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Concept# Principal bundle

Summary

In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with
# An action of G on P, analogous to (x, g)h = (x, gh) for a product space.

# A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) \mapsto x.

Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of (x,e). Likewise, there is not generally a projection onto G generalizing the projection onto the second factor, X \times G \to G that exists for the Cartesian product. They may also have a complicated topology that prevents them from being realized as a product space even

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