Classifying Space For SO(n)

In mathematics, the classifying space $\operatorname(n)$ for th
special orthogonal group
'' $\operatorname(n)$ is the base space of the universal $\operatorname(n)$ principal bundle $\operatorname(n)\rightarrow\operatorname(n)$. This means that $\operatorname(n)$ principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into $\operatorname(n)$. The isomorphism is given by pullback.

Definition

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

: $\operatorname(n) :=\widetilde\operatorname_n(\mathbb^\infty) :=\lim_\widetilde\operatorname_n(\mathbb^k).$

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

: $\widetilde\operatorname_n(\mathbb^k) =\operatorname(n+k)/(\operatorname(n)\times\operatorname(k))$

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

Simplest classifying spaces

* Since $\operatorname(1) \cong 1$ is the trivial group, $\operatorname(1) \cong\$ is the trivial topological space.

* Since $\operatorname(2) \cong\operatorname(1)$, one has $\operatorname(2) \cong\operatorname(1) \cong\mathbbP^\infty$.

Classification of principal bundles

Given a topological space $X$ the set of $\operatorname(n)$ principal bundles on it up to isomorphism is denoted $\operatorname_(X)$. If $X$ is a CW complex, then the map:

: ,\operatorname(n)rightarrow\operatorname_(X),
mapsto f^*\operatorname(n)

is bijective.

Cohomology ring

The cohomology ring of $\operatorname(n)$ with coefficients in the field $\mathbb_2$ of two elements is generated by the Stiefel–Whitney classes:Milnor & Stasheff, Theorem 12.4.Hatcher 02, Example 4D.6.

: H^*(\operatorname(n);\mathbb_2)
=\mathbb_2 _2,\ldots,w_n

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

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

: H^*(\operatorname(2n);\mathbb)
\cong\mathbb _1,\ldots,p_n,e(p_n-e^2),
: H^*(\operatorname(2n+1);\mathbb)
\cong\mathbb _1,\ldots,p_n

Infinite classifying space

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

: $\operatorname :=\lim_\operatorname(n);$
: $\operatorname :=\lim_\operatorname(n).$

$\operatorname$ is indeed the classifying space of $\operatorname$.

See also

* Classifying space for O(n)
* Classifying space for U(n)
* Classifying space for SU(n)

Literature

*
*
* {{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

Algebraic topology