# Classifying space for U(n)

# 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 Grassmannian

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*, andis theKronecker delta. The symbolis theinner producton*H*. Thus, we have that EU(*n*) is the space oforthonormal*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 bundles

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 limit

Let *Fn*(**C***k*) be the space of orthonormal families of *n* vectors in **C***k* and let *Gn*(**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 *Fn*(**C***k*) as *k* → ∞, while the base space is the direct limit of the *G**n*(**C***k*) as *k* → ∞.

Validity of the construction

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 becauseis trivial and because of thelong exact sequence of the fibration, we have`whenever. By taking`

*k*big enough, precisely for, we can repeat the process and getThis 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*iscompact, 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*)

*n*)

Proposition:The cohomology of the classifying spaceH*(BU(n)) is a ring of polynomials innvariablesc1, ...,cnwherecpis of degree 2p.

**Proof:**Let us first consider the case*n*= 1. In this case, U(1) is the circle**S**1and the universal bundle is**S**∞→**CP**∞. It is well known^{[1]}that the cohomology of**CP***k*is isomorphic to, where*c*1is theEuler classof the U(1)-bundle**S**2*k*+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 spaceis given by a point of the base spaceclassifying a complex vector space, together with a unit vectorin; together they classifywhile the splitting, trivialized by, realizes the maprepresenting direct sum with`

Applying the Gysin sequence, one has a long exact sequence

`whereis thefundamental classof the fiber. By properties of the Gysin Sequence,is a multiplicative homomorphism; by induction,is generated by elements with, wheremust be zero, and hence wheremust be surjective. It follows thatmust`

**always**be surjective: by theuniversal propertyofpolynomial rings, a choice of preimage for each generator induces a multiplicative splitting. Hence, by exactness,must always be**injective**. We therefore haveshort exact sequencessplit by a ring homomorphism`Thus we concludewhere. This completes the induction.`

K-theory of BU(*n*)

*n*)

`Consider topological complex K-theory as the cohomology theory represented by the spectrum. In this case,,`

^{[2]}andis the freemodule onandforand.^{[3]}In this description, the product structure oncomes from the H-space structure ofgiven by Whitney sum of vector bundles. This product is called thePontryagin product.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*Tn*) is numerical polynomials in *n* variables. The map *K*0(B*Tn*) → *K*0(BU(*n*)) is onto, via a splitting principle, as *Tn* is the maximal torus of U(*n*). The map is the symmetrization map

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

where

`is themultinomial coefficientandcontains`

*r*distinct integers, repeatedtimes, respectively.See also

Classifying space for O(

*n*)Topological K-theory

Atiyah–Jänich theorem

## References

*Differential Forms in Algebraic Topology*, Graduate Texts in Mathematics 82, Springer