Classifying space for O(n)

In mathematics, the classifying space for the orthogonal group O(n) may be constructed as the Grassmannian of n-planes in an infinite-dimensional real space ${\displaystyle \mathbb {R} ^{\infty }}$.

The cohomology ring of ${\displaystyle \operatorname {BO} (n)}$ with coefficients in the field ${\displaystyle \mathbb {Z} _{2}}$ of two elements is generated by the Stiefel–Whitney classes:

The canonical inclusions ${\displaystyle \operatorname {O} (n)\hookrightarrow \operatorname {O} (n+1)}$ induce canonical inclusions ${\displaystyle \operatorname {BO} (n)\hookrightarrow \operatorname {BO} (n+1)}$ on their respective classifying spaces. Their respective colimits are denoted as:

${\displaystyle \operatorname {BO} }$ is indeed the classifying space of ${\displaystyle \operatorname {O} }$.