TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Leray–Hirsch module isomorphism

Statement

Assume AC, and let $R$ be a commutative unital ring. Let $F \to E \overset{p}{\to} B$ be a Serre fibration over a path-connected CW complex, and suppose the finitely many homogeneous classes $e_{i} \in H^{*} (E; R)$ restrict to an $R$ -basis on every fiber. Then $Φ : ⨁_{i} H^{* - ∣ e_{i} ∣} (B; R) ⟶ H^{*} (E; R); (a_{i}) ⟼ \sum_{i} p^{*} a_{i} ⌣ e_{i}$ is an $H^{*} (B; R)$ -module isomorphism. It is natural for maps of such fibrations that pull the specified classes on the target to the specified classes on the source.

Facts & Assumptions

Given: The commutative unital ring, fibration, finite homogeneous family, and basis hypothesis in the statement.

[F1]

A global fiber basis trivializes Serre monodromy identifies the fiber-cohomology system as constant in the displayed basis.

[F2]

The Leray–Hirsch associated-graded isomorphism lifts without extension ambiguity lifts the associated-graded basis isomorphism for this actual map $Φ$ .

[F3]

Cohomological Serre spectral sequence identifies the $E_{2}$ page and the edge classes, while Multiplicative cohomological Serre spectral sequence identifies cup products and their pagewise products.

[A1]

The Axiom of Choice is assumed exactly as in [F1]–[F2].

Proof

technique · identify the Serre-page map and lift it

1.1F1F3

By [F1] and [F3], the $E_{2}$ page is $H^{p} (B; R) \otimes_{R} H^{q} (F; R)$ in the named basis. Each $e_{i}$ comes from total-space cohomology, so its fiber restriction is represented by the fiber edge and is a permanent cycle. The product in [F3] therefore makes the map induced by $Φ$ on $E_{2}$ the basis map $⨁_{i} H^{p} (B; R) [- ∣ e_{i} ∣] \to H^{p} (B; R) \otimes_{R} H^{q} (F; R)$ . It is an isomorphism in every bidegree.

2.1F2step 1.1

A morphism of spectral sequences that is an isomorphism on one page is an isomorphism on all subsequent pages, so step 1.1 gives the associated-graded isomorphism at $E_{\infty}$ . Applying [F2] to the actual filtered cup-product map proves that $Φ$ is an isomorphism in every total degree.

3.1F1F2A1step 1.1step 2.1∎

The formula gives $Φ (c a_{i}) = p^{*} c ⌣ Φ (a_{i})$ , so it is an $H^{*} (B; R)$ -module map. For a map of fibrations carrying every specified target $e_{i}$ to the corresponding source $e_{i}$ , contravariance of pullback and cup naturality make the two displayed formulas commute; this is the asserted, choice-of-basis-relative naturality. If the finite list is empty, all fiber cohomology is zero and both sides are zero; one basis element, the zero ring, degree-zero classes, and filtration endpoints are included in steps 1.1–2.1. No splitting or basis is selected, and AC is used exactly through [A1].

Leray–Hirsch module isomorphism

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources