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:Milnor & Stasheff, Theorem 7.1 on page 83Lawson & Michelson 90, Theorem B.7Hatcher 02, Theorem 4D.4.

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

Classifying space for U(n) Classifying space for SO(n) Classifying space for SU(n)

Literature

External links

classifying space on nLab BO(n) on nLab

References