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 ${ℝ}^{\mathrm{\infty }}$.

Cohomology ring

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

$H^{\ast }(\mathrm{BO}(n);{\mathbb{Z}}_2)={\mathbb{Z}}_2[w_1,\dots ,w_n].$

Infinite classifying space

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

$\mathrm{O}:=\underset{n\to \mathrm{\infty }}{lim}\mathrm{O}(n);$

$\mathrm{BO}:=\underset{n\to \mathrm{\infty }}{lim}\mathrm{BO}(n).$

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

See also

• Classifying space for U(n)
• Classifying space for SO(n)
• Classifying space for SU(n)

Literature

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External links

• classifying space on nLab
• BO(n) on nLab