nLab BSU(n)

Contents

Contents

Idea

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

Definition

Definition

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

BSU(n)lim kGr˜ n( k) 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

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

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

Smallest classifying space

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

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

Cohomology

Theorem

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

H *(BSU(n);)[c 2,,c n]. H^*(B SU(n);\mathbb{Z}) \cong\mathbb{Z}[c_2,\ldots,c_n].

(Hatcher 02, Example 4D.7.)

Colimit

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

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

and indeed the classifying space for SUlim nSU(n)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.