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.
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 .
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.
The cohomology ring of with coefficients in the field of two elements is generated by the StiefelâÂÂWhitney 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:
The canonical inclusions induce canonical inclusions on their respective classifying spaces. Their respective colimits are denoted as:
is indeed the classifying space of .