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Definition

There is a canonical inclusion of real oriented Grassmannians given by Gr ~ n ( R k ) ↪ Gr ~ n ( R k + 1 ) , V ↦ V × { 0 } {\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

BSO ⁡ ( n ) := Gr ~ n ( R ∞ ) := lim k → ∞ Gr ~ n ( R k ) . {\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:

Gr ~ n ( R k ) = SO ⁡ ( n + k ) / ( SO ⁡ ( n ) × SO ⁡ ( k ) ) {\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 BSO ⁡ ( n ) {\displaystyle \operatorname {BSO} (n)} .

Simplest classifying spaces

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

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

Classification of principal bundles

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

[ X , BSO ⁡ ( n ) ] → Prin SO ⁡ ( n ) ⁡ ( X ) , [ f ] ↦ f ∗ ESO ⁡ ( n ) {\displaystyle [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 BSO ⁡ ( n ) {\displaystyle \operatorname {BSO} (n)} with coefficients in the field Z 2 {\displaystyle \mathbb {Z} _{2}} of two elements is generated by the Stiefel–Whitney classes:³⁴

H ∗ ( BSO ⁡ ( n ) ; Z 2 ) = Z 2 [ w 2 , … , w n ] . {\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 char = 2 {\displaystyle \operatorname {char} =2} .

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

H ∗ ( BSO ⁡ ( 2 n ) ; Q ) ≅ Q [ p 1 , … , p n , e ] / ( p n − e 2 ) , {\displaystyle H^{*}(\operatorname {BSO} (2n);\mathbb {Q} )\cong \mathbb {Q} [p_{1},\ldots ,p_{n},e]/(p_{n}-e^{2}),} H ∗ ( BSO ⁡ ( 2 n + 1 ) ; Q ) ≅ Q [ p 1 , … , p n ] . {\displaystyle H^{*}(\operatorname {BSO} (2n+1);\mathbb {Q} )\cong \mathbb {Q} [p_{1},\ldots ,p_{n}].}

Infinite classifying space

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

SO := lim n → ∞ SO ⁡ ( n ) ; {\displaystyle \operatorname {SO} :=\lim _{n\rightarrow \infty }\operatorname {SO} (n);} BSO := lim n → ∞ BSO ⁡ ( n ) . {\displaystyle \operatorname {BSO} :=\lim _{n\rightarrow \infty }\operatorname {BSO} (n).}

BSO {\displaystyle \operatorname {BSO} } is indeed the classifying space of SO {\displaystyle \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).{{cite book}}: CS1 maint: year (link)

External links

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

References

1. Milnor & Stasheff 74, section 12.2 The Oriented Universal Bundle on page 151 ↩

2. "universal principal bundle". nLab. Retrieved 2024-03-14. https://ncatlab.org/nlab/show/universal+principal+bundle ↩

3. Milnor & Stasheff, Theorem 12.4. ↩

4. Hatcher 02, Example 4D.6. ↩