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 R ∞ {\displaystyle \mathbb {R} ^{\infty }} . == Cohomology ring == The cohomology ring of BO ⁡ ( n ) {\displaystyle \operatorname {BO} (n)} with coefficients in the field Z 2 {\displaystyle \mathbb {Z} _{2}} of two elements is generated by the Stiefel–Whitney classes: H ∗ ( BO ⁡ ( n ) ; Z 2 ) = Z 2 [ w 1 , … , w n ] .

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