Classifying space for SO(n)

In mathematics, the classifying space $\operatorname{BSO}(n)$ for the [https://en.wikipedia.org/?title=Special_orthogonal_group special orthogonal group] $\operatorname{SO}(n)$ is the base space of the universal $\operatorname{SO}(n)$ principal bundle $\operatorname{ESO}(n)\rightarrow\operatorname{BSO}(n)$ . This means that $\operatorname{SO}(n)$ principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into $\operatorname{BSO}(n)$ . The isomorphism is given by pullback.

Definition

There is a canonical inclusion of real oriented Grassmannians given by $\widetilde\operatorname{Gr}_n(\mathbb{R}^k)\hookrightarrow\widetilde\operatorname{Gr}_n(\mathbb{R}^{k+1}),
V\mapsto V\times\{0\}$ . Its colimit is:Milnor & Stasheff 74, section 12.2 The Oriented Universal Bundle on page 151

: $\operatorname{BSO}(n)
:=\widetilde\operatorname{Gr}_n(\mathbb{R}^\infty)
:=\lim_{k\rightarrow\infty}\widetilde\operatorname{Gr}_n(\mathbb{R}^k).$

Since real oriented Grassmannians can be expressed as a homogeneous space by:

: $\widetilde\operatorname{Gr}_n(\mathbb{R}^k)
=\operatorname{SO}(n+k)/(\operatorname{SO}(n)\times\operatorname{SO}(k))$

the group structure carries over to $\operatorname{BSO}(n)$ .

Simplest classifying spaces

Since $\operatorname{SO}(1)$

\cong 1 is the trivial group,

\operatorname{BSO}(1)
\cong\{*\}

is the trivial topological space.

Since $\operatorname{SO}(2)$

\cong\operatorname{U}(1), one has

\operatorname{BSO}(2)
\cong\operatorname{BU}(1)
\cong\mathbb{C}P^\infty

Classification of principal bundles

Given a topological space $X$ the set of $\operatorname{SO}(n)$ principal bundles on it up to isomorphism is denoted $\operatorname{Prin}_{\operatorname{SO}(n)}(X)$ . If $X$ is a CW complex, then the map:{{cite web |access-date=2024-03-14 |language=en |title=universal principal bundle |url=https://ncatlab.org/nlab/show/universal+principal+bundle |website=nLab}}

: $[X,\operatorname{BSO}(n)]\rightarrow\operatorname{Prin}_{\operatorname{SO}(n)}(X),
[f]\mapsto f^*\operatorname{ESO}(n)$

is bijective.

Cohomology ring

The cohomology ring of $\operatorname{BSO}(n)$ with coefficients in the field $\mathbb{Z}_2$ of two elements is generated by the Stiefel–Whitney classes:Milnor & Stasheff, Theorem 12.4.Hatcher 02, Example 4D.6.

: $H^*(\operatorname{BSO}(n);\mathbb{Z}_2)
=\mathbb{Z}_2[w_2,\ldots,w_n].$

The results holds more generally for every ring with characteristic $\operatorname{char}=2$ .

The cohomology ring of $\operatorname{BSO}(n)$ with coefficients in the field $\mathbb{Q}$ of rational numbers is generated by Pontrjagin classes and Euler class:

: $H^*(\operatorname{BSO}(2n);\mathbb{Q})
\cong\mathbb{Q}[p_1,\ldots,p_n,e]/(p_n-e^2),$

: $H^*(\operatorname{BSO}(2n+1);\mathbb{Q})
\cong\mathbb{Q}[p_1,\ldots,p_n].$

Infinite classifying space

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

: $\operatorname{SO}
:=\lim_{n\rightarrow\infty}\operatorname{SO}(n);$

: $\operatorname{BSO}
:=\lim_{n\rightarrow\infty}\operatorname{BSO}(n).$

$\operatorname{BSO}$ is indeed the classifying space of $\operatorname{SO}$ .

Literature

{{cite book|title=Characteristic classes|publisher=Princeton University Press|location=|year=1974|language=|isbn=9780691081229|url=https://www.maths.ed.ac.uk/~v1ranick/papers/milnstas.pdf|doi=10.1515/9781400881826|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=|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=|year=August 2001|isbn=|url=https://math.mit.edu/~mbehrens/18.906/prin.pdf|last=Mitchell|first=Stephen|doi=}}

External links

classifying space on nLab
BSO(n) on nLab

References

Category:Algebraic topology