Contents

Ingredients

Concepts

Constructions

Examples

Theorems

Contents

Idea

$B SU(n)$ is the classifying space for the special unitary group $SU(n)$.

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

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

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

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

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

References

Created on March 14, 2024 at 15:50:56. See the history of this page for a list of all contributions to it.