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
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", and formula_2 is the Kronecker delta. The symbol formula_3 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 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 because formula_8 is trivial and because of the long exact sequence of the fibration, we have
whenever formula_10. By taking "k" big enough, precisely for formula_11, 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 formula_15 is trivial for all "p" ≥ 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.formula_16
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").
 Proposition: The cohomology of the classifying space "H*"(BU("n")) is a ring of polynomials in "n" variables
"c"1, ..., "cn" where "cp" is of degree 2"p".
Proof: Let us first consider the case "n" = 1. In this case, U(1) is the circle S1 and the universal bundle is S∞ → CP∞. It is well known that the cohomology of CP"k" is isomorphic to formula_17, where "c"1 is the Euler class of the U(1)-bundle S2"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 space formula_19 is given by a point of the base space formula_20 classifying a complex vector space formula_21, together with a unit vector formula_22 in formula_21; together they classify formula_24 while the splitting formula_25, trivialized by formula_22, realizes the map formula_27 representing direct sum with formula_28
Applying the Gysin sequence, one has a long exact sequence
where formula_30 is the fundamental class of the fiber formula_31. By properties of the Gysin Sequence, formula_32 is a multiplicative homomorphism; by induction, formula_33 is generated by elements with formula_34, where formula_35 must be zero, and hence where formula_32 must be surjective. It follows that formula_32 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, formula_38 must always be injective. We therefore have short exact sequences split by a ring homomorphism
Thus we conclude formula_40 where formula_41. This completes the induction.
K-theory of BU("n").
Consider topological complex K-theory as the cohomology theory represented by the spectrum formula_42. In this case, formula_43, and formula_44 is the free formula_45 module on formula_46 and formula_47 for formula_48 and formula_49. In this description, the product structure on formula_50 comes from the H-space structure of formula_51 given by Whitney sum of vector bundles. This product is called the Pontryagin 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 formula_52, where formula_53, 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 the multinomial coefficient and formula_57 contains "r" distinct integers, repeated formula_58 times, respectively.