Classifying space for SU(n)

In mathematics, the classifying space

for the special unitary group

is the base space of the universal

{\displaystyle \operatorname {ESU} (n)\rightarrow \operatorname {BSU} (n)}

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.

There is a canonical inclusion of complex oriented Grassmannians given by

Since real oriented Grassmannians can be expressed as a homogeneous space by: the group structure carries over to

principal bundles on it up to isomorphism is denoted

{\displaystyle \operatorname {Prin} _{\operatorname {SU} (n)}(X)}

is a CW complex, then the map:[1] is bijective.

The cohomology ring of

with coefficients in the ring

of integers is generated by the Chern classes:[2] The canonical inclusions

induce canonical inclusions

on their respective classifying spaces.

Their respective colimits are denoted as:

is indeed the classifying space of