Classifying space for SU(n)

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

Definition

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

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

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

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

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

Simplest classifying spaces

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

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

Classification of principal bundles

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

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

is bijective.

Cohomology ring

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

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

Infinite classifying space

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

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

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

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

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).

External links

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

References

1. "universal principal bundle". nLab. Retrieved 2024-03-14.
2. Hatcher 02, Example 4D.7.