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$B SO(n)$ is the classifying space for the special orthogonal group $SO(n)$.
$B SO(n)$ is the colimit of the sequence of canonical inclusions of real orientable Grassmannians $\widetilde{Gr}_n(\mathbb{R}^k)\hookrightarrow\widetilde{Gr}_n(\mathbb{R}^{k+1})$:
the group structure carries over to $B SO(n)$.
Since $SO(1)\cong 1$ is the trivial group, the classifying space $B SO(1)$ is the trivial topological space. Since $SO(2)\cong U(1)$, once has
The cohomology ring of $B S O(n)$ with coefficients in the field $\mathbb{Z}_2$ is generated by the Stiefel-Whitney classes and given by
(Milnor & Stasheff 74, Theorem 12.4.), (Hatcher 02, Example 4D.6.)
This result holds more generally for every field with characteristic $\operatorname{char}=2$.
The cohomology ring of $B SO(n)$ with coefficients in the field $\mathbb{Q}$ of rational numbers is generated by the Pontrjagin classes as well as the Euler class and given by
These results hold more generally for every field with characteristic $\operatorname{char}\neq 2$.
The canonical inclusions $SO(n)\hookrightarrow SO(n+1)$ yield canonical inclusions $B SO(n)\hookrightarrow B SO(n+1)$ of their respective classifying spaces. The colimit is denoted as
and indeed the classifying space for $SO\coloneqq\underset{\longrightarrow}{\lim}_n SO(n)$.
Created on March 14, 2024 at 15:50:24. See the history of this page for a list of all contributions to it.