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:¹²³

${\displaystyle H^{*}(\operatorname {BO} (n);\mathbb {Z} _{2})=\mathbb {Z} _{2}[w_{1},\ldots ,w_{n}].}$

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:

${\displaystyle \operatorname {O} :=\lim _{n\rightarrow \infty }\operatorname {O} (n);}$

${\displaystyle \operatorname {BO} :=\lim _{n\rightarrow \infty }\operatorname {BO} (n).}$

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

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