Classifying space for SO(n)

In mathematics, the classifying space 'for the special orthogonal group' is the base space of the universal principal bundle. This means that principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into. The isomorphism is given by pullback. A particular application are principal SO(2)-bundles.

Definition

There is a canonical inclusion of real oriented Grassmannians given by. Its colimit is:
Since real oriented Grassmannians can be expressed as a homogeneous space by:
the group structure carries over to.

Simplest classifying spaces

Since is the trivial group, is the trivial topological space.
Since, one has.

Classification of principal bundles

Given a topological space the set of principal bundles on it up to isomorphism is denoted. If is a CW complex, then the map:
is bijective.

Cohomology ring

The cohomology ring of with coefficients in the field of two elements is generated by the Stiefel–Whitney classes:
The results holds more generally for every ring with characteristic.
The cohomology ring of with coefficients in the field of rational numbers is generated by Pontrjagin classes and Euler class:

Infinite classifying space

The canonical inclusions induce canonical inclusions on their respective classifying spaces. Their respective colimits are denoted as:
is indeed the classifying space of.