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Classifying space for SU(n)
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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.
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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 .
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Simplest classifying spaces
- Since is the trivial group, is the trivial topological space.
- Since , one has .
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Classification of principal bundles
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Perspective
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
The cohomology ring of with coefficients in the ring of integers is generated by the Chern classes:[2]
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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 .
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See also
Literature
- 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)
External links
- classifying space on nLab
- BSU(n) on nLab
References
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