Classifying space for O(n)

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In mathematics, the classifying space for the orthogonal group O(n) may be constructed as the Grassmannian of n-planes in an infinite-dimensional real space .

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:[1][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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  • 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, Theorem 7.1 on page 83
  2. ^ Hatcher 02, Theorem 4D.4.