nLab BU(n)




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



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

BU(n)lim kGr n( k) 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( k)=U(k)/(U(n)×U(kn)) Gr_n(\mathbb{C}^k) =U(k)/(U(n)\times U(k-n))

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

Smallest classifying space

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

Prin U(1)(X)[X,BU(1)]=[X,P ]=[X,K(,2)]H 2(X;). 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)Vect n(X),EE× U(n) nPrin_{U(n)}(X)\rightarrow Vect_\mathbb{C}^n(X),E\mapsto E\times_{U(n)}\mathbb{C}^n.



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

H *(BU(n);)[c 1,,c n]. 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 RR.

(Hatcher 02, page 323)


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

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

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


See also

Last revised on March 12, 2024 at 08:40:43. See the history of this page for a list of all contributions to it.