nLab BSO(n)

Idea

$B SO(n)$ is the classifying space for (principal bundles with structure group) the special orthogonal group $SO(n)$. Important examples are principal SO(2)-bundles, principal SO(3)-bundles, principal SO(4)-bundles and principal SO(6)-bundles.

Definition

Definition

$B SO(n)$ is the colimit of the sequence of canonical inclusions of real orientable Grassmannians $\widetilde{Gr}_n(\mathbb{R}^k)\hookrightarrow\widetilde{Gr}_n(\mathbb{R}^{k+1})$:

$$B SO(n) \coloneqq\underset{\longrightarrow}{\lim}_n\widetilde{Gr}_n(\mathbb{R}^k)$$

As the real orientable Grassmannian can be written as a homogenous space by

$$\widetilde{Gr}_n(\mathbb{R}^k) =SO(k)/(SO(n)\times SO(k-n))$$

the group structure carries over to $B SO(n)$.

Smallest classifying spaces

Since $SO(1)\cong 1$ is the trivial group, the classifying space $B SO(1)$ is the trivial topological space. Since $SO(2)\cong U(1)$, once has

$$B SO(2) \cong B U(1) \cong\mathbb{C}P^\infty.$$

Cohomology

Theorem

The cohomology ring of $B S O(n)$ with coefficients in the field $\mathbb{Z}_2$ is generated by the Stiefel-Whitney classes and given by

$$H^*(B SO(n);\mathbb{Z}_2) \cong\mathbb{Z}_2[w_2,\ldots,w_n].$$

(Milnor & Stasheff 74, Theorem 12.4.), (Hatcher 02, Example 4D.6.)

This result holds more generally for every field with characteristic $\operatorname{char}=2$.

Theorem

The cohomology ring of $B SO(n)$ with coefficients in the field $\mathbb{Q}$ of rational numbers is generated by the Pontrjagin classes as well as the Euler class and given by

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

$$H^*(B SO(2n+1);\mathbb{Q}) \cong\mathbb{Q}[p_1,\ldots,p_n]$$

These results hold more generally for every field with characteristic $\operatorname{char}\neq 2$.

Colimit

The canonical inclusions $SO(n)\hookrightarrow SO(n+1)$ yield canonical inclusions $B SO(n)\hookrightarrow B SO(n+1)$ of their respective classifying spaces. The colimit is denoted as

$$B SO \coloneqq\underset{\longrightarrow}{\lim}_n B SO(n).$$

and indeed the classifying space for $SO\coloneqq\underset{\longrightarrow}{\lim}_n SO(n)$.

Related concepts

• BO(n)

• BU(n)

• BSU(n)

References

• John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press (1974) (ISBN:9780691081229, doi:10.1515/9781400881826, pdf)