Classifying space for SU(n)

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

Definition

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

$\operatorname {BSU} (n):={\widetilde {\operatorname {Gr} }}_{n}(\mathbb {C} ^{\infty }):=\lim _{n\rightarrow \infty }{\widetilde {\operatorname {Gr} }}_{n}(\mathbb {C} ^{k}).$

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

${\widetilde {\operatorname {Gr} }}_{n}(\mathbb {C} ^{k})=\operatorname {SU} (n+k)/(\operatorname {SU} (n)\times \operatorname {SU} (k))$

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

Simplest classifying spaces

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

• Since $\operatorname {SU} (2)\cong \operatorname {Sp} (1)$, one has $\operatorname {BSU} (2)\cong \operatorname {BSp} (1)\cong \mathbb {H} P^{\infty }$.

Classification of principal bundles

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

$[X,\operatorname {BSU} (n)]\rightarrow \operatorname {Prin} _{\operatorname {SU} (n)}(X),[f]\mapsto f^{*}\operatorname {ESU} (n)$

is bijective.

Cohomology ring

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

$H^{*}(\operatorname {BSU} (n);\mathbb {Z} )=\mathbb {Z} [c_{2},\ldots ,c_{n}].$

Infinite classifying space

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

$\operatorname {SU} :=\lim _{n\rightarrow \infty }\operatorname {SU} (n);$

$\operatorname {BSU} :=\lim _{n\rightarrow \infty }\operatorname {BSU} (n).$

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

See also

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

Literature

• Hatcher, Allen (2002). Algebraic topology. Cambridge: Cambridge University Press. ISBN 0-521-79160-X.
• Mitchell, Stephen (August 2001). Universal principal bundles and classifying spaces (PDF).

External links

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