Classifying space for SO(n)

In mathematics, the classifying space $\operatorname {BSO} (n)$ for the 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:

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

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:

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

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:

$H^{*}(\operatorname {BSO} (n);\mathbb {Z} _{2})=\mathbb {Z} _{2}[w_{2},\ldots ,w_{n}].$

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:

$H^{*}(\operatorname {BSO} (2n);\mathbb {Q} )\cong \mathbb {Q} [p_{1},\ldots ,p_{n},e]/(p_{n}-e^{2}),$

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

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

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 $\operatorname {SO}$.

See also

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

Literature

• Milnor, John; Stasheff, James (1974). Characteristic classes (PDF). Princeton University Press. doi:10.1515/9781400881826. ISBN 9780691081229.
• Hatcher, Allen (2002). Algebraic topology. Cambridge: Cambridge University Press. ISBN 0-521-79160-X.
• Mitchell, Stephen (August 2001). Universal principal bundles and classifying spaces (PDF).

External links

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