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:Milnor & Stasheff 74, section 12.2 The Oriented Universal Bundle on page 151
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:Milnor & Stasheff, Theorem 12.4.Lawson & Michelson 90, Theorem B.8Hatcher 02, Example 4D.6.
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:Lawson & Michelson 90, Theorem B.14
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 .
See also
Classifying space for O(n) Classifying space for U(n) Classifying space for SU(n)
Literature
External links
classifying space on nLab BSO(n) on nLab
References