Classifying space for SO(n)

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

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

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

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

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

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

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

• Since ${\displaystyle \operatorname {SO} (2)\cong \operatorname {U} (1)}$, one has ${\displaystyle \operatorname {BSO} (2)\cong \operatorname {BU} (1)\cong \mathbb {C} P^{\infty }}$.

Given a topological space ${\displaystyle X}$ the set of ${\displaystyle \operatorname {SO} (n)}$ principal bundles on it up to isomorphism is denoted ${\displaystyle \operatorname {Prin} _{\operatorname {SO} (n)}(X)}$. If ${\displaystyle X}$ is a CW complex, then the map:²

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

is bijective.

The cohomology ring of ${\displaystyle \operatorname {BSO} (n)}$ with coefficients in the field ${\displaystyle \mathbb {Z} _{2}}$ of two elements is generated by the Stiefel–Whitney classes:³⁴⁵

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

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

The cohomology ring of ${\displaystyle \operatorname {BSO} (n)}$ with coefficients in the field ${\displaystyle \mathbb {Q} }$ of rational numbers is generated by Pontrjagin classes and Euler class:⁶

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

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

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

${\displaystyle \operatorname {SO} :=\lim _{n\rightarrow \infty }\operatorname {SO} (n);}$

${\displaystyle \operatorname {BSO} :=\lim _{n\rightarrow \infty }\operatorname {BSO} (n).}$

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

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• Lawson, H. Blaine; Michelsohn, Marie-Louise (1990-02-21). Spin Geometry. Princeton University Press. ISBN 9780691085425.
• 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).