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Cohomology operations are universal classes on Eilenberg--Mac Lane spaces

Statement

Assume AC. Let $A, B$ be abelian groups, let $n \geq 1$ , and let $r \in Z$ . Natural cohomology operations on based CW complexes whose basepoint is a vertex

$Θ : {\tilde{H}}^{n} (-; A) ⟶ {\tilde{H}}^{n + r} (-; B)$

are in bijection with universal classes

$u \in {\tilde{H}}^{n + r} (K (A, n); B) .$

The class belonging to $Θ$ is $u = Θ_{K (A, n)} (ι_{n})$ , and the operation belonging to $u$ is pullback of $u$ along a classifying map. No additivity is assumed. For connected CW complexes with both source and target degrees positive, this is equivalently the ordinary-cohomology statement.

For a family $(Θ_{n})_{n \geq 1}$ , commutation with reduced cohomology suspension at every positive source degree is exactly compatibility of its universal classes with cohomology suspension: if $s_{n} : Σ K (A, n) \to K (A, n + 1)$ classifies $σ ι_{n}$ , then, for every $n \geq 1$ ,

$σ u_{n} = s_{n}^{*} u_{n + 1} .$

A stable operation indexed over all integers in the earlier definition necessarily has these positive-degree identities. They do not by themselves impose its separate degree-zero suspension identity.

Facts & Assumptions

[F1]

Every $x \in {\tilde{H}}^{n} (X; A)$ has a based classifying map $f : X \to K (A, n)$ with $x = f^{*} ι_{n}$ , unique up to based homotopy (Eilenberg--Mac Lane spaces represent singular cohomology).

[F2]

Singular cohomology pullback is contravariantly functorial (Singular cohomology is contravariantly functorial). For a based homotopy $H : f ≃ g$ , the singular-chain prism satisfies $g_{#} - f_{#} = \partial P_{H} + P_{H} \partial$ in every nonnegative degree (The singular chain homotopy formula).

[F3]

Full stability means commutation with reduced cohomology suspension in every integer source degree, including zero (Stable natural cohomology operation). Its positive-degree part is the condition characterized here.

[A1]

AC is inherited from [F1]'s arbitrary-cell realization and homotopy-extension argument (The Axiom of Choice).

Proof

Given: $A, B, n, r$ , the category of based CW complexes whose basepoints are vertices, and [A1].

1.1

If $Θ$ is natural, set $u = Θ_{K (A, n)} (ι_{n})$ . For $x = f^{*} ι_{n}$ as in [F1], naturality forces

F1F2

$Θ_{X} (x) = Θ_{X} (f^{*} ι_{n}) = f^{*} Θ_{K (A, n)} (ι_{n}) = f^{*} u .$

Thus $u$ determines every value of $Θ$ . [F1, F2]

1.2

Conversely, fix $u \in {\tilde{H}}^{n + r} (K (A, n); B)$ . Given $x$ , choose its classifying map $f$ and define $Θ_{X}^{u} (x) = f^{*} u$ . If $f^{'}$ also classifies $x$ , [F1] makes $f$ and $f^{'}$ based-homotopic. For that based homotopy the prism in [F2] preserves chains of the basepoint, so precomposition with it gives a cochain homotopy on the relative singular cochains with arbitrary coefficient group $B$ . Thus $f^{*} u = f^{' *} u$ in reduced cohomology, including degree zero. Hence the definition is independent of the selected map.

F1F2

2.1

For the operation constructed in Step 1.2 and a based map $a : X^{'} \to X$ , the composite $f a$ classifies $a^{*} x$ . Therefore

F1F2step 1.2

$Θ_{X^{'}}^{u} (a^{*} x) = (f a)^{*} u = a^{*} f^{*} u = a^{*} Θ_{X}^{u} (x),$

so $Θ^{u}$ is natural. Its value on $ι_{n}$ , classified by the identity of $K (A, n)$ , is $u$ . Steps 1.1--2.1 show that the two assignments are inverse bijections. They never use an additive law. [F2, step 1.1, step 1.2]

3.1

Suppose $(Θ_{n})_{n \geq 1}$ commutes with suspension in each positive source degree $n$ , and write $u_{n} = Θ_{n} (ι_{n})$ . This hypothesis holds in particular for the positive-degree part of a fully stable operation [F3]. Apply its suspension identity to $X = K (A, n)$ and $x = ι_{n}$ . Since $s_{n}$ classifies $σ ι_{n}$ , naturality from Step 2.1 gives

F2F3step 2.1

$σ u_{n} = Θ_{n + 1} (σ ι_{n}) = Θ_{n + 1} (s_{n}^{*} ι_{n + 1}) = s_{n}^{*} u_{n + 1} .$

[F2, F3]

4.1

Conversely, assume the displayed compatibility from Step 3.1 for every $n \geq 1$ . For $x = f^{*} ι_{n}$ in positive source degree, naturality of suspension and Step 2.1 give

F2step 2.1step 3.1

$σ Θ_{n} (x) = (Σ f)^{*} σ u_{n} = (Σ f)^{*} s_{n}^{*} u_{n + 1} = Θ_{n + 1} (σ x) .$

Thus suspension commutes with the family at every positive source degree. This does not establish the $n = 0$ identity required by full stability in [F3]. For example, with $A = B = Z$ and $r = 0$ , take $Θ_{0} = id$ on ${\tilde{H}}^{0}$ , and $Θ_{n} = 0$ in all other degrees. Every positive-degree universal class is zero and satisfies the displayed compatibility, but suspension ${\tilde{H}}^{0} (S^{0}; Z) \to {\tilde{H}}^{1} (S^{1}; Z)$ is an isomorphism, so the degree-zero identity fails. Reduced and ordinary cohomology agree on connected CW complexes in every positive degree by [F1], giving the ordinary formulation when also $n + r > 0$ . At target degree zero they differ: for the one-point space, ${\tilde{H}}^{0} (*; B) = 0$ whereas $H^{0} (*; B) = B$ . Zero groups, negative target degree, the one-point space, and the zero universal class are included in the asserted reduced-cohomology result. AC is used only through [F1]. $□$ [A1, F1, F3, step 2.1, step 3.1]

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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)