Classifying space for SO(n)

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

Source: Wikipedia — Classifying space for SO(n) (CC BY-SA 4.0)