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Fiber bundles, such as the tangent bundle of a manifold and other more general vector bundles, play an important role in differential geometry and differential topology, as do principal bundles. Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps , and the class of fiber bundles forms ...
If M is a paracompact manifold and P → M is a principal G-bundle, then there exists a map f : M → BG, unique up to homotopy, such that P is isomorphic to f ∗ (EG), the pull-back of the G-bundle EG → BG by f. Proof. On one hand, the pull-back of the bundle π : EG → BG by the natural projection P × G EG → BG is the bundle P × EG.
The vector bundles associated to these principal bundles via the natural action of G on are just the tautological bundles over the Grassmannians. In other words, the Stiefel manifold V k ( F n ) {\displaystyle V_{k}(\mathbb {F} ^{n})} is the orthogonal, unitary, or symplectic frame bundle associated to the tautological bundle on a Grassmannian.
In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topology. Without this requirement, more general objects can be considered bundles. For example, one can consider a bundle π: E → B with E and B sets.
If we take the bundle in the above to be the tangent bundle of a smooth manifold, the conclusion of the above is called the Wu formula, and has the following strong consequence: since the Steenrod operations are invariant under homotopy equivalence, we conclude that the Stiefel–Whitney classes of a manifold are as well. This is an ...
The Pontryagin classes of a smooth manifold are defined to be the Pontryagin classes of its tangent bundle. Novikov proved in 1966 that if two compact, oriented, smooth manifolds are homeomorphic then their rational Pontryagin classes p k ( M , Q ) {\displaystyle p_{k}(M,\mathbf {Q} )} in H 4 k ( M , Q ) {\displaystyle H^{4k}(M,\mathbf {Q ...
It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle . [1] As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces.
Equivalently M is spin if the SO(n) principal bundle of orthonormal bases of the tangent fibers of M is a Z 2 quotient of a principal spin bundle. If the manifold has a cell decomposition or a triangulation, a spin structure can equivalently be thought of as a homotopy-class of trivialization of the tangent bundle over the 1-skeleton that ...
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