nLab BU(n)

Idea

$B U(n)$ is the classifying space for (principal bundles with structure group) the unitary group $U(n)$.

Definition

Definition

$B U(n)$ is the limit of the sequence of canonical inclusions of complex Grassmannians $Gr_n(\mathbb{C}^k)\hookrightarrow Gr_n(\mathbb{C}^{k+1})$:

$$B U(n) \coloneqq\underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{C}^k)$$

(Milnor & Stasheff 74, page 169)

As the complex Grassmannian can be written as a homogenous space by

$$Gr_n(\mathbb{C}^k) =U(k)/(U(n)\times U(k-n))$$

the group structure carries over to $B U(n)$.

Smallest classifying space

The smallest classifying space $B U(1)$ is the infinite complex projective space $\mathbb{C}P^\infty$. It is also a Eilenberg–MacLane space $K(\mathbb{Z},2)$, hence $U(1)$ principal bundles over a space are classified by its second cohomology in the ring $\mathbb{Z}$ of integers:

$$Prin_{U(1)}(X) \cong [X,B U(1)] =[X,\mathbb{C}P^\infty] =[X,K(\mathbb{Z},2)] \cong H^2(X;\mathbb{Z}).$$

Higher classifying spaces are not necessarily Eilenberg-MacLane spaces, but continuous maps into them are possible, giving the idea behind characteristic classes. But they are defined for vector bundles, which can be constructed from principal bundles using the balanced product? $Prin_{U(n)}(X)\rightarrow Vect_\mathbb{C}^n(X),E\mapsto E\times_{U(n)}\mathbb{C}^n$.

Cohomology

Theorem

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

$$H^*(B U(n);\mathbb{Z}) \cong\mathbb{Z}[c_1,\ldots,c_n].$$

(Hatcher 02, Theorem 4D.4.)

This theorem also holds if $\mathbb{Z}$ is replaced by a general ring $R$.

(Hatcher 02, page 323)

Colimit

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

$$B U \coloneqq\underset{\longrightarrow}{\lim}_n B U(n) \,,$$

which is the classifying space of the stable unitary group $U\coloneqq\underset{\longrightarrow}{\lim}_n U(n)$.

Related concepts

• BO(n)

• BSO(n)

• BSU(n)

• KU

References

• John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press (1974) (ISBN:9780691081229, doi:10.1515/9781400881826, pdf)

• Allen Hatcher, Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]

See also

• Wikipedia, Classifying space for U(n)