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
Here, H is an infinite-dimensional complex Hilbert space, the ei are vectors in H, and 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.
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.
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 Gn(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 Fn(Ck) and the quotient is the Grassmannian Gn(Ck). The map
is a fibre bundle of fibre Fn−1(Ck−1). Thus because is trivial and because of the long exact sequence of the fibration, we have
whenever . By taking k big enough, precisely for , 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).
Lemma: The group is trivial for all p ≥ 1.
Proof: Let γ : Sp → EU(n), since Sp is compact, there exists k such that γ(Sp) is included in Fn(Ck). 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 Fn(Ck) and Gn(Ck) are CW-complexes. One can find a decomposition of these spaces into CW-complexes such that the decomposition of Fn(Ck), resp. Gn(Ck), is induced by restriction of the one for Fn(Ck+1), resp. Gn(Ck+1). Thus EU(n) (and also Gn(C∞)) is a CW-complex. By Whitehead Theorem and the above Lemma, EU(n) is contractible.
Cohomology of BU(n)
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 CPk is isomorphic to , 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. This proves the Proposition for n = 1.
There are homotopy fiber sequences
Concretely, a point of the total space is given by a point of the base space classifying a complex vector space , together with a unit vector in ; together they classify while the splitting , trivialized by , realizes the map representing direct sum with
Applying the Gysin Sequence, one has a long exact sequence
where is the fundamental class of the fiber . By properties of the Gysin Sequence, is a multiplicative homomorphism; by induction, is generated by elements with , where must be zero, and hence where must be surjective. It follows that 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, must always be injective. We therefore have short exact sequences split by a ring homomorphism
Thus we conclude where . This completes the induction.
K-theory of BU(n)
Let us consider topological complex K-theory as the cohomology theory represented by the spectrum . In this case, , and is the free module on and for and . In this description, the product structure on comes from the H-space structure of given by Whitney sum of vector bundles. This product is called the Pontryagin product.
The following seems to be a computation of , where gets a ring structure from the tensor product H-space structure on . The statement needs clarification.
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 , where , 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.
and the image can be identified as the symmetric polynomials satisfying the integrality condition that
is the multinomial coefficient and contains r distinct integers, repeated times, respectively.
- R. Bott, L. W. Tu-- Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer
- Adams 1974, p. 49
- Adams 1974, p. 47
- J. F. Adams (1974), Stable Homotopy and Generalised Homology, University Of Chicago Press, ISBN 0-226-00524-0 Contains calculation of and .
- S. Ochanine; L. Schwartz (1985), "Une remarque sur les générateurs du cobordisme complex", Math. Z., 190 (4): 543–557, doi:10.1007/BF01214753 Contains a description of as a -comodule for any compact, connected Lie group.
- L. Schwartz (1983), "K-théorie et homotopie stable", Thesis, Université de Paris–VII Explicit description of
- A. Baker; F. Clarke; N. Ray; L. Schwartz (1989), "On the Kummer congruences and the stable homotopy of BU", Trans. Amer. Math. Soc., American Mathematical Society, 316 (2): 385–432, doi:10.2307/2001355, JSTOR 2001355