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$B U(n)$ is the classifying space for (principal bundles with structure group) the unitary group $U(n)$.
$B U(n)$ is the limit of the sequence of canonical inclusions of complex Grassmannians $Gr_n(\mathbb{C}^k)\hookrightarrow Gr_n(\mathbb{C}^{k+1})$:
(Milnor & Stasheff 74, page 169)
As the complex Grassmannian can be written as a homogenous space by
the group structure carries over to $B U(n)$.
The smallest classifying space $B U(1)$ is the infinite complex projective space $\mathbb{C}P^\infty$. It is also a Eilenberg–MacLane space $K(\mathbb{Z},2)$, hence $U(1)$ principal bundles over a space are classified by its second cohomology in the ring $\mathbb{Z}$ of integers:
Higher classifying spaces are not necessarily Eilenberg-MacLane spaces, but continuous maps into them are possible, giving the idea behind characteristic classes. But they are defined for vector bundles, which can be constructed from principal bundles using the balanced product? $Prin_{U(n)}(X)\rightarrow Vect_\mathbb{C}^n(X),E\mapsto E\times_{U(n)}\mathbb{C}^n$.
The cohomology ring of $B U(n)$ with coefficients in the ring $\mathbb{Z}$ of integers is generated by the Chern classes and given by
This theorem also holds if $\mathbb{Z}$ is replaced by a general ring $R$.
The canonical inclusions $U(n)\hookrightarrow U(n+1)$ yield canonical inclusions $B U(n)\hookrightarrow B U(n+1)$ of their respective classifying spaces. The colimit is denoted
which is the classifying space of the stable unitary group $U\coloneqq\underset{\longrightarrow}{\lim}_n U(n)$.
John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press (1974) (ISBN:9780691081229, doi:10.1515/9781400881826, pdf)
Allen Hatcher, Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]
See also
Last revised on March 12, 2024 at 08:40:43. See the history of this page for a list of all contributions to it.