Classifying space for SU(n)

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

Definition

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

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

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

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

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

Simplest classifying spaces

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

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

Classification of principal bundles

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

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

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

Infinite classifying space

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