nLab BSU(n)

Idea

$B SU(n)$ is the classifying space for (principal bundles with structure group) the special unitary group $SU(n)$. Important examples are principal SU(2)-bundles, principal SU(3)-bundles and principal SU(4)-bundles.

Definition

Definition

$B SU(n)$ is the limit of the sequence of canonical inclusions of complex orientable Grassmannians $\widetilde{Gr}_n(\mathbb{C}^k)\hookrightarrow\widetilde{Gr}_n(\mathbb{C}^{k+1})$:

$$B SU(n) \coloneqq\underset{\longrightarrow}{\lim}_k\widetilde{Gr}_n(\mathbb{C}^k)$$

As the complex orientable Grassmannian can be written as a homogenous space by

$$\widetilde{Gr}_n(\mathbb{C}^k) =SU(k)/(SU(n)\times SU(k-n))$$

the group structure carries over to $B SU(n)$.

Smallest classifying space

Since $SU(1)\cong 1$ is the trivial group, the classifying space $B SU(1)$ is the trivial topological space. Since $SU(2)\cong Sp(1)$, once has

$$B SU(2) \cong B Sp(1) \cong\mathbb{H}P^\infty.$$

This result is important for the classification of principal SU(2)-bundles.

Cohomology

Theorem

The cohomology ring of $B SU(n)$ with coefficients in the ring $\mathbb{Z}$ of integers is generated by the Chern classes and given by

$$H^*(B SU(n);\mathbb{Z}) \cong\mathbb{Z}[c_2,\ldots,c_n].$$

(Hatcher 02, Example 4D.7.)

Colimit

The canonical inclusions $S U(n)\hookrightarrow S U(n+1)$ yield canonical inclusions $B S U(n)\hookrightarrow B S U(n+1)$ of their respective classifying spaces. The colimit is denoted as

$$B SU \coloneqq\underset{\longrightarrow}{\lim}_n B SU(n).$$

and indeed the classifying space for $S U\coloneqq\underset{\longrightarrow}{\lim}_n SU(n)$.

Related concepts

• BO(n)

• BU(n)

• BSO(n)

References

• Allen Hatcher, Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]