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

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

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:


 * $$[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:


 * $$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:


 * $$\operatorname{SU}
 * =\lim_{n\rightarrow\infty}\operatorname{SU}(n);$$
 * $$\operatorname{BSU}
 * =\lim_{n\rightarrow\infty}\operatorname{BSU}(n).$$

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