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 $$\mathbb{R}^\infty$$.

Cohomology ring
The cohomology ring of $$\operatorname{BO}(n)$$ with coefficients in the field $$\mathbb{Z}_2$$ of two elements is generated by the Stiefel–Whitney classes:


 * $$H^*(\operatorname{BO}(n);\mathbb{Z}_2)

=\mathbb{Z}_2[w_1,\ldots,w_n].$$

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


 * $$\operatorname{O}
 * =\lim_{n\rightarrow\infty}\operatorname{O}(n);$$
 * $$\operatorname{BO}
 * =\lim_{n\rightarrow\infty}\operatorname{BO}(n).$$

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