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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

  1. the Grassmannian of n-planes in an infinite-dimensional complex Hilbert space; or,

  2. 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

Here, H denotes an infinite-dimensional complex Hilbert space, the eiare vectors in H, andis theKronecker delta. The symbolis theinner producton H. Thus, we have that EU(n) is the space oforthonormaln-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 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

is a fibre bundle of fibre Fn−1(Ck−1). Thus becauseis trivial and because of thelong 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 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.

Proof: Let γ : Sp→ EU(n), since Spiscompact, 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 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.

Proof: Let us first consider the case n = 1. In this case, U(1) is the circle S1and the universal bundle is S∞→ CP∞. It is well known[1] that the cohomology of CPkis isomorphic to, where c1is theEuler classof 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 spaceis given by a point of the base spaceclassifying a complex vector space, together with a unit vectorin; together they classifywhile the splitting, trivialized by, realizes the maprepresenting direct sum with

Applying the Gysin sequence, one has a long exact sequence

whereis thefundamental classof the fiber. By properties of the Gysin Sequence,is a multiplicative homomorphism; by induction,is generated by elements with, wheremust be zero, and hence wheremust be surjective. It follows thatmust always be surjective: by theuniversal propertyofpolynomial rings, a choice of preimage for each generator induces a multiplicative splitting. Hence, by exactness,must always be injective. We therefore haveshort exact sequencessplit by a ring homomorphism
Thus we concludewhere. This completes the induction.

K-theory of BU(n)

Consider topological complex K-theory as the cohomology theory represented by the spectrum. In this case,,[2] andis the freemodule onandforand.[3] In this description, the product structure oncomes from the H-space structure ofgiven by Whitney sum of vector bundles. This product is called thePontryagin 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, 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.

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 themultinomial coefficientandcontains r distinct integers, repeatedtimes, respectively.

See also

  • Classifying space for O(n)

  • Topological K-theory

  • Atiyah–Jänich theorem

References

[1]
Citation Linkopenlibrary.orgR. Bott, L. W. Tu-- Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer
Sep 29, 2019, 5:36 PM
[2]
Citation Linkopenlibrary.orgAdams 1974, p. 49
Sep 29, 2019, 5:36 PM
[3]
Citation Linkopenlibrary.orgAdams 1974, p. 47
Sep 29, 2019, 5:36 PM
[4]
Citation Link//doi.org/10.1007%2FBF0121475310.1007/BF01214753
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Citation Link//www.jstor.org/stable/20013552001355
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Citation Linkdoi.org10.2307/2001355
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Citation Linken.wikipedia.orgThe original version of this page is from Wikipedia, you can edit the page right here on Everipedia.Text is available under the Creative Commons Attribution-ShareAlike License.Additional terms may apply.See everipedia.org/everipedia-termsfor further details.Images/media credited individually (click the icon for details).
Sep 29, 2019, 5:36 PM