Classifying space for SO(n)

In mathematics, the classifying space

for the special orthogonal group

is the base space of the universal

{\displaystyle \operatorname {ESO} (n)\rightarrow \operatorname {BSO} (n)}

principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into

There is a canonical inclusion of real oriented Grassmannians given by

Its colimit is:[1] Since real oriented Grassmannians can be expressed as a homogeneous space by: the group structure carries over to

Given a topological space

principal bundles on it up to isomorphism is denoted

is a CW complex, then the map:[2] is bijective.

with coefficients in the field

of two elements is generated by the Stiefel–Whitney classes:[3][4] The results holds more generally for every ring with characteristic

with coefficients in the field

of rational numbers is generated by Pontrjagin classes and Euler class: The canonical inclusions

induce canonical inclusions

on their respective classifying spaces.

Their respective colimits are denoted as: