Classifying space for O(n)

{{DISPLAYTITLE: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:Milnor & Stasheff, Theorem 7.1 on page 83Hatcher 02, Theorem 4D.4.

: $H^*(\operatorname{BO}(n);\mathbb{Z}_2)
=\mathbb{Z}_2[w_1,\ldots,w_n].$

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 $\operatorname{O}$ .

Literature

{{cite book|title=Characteristic classes|publisher=Princeton University Press|location=|year=1974|language=en|isbn=9780691081229|doi=10.1515/9781400881826|url=https://www.maths.ed.ac.uk/~v1ranick/papers/milnstas.pdf|last=Milnor|first=John|author-link=John Milnor|last2=Stasheff|first2=James|author-link2=James Stasheff}}
{{cite book|last=Hatcher|first=Allen|title=Algebraic topology|publisher=Cambridge University Press|location=Cambridge|year=2002|language=en|isbn=0-521-79160-X|url=https://pi.math.cornell.edu/~hatcher/AT/ATpage.html}}
{{cite book|title=Universal principal bundles and classifying spaces|publisher=|location=|isbn=|url=https://math.mit.edu/~mbehrens/18.906/prin.pdf|doi=|last=Mitchell|first=Stephen|year=August 2001}}

External links

classifying space on nLab
BO(n) on nLab

References

Category:Algebraic topology