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
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 Fn(Ck) be the space of orthonormal families of n vectors in Ck and let Gn(Ck) be the Grassmannian of n-dimensional subvector spaces of Ck. 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 G**n(Ck) 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(Ck) and the quotient is the Grassmannian G**n(Ck). The map
This last group is trivial for k > n + p. Let
be the direct limit of all the F**n(Ck) (with the induced topology). Let
be the direct limit of all the G**n(Ck) (with the induced topology).
Lemma: The group is trivial for all p ≥ 1.
In addition, U(n) acts freely on EU(n). The spaces F**n(Ck) and G**n(Ck) are CW-complexes. One can find a decomposition of these spaces into CW-complexes such that the decomposition of F**n(Ck), resp. G**n(Ck), is induced by restriction of the one for F**n(Ck+1), resp. G**n(Ck+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, ..., cn where cp is of degree 2p.
There are homotopy fiber sequences
Applying the Gysin sequence, one has a long exact sequence
K-theory of BU(n)
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
See also
Classifying space for O(n)
Topological K-theory
Atiyah–Jänich theorem