Langbahn Team – Weltmeisterschaft

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

Cohomology ring

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

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

Infinite classifying space

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

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

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

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

See also

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

References

^ Milnor & Stasheff, Theorem 7.1 on page 83
^ Hatcher 02, Theorem 4D.4.

Quelle: http://en.wikipedia.org/wiki/Classifying_space_for_O(n)