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A continuously varying family of vector spaces of the same dimension over a topological space. If the vector spaces are one-dimensional, the term line bundle is used and has the associated tag line-bundles.

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Extending the natural thom form of a vector bundle from the boundary of a manifold

(Edited after taking into account Tom Goodwillie's answer.) Let $E \rightarrow X$ be an orientable vector bundle. In this MO answer it is explained how to obtain a representative of the Thom class (ca …