Classifying space for U(n)
Exact homotopy case / From Wikipedia, the free encyclopedia
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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
- the Grassmannian of n-planes in an infinite-dimensional complex Hilbert space; or,
- the direct limit, with the induced topology, of Grassmannians of n planes.
Both constructions are detailed here.