nLab BO(n)

Idea

$B O(n)$ is the classifying space for (principal bundles with structure group) the orthogonal group $O(n)$.

Definition

Definition

$B O(n)$ is the limit of the sequence of canonical inclusions of real Grassmannians $Gr_n(\mathbb{R}^k)\hookrightarrow Gr_n(\mathbb{R}^{k+1})$:

$$B O(n) \coloneqq\underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{R}^k)$$

(Milnor & Stasheff 74, page 151)

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

$$Gr_n(\mathbb{R}^k) =O(k)/(O(n)\times O(k-n))$$

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

Smallest classifying space

The smallest classifying space $B O(1)$ is the infinite real projective space $\mathbb{R}P^\infty$. It is also a Eilenberg–MacLane space $K(\mathbb{Z}_2,1)$, hence $U(1)$ principal bundles over a space are classified by its first cohomology in the field $\mathbb{Z}_2$ of two elements:

$$Prin_{O(1)}(X) =[X,B O(1)] =[X,\mathbb{R}P^\infty] =[X,K(\mathbb{Z}_2,1)] \cong H^1(X;\mathbb{Z}_2)$$

Higher classifying spaces are not necessarily Eilenberg-MacLane spaces, but continuous maps into them are possible, giving the idea behind characteristic classes. But they are defined for vector bundles, which can be constructed from principal bundles using the balanced product? $Prin_{O(n)}(X)\rightarrow Vect_\mathbb{R}^n(X),E\mapsto E\times_{O(n)}\mathbb{R}^n$.

Cohomology

Theorem

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

$$H^*(B O(n);\mathbb{Z}_2) \cong\mathbb{Z}_2[w_1,\ldots,w_n].$$

(Milnor & Stasheff 74, Theorem 7.1.), (Hatcher 02, Theorem 4D.4.)

Theorem

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

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

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

Colimit

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

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

and indeed the classifying space for the stable orthogonal group $O\coloneqq\underset{\longrightarrow}{\lim}_n O(n)$.

Related concepts

• BSO(n)

• BU(n)

• BSU(n)

• KO

References

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

• Allen Hatcher, Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]

See also

• Wikipedia, Classifying space for O(n)