Classifying space for U(n)

Construction as an infinite Grassmannian

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

Here, H is an infinite-dimensional complex Hilbert space, the e_i are vectors in H, and δ i j 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 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 BT.

The topological K-theory K₀(BT) is given by numerical polynomials; more details below.

Construction as an inductive limit

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 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 π p ( S 2 k − 1 ) is trivial and because of the long exact sequence of the fibration, we have

whenever p ≤ 2 k − 2 . By taking k big enough, precisely for k > 1 2 p + n − 1 , 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 π p ( E U ( n ) ) 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. ◻

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 c₁, ..., c_n where c_p is of degree 2p.

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 R [ c 1 ] / c 1 k + 1 , where c₁ 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

S 2 n − 1 → B U ( n − 1 ) → B U ( n )

Concretely, a point of the total space B U ( n − 1 ) is given by a point of the base space B U ( n ) classifying a complex vector space V , together with a unit vector u in V ; together they classify u ⊥ < V while the splitting V = ( C u ) ⊕ u ⊥ , trivialized by u , realizes the map B U ( n − 1 ) → B U ( n ) representing direct sum with C .

Applying the Gysin Sequence, one has a long exact sequence

H p ( B U ( n ) ) → ⌣ d 2 n η H p + 2 n ( B U ( n ) ) → j ∗ H p + 2 n ( B U ( n − 1 ) ) → ∂ H p + 1 ( B U ( n ) ) …

where η is the fundamental class of the fiber S 2 n − 1 . By properties of the Gysin Sequence, j ∗ is a multiplicative homomorphism; by induction, H ∗ B U ( n − 1 ) is generated by elements with p < − 1 , where ∂ must be zero, and hence where j ∗ must be surjective. It follows that j ∗ 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, ⌣ d 2 n η must always be injective. We therefore have short exact sequences split by a ring homomorphism

0 → H p ( B U ( n ) ) → ⌣ d 2 n η H p + 2 n ( B U ( n ) ) → j ∗ H p + 2 n ( B U ( n − 1 ) ) → 0

Thus we conclude H ∗ ( B U ( n ) ) = H ∗ ( B U ( n − 1 ) ) [ c 2 n ] where c 2 n = d 2 n η . This completes the induction.

K-theory of BU(n)

Let us consider topological complex K-theory as the cohomology theory represented by the spectrum K U . In this case, K U ∗ ( B U ( n ) ) ≅ Z [ t , t − 1 ] [ [ c 1 , . . . , c n ] ] , and K U ∗ ( B U ( n ) ) is the free Z [ t , t − 1 ] module on β 0 and β i 1 … β i r for n ≥ i j > 0 and r ≤ n . In this description, the product structure on K U ∗ ( B U ( n ) ) comes from the H-space structure of B U 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 K₀, 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 K ∗ ( X ) = π ∗ ( K ) ⊗ K 0 ( X ) , where π ∗ ( K ) = Z [ t , t − 1 ] , where t is the Bott generator.

K₀(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, K₀(BTⁿ) is numerical polynomials in n variables. The map K₀(BTⁿ) → K₀(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 k 1 , … , k n contains r distinct integers, repeated n 1 , … , n r times, respectively.