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Classifying space for SO(n)

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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.

Definition

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There is a canonical inclusion of real oriented Grassmannians given by . Its colimit is:[1]

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:[2]

is bijective.

Cohomology ring

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The cohomology ring of with coefficients in the field of two elements is generated by the Stiefel–Whitney classes:[3][4]

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:

The results holds more generally for every ring with characteristic .

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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  • Milnor, John; Stasheff, James (1974). Characteristic classes (PDF). Princeton University Press. doi:10.1515/9781400881826. ISBN 9780691081229.
  • 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)
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References

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  1. ^ Milnor & Stasheff 74, section 12.2 The Oriented Universal Bundle on page 151
  2. ^ "universal principal bundle". nLab. Retrieved 2024-03-14.
  3. ^ Milnor & Stasheff, Theorem 12.4.
  4. ^ Hatcher 02, Example 4D.6.