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 ${\displaystyle \mathbb {R} ^{\infty }}$.

Cohomology ring

The cohomology ring of ${\displaystyle \operatorname {BO} (n)}$ with coefficients in the field ${\displaystyle \mathbb {Z} _{2}}$ of two elements is generated by the Stiefel–Whitney classes:^[1][2]

${\displaystyle H^{*}(\operatorname {BO} (n);\mathbb {Z} _{2})=\mathbb {Z} _{2}[w_{1},\ldots ,w_{n}].}$

Infinite classifying space

The canonical inclusions ${\displaystyle \operatorname {O} (n)\hookrightarrow \operatorname {O} (n+1)}$ induce canonical inclusions ${\displaystyle \operatorname {BO} (n)\hookrightarrow \operatorname {BO} (n+1)}$ on their respective classifying spaces. Their respective colimits are denoted as:

${\displaystyle \operatorname {O}$ :=\lim _{n\rightarrow \infty }\operatorname {O} (n);}

${\displaystyle \operatorname {BO}$ :=\lim _{n\rightarrow \infty }\operatorname {BO} (n).}

${\displaystyle \operatorname {BO} }$ is indeed the classifying space of ${\displaystyle \operatorname {O} }$.

See also

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

Literature

• Milnor, John; Stasheff, James (1974). 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).{{cite book}}: CS1 maint: year (link)

External links

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