Homotopy theory

Classifying space for U(n)

In mathematics, the classifying space for the unitary group U(n) is a space BU(n) together with a universal bundle EU(n) such that any hermitian bundle on a paracompact space X is the pull-back of EU(n) by a map X → BU(n) unique up to homotopy. This space with its universal fibration may be constructed as either 1. * the Grassmannian of n-planes in an infinite-dimensional complex Hilbert space; or, 2. * the direct limit, with the induced topology, of Grassmannians of n planes. Both constructions are detailed here. (Wikipedia).

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Related pages

Homotopy group | Unitary group | Maximal torus | Kronecker delta | CW complex | Isomorphism class | Symmetric polynomial | Cohomology | Grassmannian | Classifying space for O(n) | Classifying space | Polynomial ring | Euler class | Circle bundle | Direct limit | Torus | Mathematics | Fundamental class | Ring (mathematics) | Pontryagin product | Compact space | Manifold | Hilbert space | Whitehead theorem | Universal bundle | Bott periodicity theorem | Projective unitary group | Universal property | Splitting principle | Complex projective space | Paracompact space | Topological K-theory