group--are classified up to fiber homotopy equivalence by

$k (q) \in H^{n + 1} (B; A) .$

For $k \in H^{n + 1} (B; A)$ , choose $κ : B \to K (A, n + 1)$ with $κ^{*} ι_{n + 1} = k$ ; the corresponding stage is $E_{k} = hofib (κ)$ . If the marking $π_{n} (K (A, n)) ≅ A$ is forgotten, $Aut (A)$ acts on the classification, and unmarked stages over the fixed base are classified by the resulting orbits.

Facts & Assumptions

[F1]

Representability gives a based map $κ$ for every $k$ , unique up to based homotopy (Eilenberg--Mac Lane spaces represent singular cohomology).

[F2]

Apply Mapping path factorization to the inclusion $* \to K (A, n + 1)$ : its endpoint projection from paths starting at $*$ is a Hurewicz fibration, and its total path space contracts by the explicit reparametrization in that theorem. Precomposition with $t \mapsto 1 - t$ is a homeomorphism of the compact-open path space (and its kification), from the terminal-point-fixed model $P^{-} K = {γ : γ (1) = *}$ used below to the initial-point-fixed model; it identifies the projection $γ \mapsto γ (0)$ with endpoint evaluation and acts on the loop fiber by inversion.

[F3]

A fibration long exact sequence computes homotopy groups of a pullback homotopy fiber (Long exact sequence of homotopy groups of a fibration).

[F4]

The homotopy-fiber definition fixes the endpoint convention used in $E_{k}$ (Homotopy fiber of a map).

[F5]

The primary section obstruction is the marked $k$ -invariant and is preserved by marked fiber homotopy equivalence (Postnikov k-invariant, Obstruction theory for lifting through a fibration). May--Ponto, Lemma 3.4.2, identifies $Ω K (A, n + 1)$ up to homotopy with $K (A, n)$ , proves the low-degree cohomological transgression calculation, and constructs a fiber-homotopy equivalence from a fibration with $K (A, n)$ fiber and trivial monodromy to the path-fibration pullback representing its transgression. Its proof also corrects the induced fiber endomorphism to the specified marking.

[F6]

For a Hurewicz fibration, CW-type base and fiber imply CW-type total space (Schon, Theorem 2), while CW-type total space and base imply CW-type fiber (Schon, Proposition 3). The term Hurewicz has the ordinary homotopy-lifting meaning in Hurewicz and serre fibrations. Thus the representability and fiberwise Whitehead steps used below apply to the stated stage presentations and to the terminal-path model.

[A1]

AC is inherited from representability, obstruction realization, and homotopy uniqueness of the fiber models (The Axiom of Choice, Existence and homotopy uniqueness of Eilenberg--Mac Lane spaces).

Proof

Given: $B, A, n, k$ and [A1] as in the statement.

1.1

Choose $κ$ by [F1] and form the pullback

F1F2F4

$E_{k} = {(b, γ) : γ (0) = κ (b), γ (1) = *} ⟶ B .$

Path reversal in [F2] identifies this terminal-point-fixed construction with a pullback of the published initial-point-fixed path fibration, so it is Hurewicz. Its fiber is $Ω K (A, n + 1)$ , whose homotopy groups satisfy $π_{i} ≅ π_{i + 1} K (A, n + 1)$ by [F3]. The full initial-path space contracts by [F2], and its base $K (A, n + 1)$ is CW; hence Schon [F6] gives CW type to its loop fiber. May--Ponto [F5] identifies this CW-type fiber up to homotopy with $K (A, n)$ . Use the literal terminal-path loop coordinate for its marking; reversal changes that marking by inversion relative to the initial-path model. [A1, F1, F2, F3, F4, F5, F6]

1.2

We record the exact low-degree calculation needed for completeness, rather than postulate a fibration of spaces of fiberwise equivalences. Let $q : E \to B$ be a marked stage in the Statement, with fiber $F ≃ K (A, n)$ over the chosen basepoint. Since the base and fiber have CW type and $q$ is Hurewicz, [F6] gives CW type for $E$ . The fiber fundamental class identifies $H^{n} (F; A)$ with $End (A)$ : its evaluation on $π_{n} (F) = A$ is the indicated endomorphism, and the chosen marking makes the fundamental class ${id}_{A}$ . Trivial monodromy makes this a constant coefficient group over $B$ .

F1F5F6

Filter the cochains of $E$ by the inverse images of the CW skeleta of $B$ , as in the low-degree calculation proved by May--Ponto [F5]. The resulting cohomological Serre page has $E_{2}^{p, r} = H^{p} (B; H^{r} (F; A))$ . Because $H^{r} (F; A) = 0$ for $0 < r < n$ , the first possible differential from $E_{2}^{0, n} = End (A)$ is the transgression $d_{n + 1}$ into $E_{n + 1}^{n + 1, 0} = H^{n + 1} (B; A)$ ; no other differential can enter that latter group in these degrees. The filtration edge maps therefore give the exact segment [F5]

$H^{n} (E; A) \overset{i^{*}}{\to} End (A) \overset{τ_{q}}{\to} H^{n + 1} (B; A) \overset{q^{*}}{\to} H^{n + 1} (E; A) .$

Fix the sign of $τ$ by the library's terminal-path convention: the path fibration with paths from the variable point to $*$ sends ${id}_{A}$ to $+ ι_{n + 1}$ . May--Ponto computes $d_{n + 1} ({id}_{A}) = + ι_{n + 1}$ for paths in the reverse direction. Path reversal changes the literal loop-fiber marking by inversion, so in our terminal-path coordinates the same differential sends ${id}_{A}$ to $- ι_{n + 1}$ ; put $τ_{q} = - d_{n + 1}$ uniformly. This sign change does not alter the exact segment. On an oriented relative $(n + 1)$ -cell, $τ_{q} ({id}_{A})$ evaluates its attaching $n$ -sphere in $π_{n} (F) = A$ with the primary section-obstruction sign of [F5]. Thus [F5]

$τ_{q} ({id}_{A}) = k (q), τ_{E_{κ}} ({id}_{A}) = κ^{*} ι_{n + 1} .$

2.1

Since $B$ has no homotopy above $n - 1$ , the long exact sequence applied to the construction in Step 1.1 gives $π_{i} (E_{k}) ≅ π_{i} (B)$ for $i < n$ , $π_{n} (E_{k}) ≅ A$ , and $π_{i} (E_{k}) = 0$ for $i > n$ . Thus $E_{k} \to B$ is a Postnikov stage with the required marking.

F3step 1.1

2.2

For the stage constructed in Step 1.1, the terminal-path normalization in Step 1.2 identifies the primary section obstruction of the universal path fibration with $+ ι_{n + 1}$ . A section over a subcomplex is a nullhomotopy of the inclusion there; on an attaching cell its failure is the same oriented sphere evaluated by the transgression. Naturality under pullback gives

F1F2F5step 1.1step 1.2

$k (E_{k} \to B) = κ^{*} ι_{n + 1} = k .$

[F1, F2, F5, step 1.2]

2.3

If two representatives of the construction in Step 1.1 are homotopic, pull the path fibration back over their homotopy $B \times I \to K (A, n + 1)$ . Homotopy lifting along $I$ gives mutually inverse maps between the endpoint pullbacks over $B$ , with composites fiberwise homotopic to the identities. Hence the fiber homotopy type of $E_{k}$ depends only on $k$ .

F2step 1.1

2.4

Now start with an arbitrary marked Hurewicz stage $q : E \to B$ . Set $k = k (q)$ and choose a based representative $κ : B \to K (A, n + 1)$ by [F1]. Exactness at $H^{n + 1} (B; A)$ in Step 1.2 gives $q^{*} k = 0$ ; geometrically, the pullback of $q$ along itself has its diagonal section, so its section obstruction vanishes. Since [F6] gives $E$ CW type, [F1] makes $κ q : E \to K (A, n + 1)$ based nullhomotopic. Choose a based nullhomotopy $h (e, -)$ running from $κ (q (e))$ to $*$ . The endpoint condition in [F4] then defines a continuous map over $B$

A1F1F4F5F6step 1.2

$λ_{0} : E ⟶ E_{κ}, e ⟼ (q (e), h (e, -)) .$

Its restriction to the marked fiber induces some endomorphism $c : A \to A$ . No claim that $c$ is yet invertible is made. [F4, step 1.2]

3.1

We correct the fiber marking of $λ_{0}$ explicitly. Naturality of the transgression in Step 1.2 for the map $λ_{0}$ over $B$ says

F5step 1.2step 2.4

$τ_{q} (c) = τ_{E_{κ}} ({id}_{A}) = k = τ_{q} ({id}_{A}) .$

Hence ${id}_{A} - c \in \ker τ_{q}$ . Exactness in Step 1.2 supplies a class $ℓ \in H^{n} (E; A)$ whose restriction to $F$ is ${id}_{A} - c$ . By [F1, F5, F6], represent $ℓ$ by a based map $L : E \to Ω K (A, n + 1)$ , using the literal terminal-loop identification of that CW-type space with $K (A, n)$ . Append the loop $L (e)$ to the path $h (e, -)$ in Step 2.4, using a fixed linear reparametrization of their two halves. This changes no starting point or endpoint, and gives a continuous map over $B$ [F1, F2, F4, F5, F6, step 1.2, step 2.4]

$λ (e) = (q (e), h (e, -) * L (e)) : E ⟶ E_{κ} .$

Loop multiplication represents addition of the corresponding degree- $n$ cohomology classes, as in May--Ponto's proof cited in [F5]. Therefore the restriction of $λ$ to $F$ induces $c + ({id}_{A} - c) = {id}_{A}$ on $π_{n} (F) = A$ . Both fibers have the homotopy type $K (A, n)$ , so this restriction is a homotopy equivalence. May--Ponto's mapping-path lifting argument then promotes $λ$ to a fiber homotopy equivalence over the CW base; it does not merely infer a fiberwise inverse from a pointwise weak equivalence. This proves that every marked stage is represented by the path pullback of its own $k (q)$ . [F5, F6, step 1.2, step 2.4]

4.1

If two marked stages have the same $k$ -invariant, [F1] makes their representing maps $κ$ based homotopic. Step 2.3 identifies the corresponding path pullbacks by a fiber homotopy equivalence, and Step 3.1 identifies each original stage with its path pullback through a fiber map inducing the fixed identity marking. Thus the two original stages are marked fiber homotopy equivalent. Conversely, the exact segment of Step 1.2 is natural under a marked fiber homotopy equivalence, so the transgression of ${id}_{A}$ , hence $k (q)$ , is preserved. This proves injectivity as well as surjectivity; no unconstructed comparison-space fibration is used.

F1F5step 1.2step 2.3step 3.1

5.1

Step 2.2 proves every cohomology class occurs, and Step 4.1 proves the claimed bijection. Replacing the fiber marking by $α \in Aut (A)$ postcomposes each obstruction value by $α$ , so it sends $k$ to $α_{*} k$ . Therefore forgetting the marking takes precisely the $Aut (A)$ -orbits. A base self-equivalence would additionally act by pullback, but the statement fixes the base. For $A = 0$ the exact segment has a zero endomorphism group and Step 3.1 still identifies every stage with the product-stage homotopy type. Nontrivial monodromy would require local coefficients and lies outside this untwisted theorem. $□$

A1F5step 2.2step 4.1

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Sources

James Davis and Paul Kirk, Lecture Notes in Algebraic Topology (standard reference, not scraped)
Haynes Miller, MIT 18.906 Algebraic Topology II lecture notes (standard reference, not scraped)
J. P. May and Kate Ponto, More Concise Algebraic Topology (standard reference, not scraped)
Rolf Schon, Fibrations Over a CWh-Base (standard reference, not scraped)