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The space of immersions of a manifold in another manifold is homotopically equivalent to the space of bundle injections from the tangent space of the first to the tangent ...

Two vector bundles are stably equivalent iff isomorphic vector bundles are obtained upon Whitney summing each vector bundle with a trivial vector bundle.

The space E of a fiber bundle given by the map f:E->B, where B is the base space of the fiber bundle.

A fiber space, depending on context, means either a fiber bundle or a fibration.

A hypersphere S^n is parallelizable if there are n vector fields that are linearly independent at each point. There exist only three parallelizable spheres: S^1, S^3, and S^7 ...

A foliation F of dimension p on a manifold M is transversely orientable if it is integral to a p-plane distribution D on M whose normal bundle Q is orientable. A p-plane ...

Roughly speaking, a tangent vector is an infinitesimal displacement at a specific point on a manifold. The set of tangent vectors at a point P forms a vector space called the ...

Characteristic classes are cohomology classes in the base space of a vector bundle, defined through obstruction theory, which are (perhaps partial) obstructions to the ...

A gadget defined for complex vector bundles. The Chern classes of a complex manifold are the Chern classes of its tangent bundle. The ith Chern class is an obstruction to the ...

If f:E->B is a fiber bundle with B a paracompact topological space, then f satisfies the homotopy lifting property with respect to all topological spaces. In other words, if ...