Contents

Ingredients

Concepts

Constructions

Examples

Theorems

# Contents

## Idea

$B SO(n)$ is the classifying space for the special orthogonal group $SO(n)$.

## Definition

###### Definition

$B SO(n)$ is the colimit of the sequence of canonical inclusions of real orientable Grassmannians $\widetilde{Gr}_n(\mathbb{R}^k)\hookrightarrow\widetilde{Gr}_n(\mathbb{R}^{k+1})$:

$B SO(n) \coloneqq\underset{\longrightarrow}{\lim}_n\widetilde{Gr}_n(\mathbb{R}^k)$

As the real orientable Grassmannian can be written as a homogenous space by

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

the group structure carries over to $B SO(n)$.

## Smallest classifying spaces

Since $SO(1)\cong 1$ is the trivial group, the classifying space $B SO(1)$ is the trivial topological space. Since $SO(2)\cong U(1)$, once has

$B SO(2) \cong B U(1) \cong\mathbb{C}P^\infty.$

## Cohomology

###### Theorem

The cohomology ring of $B S O(n)$ with coefficients in the field $\mathbb{Z}_2$ is generated by the Stiefel-Whitney classes and given by

$H^*(B SO(n);\mathbb{Z}_2) \cong\mathbb{Z}_2[w_2,\ldots,w_n].$

This result holds more generally for every field with characteristic $\operatorname{char}=2$.

###### Theorem

The cohomology ring of $B SO(n)$ with coefficients in the field $\mathbb{Q}$ of rational numbers is generated by the Pontrjagin classes as well as the Euler class and given by

$H^*(B SO(2n);\mathbb{Q}) \cong\mathbb{Q}[p_1,\ldots,p_n,e]/(p_n-e^2)$
$H^*(B SO(2n+1);\mathbb{Q}) \cong\mathbb{Q}[p_1,\ldots,p_n]$

These results hold more generally for every field with characteristic $\operatorname{char}\neq 2$.

## Colimit

The canonical inclusions $SO(n)\hookrightarrow SO(n+1)$ yield canonical inclusions $B SO(n)\hookrightarrow B SO(n+1)$ of their respective classifying spaces. The colimit is denoted as

$B SO \coloneqq\underset{\longrightarrow}{\lim}_n B SO(n).$

and indeed the classifying space for $SO\coloneqq\underset{\longrightarrow}{\lim}_n SO(n)$.

## References

Created on March 14, 2024 at 15:50:24. See the history of this page for a list of all contributions to it.