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$B O(n)$ is the classifying space for (principal bundles with structure group) the orthogonal group $O(n)$.
$B O(n)$ is the limit of the sequence of canonical inclusions of real Grassmannians $Gr_n(\mathbb{R}^k)\hookrightarrow Gr_n(\mathbb{R}^{k+1})$:
(Milnor & Stasheff 74, page 151)
As the real Grassmannian can be written as a homogenous space by
the group structure carries over to $B O(n)$.
The smallest classifying space $B O(1)$ is the infinite real projective space $\mathbb{R}P^\infty$. It is also a Eilenberg–MacLane space $K(\mathbb{Z}_2,1)$, hence $U(1)$ principal bundles over a space are classified by its first cohomology in the field $\mathbb{Z}_2$ of two elements:
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? $Prin_{O(n)}(X)\rightarrow Vect_\mathbb{R}^n(X),E\mapsto E\times_{O(n)}\mathbb{R}^n$.
The cohomology ring of $B O(n)$ with coefficients in the field $\mathbb{Z}_2$ is generated by the Stiefel-Whitney classes and given by
(Milnor & Stasheff 74, Theorem 7.1.), (Hatcher 02, Theorem 4D.4.)
The cohomology ring of $B O(n)$ with coefficients in the field $\mathbb{Q}$ of rational numbers is generated by the Pontrjagin classes and given by
The canonical inclusions $O(n)\hookrightarrow O(n+1)$ yield canonical inclusions $B O(n)\hookrightarrow B O(n+1)$ of their respective classifying spaces. The colimit is denoted as
and indeed the classifying space for the stable orthogonal group $O\coloneqq\underset{\longrightarrow}{\lim}_n O(n)$.
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 08:40:35. See the history of this page for a list of all contributions to it.