Classifying space for SO(n)

In mathematics, the classifying space $\mathrm{BSO}(n)$ for the special orthogonal group $\mathrm{SO}(n)$ is the base space of the universal $\mathrm{SO}(n)$ principal bundle $\mathrm{ESO}(n)\to \mathrm{BSO}(n)$. This means that $\mathrm{SO}(n)$ principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into $\mathrm{BSO}(n)$. The isomorphism is given by pullback. A particular application are principal SO(2)-bundles.

Definition

There is a canonical inclusion of real oriented Grassmannians given by ${\widetilde{\mathrm{Gr}}}_n({ℝ}^k)\hookrightarrow {\widetilde{\mathrm{Gr}}}_n({ℝ}^{k+1}),V\mapsto V\times \{0\}$. Its colimit is:

$\mathrm{BSO}(n):={\widetilde{\mathrm{Gr}}}_n({ℝ}^{\mathrm{\infty }}):=\underset{k\to \mathrm{\infty }}{lim}{\widetilde{\mathrm{Gr}}}_n({ℝ}^k).$

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

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

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

Simplest classifying spaces

• Since $\mathrm{SO}(1)\cong 1$ is the trivial group, $\mathrm{BSO}(1)\cong \{\ast \}$ is the trivial topological space.

• Since $\mathrm{SO}(2)\cong \mathrm{U}(1)$, one has $\mathrm{BSO}(2)\cong \mathrm{BU}(1)\cong ℂP^{\mathrm{\infty }}$.

Classification of principal bundles

Given a topological space $X$ the set of $\mathrm{SO}(n)$ principal bundles on it up to isomorphism is denoted ${\mathrm{Prin}}_{\mathrm{SO}(n)}(X)$. If $X$ is a CW complex, then the map:

$[X,\mathrm{BSO}(n)]\to {\mathrm{Prin}}_{\mathrm{SO}(n)}(X),[f]\mapsto f^{\ast }\mathrm{ESO}(n)$

is bijective.

Cohomology ring

The cohomology ring of $\mathrm{BSO}(n)$ with coefficients in the field ${\mathbb{Z}}_2$ of two elements is generated by the Stiefel–Whitney classes:

$H^{\ast }(\mathrm{BSO}(n);{\mathbb{Z}}_2)={\mathbb{Z}}_2[w_2,\dots ,w_n].$

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

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

$H^{\ast }(\mathrm{BSO}(2n);\mathbb{Q})\cong \mathbb{Q}[p_1,\dots ,p_n,e]/(p_n-e^2),$

$H^{\ast }(\mathrm{BSO}(2n+1);\mathbb{Q})\cong \mathbb{Q}[p_1,\dots ,p_n].$

Infinite classifying space

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

$\mathrm{SO}:=\underset{n\to \mathrm{\infty }}{lim}\mathrm{SO}(n);$

$\mathrm{BSO}:=\underset{n\to \mathrm{\infty }}{lim}\mathrm{BSO}(n).$

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

See also

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

Literature

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External links

• classifying space on nLab
• BSO(n) on nLab