Jump to content

Classifying space for SU(n)

From Wikipedia, the free encyclopedia
(Redirected from Classifying space for SU)

In mathematics, the classifying space for the special unitary 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.

Definition

[edit]

There is a canonical inclusion of complex 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

[edit]
  • Since is the trivial group, is the trivial topological space.
  • Since , one has .

Classification of principal bundles

[edit]

Given a topological space the set of principal bundles on it up to isomorphism is denoted . If is a CW complex, then the map:[1]

is bijective.

Cohomology ring

[edit]

The cohomology ring of with coefficients in the ring of integers is generated by the Chern classes:[2]

Infinite classifying space

[edit]

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

[edit]

Literature

[edit]
  • Hatcher, Allen (2002). Algebraic topology. Cambridge: Cambridge University Press. ISBN 0-521-79160-X.
  • Mitchell, Stephen (August 2001). Universal principal bundles and classifying spaces (PDF).{{cite book}}: CS1 maint: year (link)
[edit]

References

[edit]
  1. ^ "universal principal bundle". nLab. Retrieved 2024-03-14.
  2. ^ Hatcher 02, Example 4D.7.