# Classifying space for U(*n*)

In mathematics, the **classifying space for the unitary group** U(*n*) is a space BU(*n*) together with a universal bundle EU(*n*) such that any hermitian bundle on a paracompact space *X* is the pull-back of EU(*n*) by a map *X* → BU(*n*) unique up to homotopy.

This space with its universal fibration may be constructed as either

- the Grassmannian of
*n*-planes in an infinite-dimensional complex Hilbert space; or, - the direct limit, with the induced topology, of Grassmannians of
*n*planes.

Both constructions are detailed here.

## Construction as an infinite GrassmannianEdit

The total space EU(*n*) of the universal bundle is given by

Here, *H* denotes an infinite-dimensional complex Hilbert space, the *e*_{i} are vectors in *H*, and is the Kronecker delta. The symbol is the inner product on *H*. Thus, we have that EU(*n*) is the space of orthonormal *n*-frames in *H*.

The group action of U(*n*) on this space is the natural one. The base space is then

and is the set of Grassmannian *n*-dimensional subspaces (or *n*-planes) in *H*. That is,

so that *V* is an *n*-dimensional vector space.

### Case of line bundlesEdit

For *n* = 1, one has EU(1) = **S**^{∞}, which is known to be a contractible space. The base space is then BU(1) = **CP**^{∞}, the infinite-dimensional complex projective space. Thus, the set of isomorphism classes of circle bundles over a manifold *M* are in one-to-one correspondence with the homotopy classes of maps from *M* to **CP**^{∞}.

One also has the relation that

that is, BU(1) is the infinite-dimensional projective unitary group. See that article for additional discussion and properties.

For a torus *T*, which is abstractly isomorphic to U(1) × ... × U(1), but need not have a chosen identification, one writes B*T*.

The topological K-theory *K*_{0}(B*T*) is given by numerical polynomials; more details below.

## Construction as an inductive limitEdit

Let *F _{n}*(

**C**

^{k}) be the space of orthonormal families of

*n*vectors in

**C**

^{k}and let

*G*(

_{n}**C**

^{k}) be the Grassmannian of

*n*-dimensional subvector spaces of

**C**

^{k}. The total space of the universal bundle can be taken to be the direct limit of the

*F*(

_{n}**C**

^{k}) as

*k*→ ∞, while the base space is the direct limit of the

*G*

_{n}(

**C**

^{k}) as

*k*→ ∞.

### Validity of the constructionEdit

In this section, we will define the topology on EU(*n*) and prove that EU(*n*) is indeed contractible.

The group U(*n*) acts freely on *F*_{n}(**C**^{k}) and the quotient is the Grassmannian *G*_{n}(**C**^{k}). The map

is a fibre bundle of fibre *F*_{n−1}(**C**^{k−1}). Thus because is trivial and because of the long exact sequence of the fibration, we have

whenever . By taking *k* big enough, precisely for , we can repeat the process and get

This last group is trivial for *k* > *n* + *p*. Let

be the direct limit of all the *F*_{n}(**C**^{k}) (with the induced topology). Let

be the direct limit of all the *G*_{n}(**C**^{k}) (with the induced topology).

Lemma:The group is trivial for allp≥ 1.

**Proof:** Let γ : **S**^{p} → EU(*n*), since **S**^{p} is compact, there exists *k* such that γ(**S**^{p}) is included in *F*_{n}(**C**^{k}). By taking *k* big enough, we see that γ is homotopic, with respect to the base point, to the constant map.

In addition, U(*n*) acts freely on EU(*n*). The spaces *F*_{n}(**C**^{k}) and *G*_{n}(**C**^{k}) are CW-complexes. One can find a decomposition of these spaces into CW-complexes such that the decomposition of *F*_{n}(**C**^{k}), resp. *G*_{n}(**C**^{k}), is induced by restriction of the one for *F*_{n}(**C**^{k+1}), resp. *G*_{n}(**C**^{k+1}). Thus EU(*n*) (and also *G*_{n}(**C**^{∞})) is a CW-complex. By Whitehead Theorem and the above Lemma, EU(*n*) is contractible.

## Cohomology of BU(*n*)Edit

Proposition:The cohomology of the classifying spaceH*(BU(n)) is a ring of polynomials innvariablesc_{1}, ...,cwhere_{n}cis of degree 2_{p}p.

**Proof:** Let us first consider the case *n* = 1. In this case, U(1) is the circle **S**^{1} and the universal bundle is **S**^{∞} → **CP**^{∞}. It is well known^{[1]} that the cohomology of **CP**^{k} is isomorphic to , where *c*_{1} is the Euler class of the U(1)-bundle **S**^{2k+1} → **CP**^{k}, and that the injections **CP**^{k} → **CP**^{k+1}, for *k* ∈ **N***, are compatible with these presentations of the cohomology of the projective spaces. This proves the Proposition for *n* = 1.

There are homotopy fiber sequences

Concretely, a point of the total space is given by a point of the base space classifying a complex vector space , together with a unit vector in ; together they classify while the splitting , trivialized by , realizes the map representing direct sum with

Applying the Gysin sequence, one has a long exact sequence

where is the fundamental class of the fiber . By properties of the Gysin Sequence^{[citation needed]}, is a multiplicative homomorphism; by induction, is generated by elements with , where must be zero, and hence where must be surjective. It follows that must **always** be surjective: by the universal property of polynomial rings, a choice of preimage for each generator induces a multiplicative splitting. Hence, by exactness, must always be **injective**. We therefore have short exact sequences split by a ring homomorphism

Thus we conclude where . This completes the induction.

## K-theory of BU(*n*)Edit

Consider topological complex K-theory as the cohomology theory represented by the spectrum . In this case, ,^{[2]} and is the free module on and for and .^{[3]} In this description, the product structure on comes from the H-space structure of given by Whitney sum of vector bundles. This product is called the Pontryagin product.

The following seems to be a computation of , where gets a ring structure from the tensor product H-space structure on . The statement needs clarification. |

The topological K-theory is known explicitly in terms of numerical symmetric polynomials.

The K-theory reduces to computing *K*_{0}, since K-theory is 2-periodic by the Bott periodicity theorem, and BU(*n*) is a limit of complex manifolds, so it has a CW-structure with only cells in even dimensions, so odd K-theory vanishes.

Thus , where , where *t* is the Bott generator.

*K*_{0}(BU(1)) is the ring of numerical polynomials in *w*, regarded as a subring of *H*_{∗}(BU(1); **Q**) = **Q**[*w*], where *w* is element dual to tautological bundle.

For the *n*-torus, *K*_{0}(B*T ^{n}*) is numerical polynomials in

*n*variables. The map

*K*

_{0}(B

*T*) →

^{n}*K*

_{0}(BU(

*n*)) is onto, via a splitting principle, as

*T*is the maximal torus of U(

^{n}*n*). The map is the symmetrization map

and the image can be identified as the symmetric polynomials satisfying the integrality condition that

where

is the multinomial coefficient and contains *r* distinct integers, repeated times, respectively.

## See alsoEdit

## NotesEdit

## ReferencesEdit

- J. F. Adams (1974),
*Stable Homotopy and Generalised Homology*, University Of Chicago Press, ISBN 0-226-00524-0 Contains calculation of and . - S. Ochanine; L. Schwartz (1985), "Une remarque sur les générateurs du cobordisme complex",
*Math. Z.*,**190**(4): 543–557, doi:10.1007/BF01214753 Contains a description of as a -comodule for any compact, connected Lie group. - L. Schwartz (1983), "K-théorie et homotopie stable",
*Thesis*, Université de Paris–VII Explicit description of - A. Baker; F. Clarke; N. Ray; L. Schwartz (1989), "On the Kummer congruences and the stable homotopy of
*BU*",*Trans. Amer. Math. Soc.*, American Mathematical Society,**316**(2): 385–432, doi:10.2307/2001355, JSTOR 2001355