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:^[1]

\operatorname {BSO} (n):={\widetilde {\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

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

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

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:^[2]

[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:^[3]^[4]

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: