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

Both constructions are detailed here.

The total space EU(n) of the universal bundle is given by

Here, H is an infinite-dimensional complex Hilbert space, the e_i are vectors in H, and ${\displaystyle \delta _{ij}}$ is the Kronecker delta. The symbol ${\displaystyle (\cdot ,\cdot )}$ is the inner product on H. Thus, we have that EU(n) is the space of orthonormal n-frames in H.

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