Classifying space for SO(n)

In mathematics, the classifying space $$\operatorname{BSO}(n)$$ for the special orthogonal group $$\operatorname{SO}(n)$$ is the base space of the universal $$\operatorname{SO}(n)$$ principal bundle $$\operatorname{ESO}(n)\rightarrow\operatorname{BSO}(n)$$. This means that $$\operatorname{SO}(n)$$ principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into $$\operatorname{BSO}(n)$$. The isomorphism is given by pullback.

Definition
There is a canonical inclusion of real oriented Grassmannians given by $$\widetilde\operatorname{Gr}_n(\mathbb{R}^k)\hookrightarrow\widetilde\operatorname{Gr}_n(\mathbb{R}^{k+1}), V\mapsto V\times\{0\}$$. Its colimit is:


 * $$\operatorname{BSO}(n)
 * =\operatorname{Gr}_n(\mathbb{R}^\infty)
 * =\lim_{k\rightarrow\infty}\widetilde\operatorname{Gr}_n(\mathbb{R}^k).$$

Since real oriented Grassmannians can be expressed as a homogeneous space by:


 * $$\widetilde\operatorname{Gr}_n(\mathbb{R}^k)

=\operatorname{SO}(n+k)/(\operatorname{SO}(n)\times\operatorname{SO}(k))$$

the group structure carries over to $$\operatorname{BSO}(n)$$.

Simplest classifying spaces
\cong 1$$ is the trivial group, $$\operatorname{BSO}(1) \cong\{*\}$$ is the trivial topological space.
 * Since $$\operatorname{SO}(1)

\cong\operatorname{U}(1)$$, one has $$\operatorname{BSO}(2) \cong\operatorname{BU}(1) \cong\mathbb{C}P^\infty$$.
 * Since $$\operatorname{SO}(2)

Classification of principal bundles
Given a topological space $$X$$ the set of $$\operatorname{SO}(n)$$ principal bundles on it up to isomorphism is denoted $$\operatorname{Prin}_{\operatorname{SO}(n)}(X)$$. If $$X$$ is a CW complex, then the map:


 * $$[X,\operatorname{BSO}(n)]\rightarrow\operatorname{Prin}_{\operatorname{SO}(n)}(X),

[f]\mapsto f^*\operatorname{ESO}(n)$$

is bijective.

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


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

=\mathbb{Z}_2[w_2,\ldots,w_n].$$

The results holds more generally for every ring with characteristic $$\operatorname{char}=2$$.

The cohomology ring of $$\operatorname{BSO}(n)$$ with coefficients in the field $$\mathbb{Q}$$ of rational numbers is generated by Pontrjagin classes and Euler class:


 * $$H^*(\operatorname{BSO}(2n);\mathbb{Q})

\cong\mathbb{Q}[p_1,\ldots,p_n,e]/(p_n-e^2),$$
 * $$H^*(\operatorname{BSO}(2n+1);\mathbb{Q})

\cong\mathbb{Q}[p_1,\ldots,p_n].$$

The results holds more generally for every ring with characteristic $$\operatorname{char}\neq 2$$.

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


 * $$\operatorname{SO}
 * =\lim_{n\rightarrow\infty}\operatorname{SO}(n);$$
 * $$\operatorname{BSO}
 * =\lim_{n\rightarrow\infty}\operatorname{BSO}(n).$$

$$\operatorname{BSO}$$ is indeed the classifying space of $$\operatorname{SO}$$.