TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-14

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Vector bundles are glued from transition cocycles

Statement

Let $(U_{i})_{i \in I}$ be an open cover of $X$ , and let $g_{j i} : U_{i} \cap U_{j} \to {GL}_{n} (F)$ be continuous maps such that

$g_{i i} = I, g_{k i} = g_{k j} g_{j i} .$

The quotient of $∐_{i} U_{i} \times F^{n}$ by $(x, v, i) \sim (x, g_{j i} (x) v, j)$ is a rank- $n$ $F$ -vector bundle. Replacing $g_{j i}$ by $g_{j i}^{'} = h_{j} g_{j i} h_{i}^{- 1}$ for continuous $h_{i} : U_{i} \to {GL}_{n} (F)$ gives an isomorphic bundle. Every rank- $n$ bundle is recovered from the cocycle of any linear atlas.

Facts & Assumptions

Given: The cover and cocycle in the statement.

[F1]

A vector bundle is locally a product by fiberwise-linear charts, and its transition order is $g_{k i} = g_{k j} g_{j i}$ (Real and complex topological vector bundles).

[F2]

A map out of a quotient is continuous exactly when its composite with the quotient map is continuous (For a quotient map $q : X \to Y$ , a map out of $Y$ is continuous iff its composite with $q$ is; a continuous map on $X$ constant on the fibres of $q$ factors uniquely through $q$ ; and a composite of quotient maps is a quotient map).

Proof

technique · direct

1.1

The cocycle with $k = i$ gives $g_{i j} g_{j i} = I$ . Hence the displayed relation is reflexive, symmetric, and transitive: the transitive calculation is $g_{k j} (g_{j i} v) = g_{k i} v$ . It therefore defines a quotient $q : ∐_{i} U_{i} \times F^{n} \to E$ and a map $p : E \to X$ by $p [x, v, i] = x$ . Since $p q (x, v, i) = x$ is continuous on every summand, the quotient property [F2] makes $p$ continuous.

F2givenalgebra

2.1

The quotient map $q$ is open. Indeed, if $O$ is open in the coproduct, then the part of its saturation in the $j$ th summand is the union over $k$ of the images of $O \cap ((U_{j} \cap U_{k}) \times F^{n} \times {k})$ under the homeomorphism $(x, v, k) \mapsto (x, g_{j k} (x) v, j)$ ; its inverse uses $g_{k j} = g_{j k}^{- 1}$ . Hence every such part is open. The restriction of $q$ over the saturated open set $p^{- 1} (U_{i})$ is therefore again a quotient map. Define $Φ_{i} [x, v, k] = (x, g_{i k} (x) v)$ . The cocycle makes this independent of the representative, and its composite with the restricted quotient map is continuous on every summand, so [F2] makes $Φ_{i}$ continuous. Its inverse is $(x, w) \mapsto [x, w, i]$ and is continuous as the $i$ th-summand inclusion followed by $q$ . Thus $Φ_{i} : p^{- 1} (U_{i}) ≅ U_{i} \times F^{n}$ is a fiberwise-linear chart. Its overlap from $i$ to $j$ is $g_{j i}$ , so [F1] proves that $E$ is the claimed bundle.

F1F2step 1.1

3.1

For the primed cocycle, the maps on summands $(x, v, i) \mapsto [x, h_{i} (x) v, i]^{'}$ respect the relation because $g_{j i}^{'} h_{i} = h_{j} g_{j i}$ . By [F2] they descend to a continuous fiberwise-linear map $E \to E^{'}$ . Replacing $h_{i}$ by $h_{i}^{- 1}$ gives its continuous inverse, so it is a bundle isomorphism.

F2step 2.1algebra

4.1

Finally, a linear atlas of a rank- $n$ bundle supplies the functions $g_{j i}$ and their cocycle law by [F1]. Sending the quotient class $[x, v, i]$ to $ϕ_{i}^{- 1} (x, v)$ is well-defined, continuous by [F2], and in each chart is the identity map on $U_{i} \times F^{n}$ . It is therefore a bundle isomorphism from the reconstructed quotient to the original bundle.

F1F2step 2.1∎

Depends on

Used by

Clutching construction for bundles over a suspension Definition
Stiefel spaces, Grassmannians, and tautological bundles Definition
Whitney sum, tensor, dual, Hom, and exterior-power bundles Definition
Normalized clutching data for bundles over X×S² Lemma
Polynomial clutching families stabilize to linear clutching Lemma

Dependency tree · two levels

10 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, §§2–3 (standard reference, not scraped)