higher geometry / derived geometry
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$B SU(n)$ is the classifying space for the special unitary group $SU(n)$.
$B SU(n)$ is the limit of the sequence of canonical inclusions of complex orientable Grassmannians $\widetilde{Gr}_n(\mathbb{C}^k)\hookrightarrow\widetilde{Gr}_n(\mathbb{C}^{k+1})$:
the group structure carries over to $B SU(n)$.
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
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
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
and indeed the classifying space for $S U\coloneqq\underset{\longrightarrow}{\lim}_n SU(n)$.
Created on March 14, 2024 at 15:50:56. See the history of this page for a list of all contributions to it.