Classifying space for SU(n)

In mathematics, the classifying space $\mathrm{BSU}(n)$ for the special unitary group $\mathrm{SU}(n)$ is the base space of the universal $\mathrm{SU}(n)$ principal bundle $\mathrm{ESU}(n)\to \mathrm{BSU}(n)$. This means that $\mathrm{SU}(n)$ principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into $\mathrm{BSU}(n)$. The isomorphism is given by pullback. A particular application are principal SU(2)-bundles.

Definition

There is a canonical inclusion of complex oriented Grassmannians given by ${\widetilde{\mathrm{Gr}}}_n({ℂ}^k)\hookrightarrow {\widetilde{\mathrm{Gr}}}_n({ℂ}^{k+1}),V\mapsto V\times \{0\}$. Its colimit is:

$\mathrm{BSU}(n):={\widetilde{\mathrm{Gr}}}_n({ℂ}^{\mathrm{\infty }}):=\underset{k\to \mathrm{\infty }}{lim}{\widetilde{\mathrm{Gr}}}_n({ℂ}^k).$

Since real oriented Grassmannians can be expressed as a homogeneous space by:

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

the group structure carries over to $\mathrm{BSU}(n)$.

Simplest classifying spaces

• Since $\mathrm{SU}(1)\cong 1$ is the trivial group, $\mathrm{BSU}(1)\cong \{\ast \}$ is the trivial topological space.

• Since $\mathrm{SU}(2)\cong \mathrm{Sp}(1)$, one has $\mathrm{BSU}(2)\cong \mathrm{BSp}(1)\cong ℍP^{\mathrm{\infty }}$.

Classification of principal bundles

Given a topological space $X$ the set of $\mathrm{SU}(n)$ principal bundles on it up to isomorphism is denoted ${\mathrm{Prin}}_{\mathrm{SU}(n)}(X)$. If $X$ is a CW complex, then the map:

$[X,\mathrm{BSU}(n)]\to {\mathrm{Prin}}_{\mathrm{SU}(n)}(X),[f]\mapsto f^{\ast }\mathrm{ESU}(n)$

is bijective.

Cohomology ring

The cohomology ring of $\mathrm{BSU}(n)$ with coefficients in the ring $\mathbb{Z}$ of integers is generated by the Chern classes:

$H^{\ast }(\mathrm{BSU}(n);\mathbb{Z})=\mathbb{Z}[c_2,\dots ,c_n].$

Infinite classifying space

The canonical inclusions $\mathrm{SU}(n)\hookrightarrow \mathrm{SU}(n+1)$ induce canonical inclusions $\mathrm{BSU}(n)\hookrightarrow \mathrm{BSU}(n+1)$ on their respective classifying spaces. Their respective colimits are denoted as:

$\mathrm{SU}:=\underset{n\to \mathrm{\infty }}{lim}\mathrm{SU}(n);$

$\mathrm{BSU}:=\underset{n\to \mathrm{\infty }}{lim}\mathrm{BSU}(n).$

$\mathrm{BSU}$ is indeed the classifying space of $\mathrm{SU}$.

See also

• Classifying space for O(n)
• Classifying space for SO(n)
• Classifying space for U(n)

Literature

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External links

• classifying space on nLab
• BSU(n) on nLab