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 ] . {\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; Stasheff, James (1974). (PDF). Princeton University Press. doi:. ISBN 9780691081229.
• Lawson, H. Blaine; Michelsohn, Marie-Louise (1990-02-21). Spin Geometry. Princeton University Press. ISBN 9780691085425.
• Hatcher, Allen (2002). . Cambridge: Cambridge University Press. ISBN 0-521-79160-X.
• Mitchell, Stephen (August 2001). (PDF).

External links

• classifying space on nLab
• BO(n) on nLab