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.
The total space EU(n) of the universal bundle is given by Here, H denotes an infinite-dimensional complex Hilbert space, the ei are vectors in H, and
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.
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.
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.
The total space of the universal bundle can be taken to be the direct limit of the Fn(Ck) as k → ∞, while the base space is the direct limit of the Gn(Ck) as k → ∞.
is trivial and because of the long exact sequence of the fibration, we have whenever
, we can repeat the process and get This last group is trivial for k > n + p. Let be the direct limit of all the Fn(Ck) (with the induced topology).
Let be the direct limit of all the Gn(Ck) (with the induced topology).
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).
Gn(Ck), is induced by restriction of the one for Fn(Ck+1), resp.
By Whitehead Theorem and the above Lemma, EU(n) is contractible.
of integers is generated by the Chern classes:[1] 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∞.
, where c1 is the Euler class of the U(1)-bundle S2k+1 → CPk, and that the injections CPk → CPk+1, for k ∈ N*, are compatible with these presentations of the cohomology of the projective spaces.
There are homotopy fiber sequences Concretely, a point of the total space
classifying a complex vector space
By properties of the Gysin Sequence[citation needed],
must always be surjective: by the universal property of polynomial rings, a choice of preimage for each generator induces a multiplicative splitting.
We therefore have short exact sequences split by a ring homomorphism Thus we conclude
Consider topological complex K-theory as the cohomology theory represented by the spectrum
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.
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