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

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

[ 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:

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

Literature

External links

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