nLab BSU(n)




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



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.



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.)


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).


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