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Classifying space for SU(n)

From Wikipedia, the free encyclopedia

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

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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

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

Classification of principal bundles

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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

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

Infinite classifying space

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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

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Literature

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  • 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).
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References

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