Classifying space for O(n)

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

The cohomology ring of with coefficients in the field of two elements is generated by the Stiefel–Whitney classes:[1][2]

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 .

See also

Literature

  • 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).

References

  1. Milnor & Stasheff, Theorem 7.1 on page 83
  2. Hatcher 02, Theorem 4D.4.
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