DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Pullback vector bundles and sections

Definition

Given a continuous map $f : X \to Y$ and a vector bundle $q : E \to Y$ , its pullback is

$f^{*} E = {(x, e) \in X \times E : f (x) = q (e)}, (x, e) ⟼ x,$

with the subspace topology of Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace and fiberwise operations inherited from $E$ . A linear chart $q^{- 1} (U) ≅ U \times F^{n}$ pulls back to $f^{- 1} (U) \times F^{n}$ , so this is a vector bundle in the sense of Real and complex topological vector bundles.

The map $(x, e) \mapsto e$ is the canonical bundle map $f^{*} E \to E$ over $f$ in the sense of Bundle maps, sections, subbundles, and isomorphisms. If $s : Y \to E$ is a section, its pullback is

$f^{*} s : X ⟶ f^{*} E, x ⟼ (x, s (f (x))) .$

Depends on

Used by

General Thom isomorphism from the relative Serre spectral sequence Lemma
Vector-bundle pullback is canonically functorial Proposition
Homotopy invariance of vector-bundle pullback Theorem
Naturality and uniqueness of Thom classes Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hatcher, Vector Bundles & K-Theory, §1.1 (standard reference, not scraped)
Milnor and Stasheff, Characteristic Classes, §3 (standard reference, not scraped)