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 ] . {\displaystyle H^{*}(\operatorname {BO} (n);\mathbb {Z} _{2})=\mathbb {Z} _{2}[w_{1},\ldots ,w_{n}].}
Infinite classifying space The canonical inclusions O ( n ) ↪ O ( n + 1 ) {\displaystyle \operatorname {O} (n)\hookrightarrow \operatorname {O} (n+1)} induce canonical inclusions BO ( n ) ↪ BO ( n + 1 ) {\displaystyle \operatorname {BO} (n)\hookrightarrow \operatorname {BO} (n+1)} on their respective classifying spaces. Their respective colimits are denoted as:
O := lim n → ∞ O ( n ) ; {\displaystyle \operatorname {O} :=\lim _{n\rightarrow \infty }\operatorname {O} (n);}
BO := lim n → ∞ BO ( n ) . {\displaystyle \operatorname {BO} :=\lim _{n\rightarrow \infty }\operatorname {BO} (n).}
BO {\displaystyle \operatorname {BO} } is indeed the classifying space of O {\displaystyle \operatorname {O} } .
See also Classifying space for U(n) Classifying space for SO(n) Classifying space for SU(n)
Literature Milnor, John W.; Stasheff, James D. (1974). Characteristic Classes. Annals of Mathematics Studies. Vol. 76. Princeton University Press; University of Tokyo Press. ISBN 978-0-691-08122-9. MR 0440554. Lawson, H. Blaine; Michelsohn, Marie-Louise (1990-02-21). Spin Geometry. Princeton University Press. ISBN 9780691085425. 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).
External links classifying space on nLab BO(n) on nLab
References