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Definition

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

BSU ⁡ ( n ) := Gr ~ n ( C ∞ ) := lim n → ∞ Gr ~ n ( C k ) . {\displaystyle \operatorname {BSU} (n):={\widetilde {\operatorname {Gr} }}_{n}(\mathbb {C} ^{\infty }):=\lim _{n\rightarrow \infty }{\widetilde {\operatorname {Gr} }}_{n}(\mathbb {C} ^{k}).}

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

Gr ~ n ( C k ) = SU ⁡ ( n + k ) / ( SU ⁡ ( n ) × SU ⁡ ( k ) ) {\displaystyle {\widetilde {\operatorname {Gr} }}_{n}(\mathbb {C} ^{k})=\operatorname {SU} (n+k)/(\operatorname {SU} (n)\times \operatorname {SU} (k))}

the group structure carries over to BSU ⁡ ( n ) {\displaystyle \operatorname {BSU} (n)} .

Simplest classifying spaces

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

• Since SU ⁡ ( 2 ) ≅ Sp ⁡ ( 1 ) {\displaystyle \operatorname {SU} (2)\cong \operatorname {Sp} (1)} , one has BSU ⁡ ( 2 ) ≅ BSp ⁡ ( 1 ) ≅ H P ∞ {\displaystyle \operatorname {BSU} (2)\cong \operatorname {BSp} (1)\cong \mathbb {H} P^{\infty }} .

Classification of principal bundles

Given a topological space X {\displaystyle X} the set of SU ⁡ ( n ) {\displaystyle \operatorname {SU} (n)} principal bundles on it up to isomorphism is denoted Prin SU ⁡ ( n ) ⁡ ( X ) {\displaystyle \operatorname {Prin} _{\operatorname {SU} (n)}(X)} . If X {\displaystyle X} is a CW complex, then the map:¹

[ X , BSU ⁡ ( n ) ] → Prin SU ⁡ ( n ) ⁡ ( X ) , [ f ] ↦ f ∗ ESU ⁡ ( n ) {\displaystyle [X,\operatorname {BSU} (n)]\rightarrow \operatorname {Prin} _{\operatorname {SU} (n)}(X),[f]\mapsto f^{*}\operatorname {ESU} (n)}

is bijective.

Cohomology ring

The cohomology ring of BSU ⁡ ( n ) {\displaystyle \operatorname {BSU} (n)} with coefficients in the ring Z {\displaystyle \mathbb {Z} } of integers is generated by the Chern classes:²

H ∗ ( BSU ⁡ ( n ) ; Z ) = Z [ c 2 , … , c n ] . {\displaystyle H^{*}(\operatorname {BSU} (n);\mathbb {Z} )=\mathbb {Z} [c_{2},\ldots ,c_{n}].}

Infinite classifying space

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

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

BSU {\displaystyle \operatorname {BSU} } is indeed the classifying space of SU {\displaystyle \operatorname {SU} } .

See also

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

Literature

• 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
• BSU(n) on nLab

References

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

2. Hatcher 02, Example 4D.7. ↩