Classifying space for U(n) explained

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

EU(n) = \left \{e _{1,\ldots,e} _{n} : (e _{i,e} _{j)=\delta} _{ij} , e _{i\in} H \right \}.

Here, H denotes an infinite-dimensional complex Hilbert space, the e_i are vectors in H, and

\delta _{ij} 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

BU(n) = EU(n)/U(n)

and is the set of Grassmannian n-dimensional subspaces (or n-planes) in H. That is,

BU(n) = \{ V \subset H : \dim V = n \}

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

BU( 1 ) = PU(H),

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

\begin{align} F ^{k)} _{n( C} & \longrightarrow S ^{2k-1} \\ (e _{1,\ldots,e} _{n)} & \longmapsto e _{n \end{align}}

is a fibre bundle of fibre F_n−1(C^k−1). Thus because

\pi ^{2 k -1} _{p( S} ) is trivial and because of the long exact sequence of the fibration, we have\pi _{p(F} ^{k)) = \pi} ^{p(F} _{n-1} ( C ^{k-1} ))

whenever

p\leq 2 k -2 . By taking k big enough, precisely for k>\tfrac{ 1 }{ 2 }p + n -1 , we can repeat the process and get\pi _{p(F} ^{k))} ^{n( C} = \pi _{p(F} _{n-1} ( C ^{k-1} )) = … = \pi _{p(F} ^{k +1- n} ^{1 ( C} )) = \pi ^{k - n} _{p( S} ).

This last group is trivial for k > n + p. Let

EU(n) = {\lim _{\to} }\;_F_n(\mathbf^k)

be the direct limit of all the F_n(C^k) (with the induced topology). Let

G ^{inf ty) = {\lim} _{\to}} _{k\toinfty} G ^{k)} _{n( C}

be the direct limit of all the G_n(C^k) (with the induced topology).

Lemma: The group

\pi _{p(EU(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.

\Box

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 ring of

\operatorname{BU}(n) with coefficients in the ring Z of integers is generated by the Chern classes:^[1] H ^{*(\operatorname{BU}(n); Z ) = Z [c} _{1 ,\ldots,c} _{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^[2] that the cohomology of CP^k is isomorphic to

R \lbrack c _{1\rbrack/c} ^{k +1} ^{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 ^{2n-1} \to B U(n -1 ) \to B U(n)

Concretely, a point of the total space

BU(n -1 ) is given by a point of the base space BU(n) classifying a complex vector space V , together with a unit vector u in V ; together they classify u ^{\perp} < V while the splitting V = ( C u) ⊕ u ^{\perp} , trivialized by u , realizes the map B U(n -1 ) \to B U(n) representing direct sum with C .

Applying the Gysin sequence, one has a long exact sequence

H ^{p} ( BU(n) ) \overset{\smile d _{2n} η }{\longrightarrow} H ^{p+2n} ( BU(n) ) \overset{j ^{*}{\longrightarrow}} H ^{p+2n} (BU(n -1 )) \overset{\partial}{\longrightarrow} H ^{p+1} (BU(n)) \longrightarrow …

where

η is the fundamental class of the fiber S ^{2n-1} . By properties of the Gysin Sequence, j ^{*} is a multiplicative homomorphism; by induction, H ^{*BU(n-1)} is generated by elements with p < -1 , where \partial 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, \smile d _{2n} η must always be injective. We therefore have short exact sequences split by a ring homomorphism0 \to H ^{p} ( BU(n) ) \overset{\smile d _{2n} η }{\longrightarrow} H ^{p+2n} ( BU(n) ) \overset{j ^{*}{\longrightarrow}} H ^{p+2n} (BU(n -1 )) \to 0

Thus we conclude

H ^{*(BU(n))} = H ^{*(BU(n -1 ))[c} _{2 n} ] where c _{2n} = d _{2n} η . This completes the induction.

K-theory of BU(n)

Consider topological complex K-theory as the cohomology theory represented by the spectrum

KU . In this case, KU ^{*(BU(n))\cong} Z [t,t ^{-1} ][[c _{1,...,c} _{n]]} ,^[3] and KU _{*(BU(n))} is the free Z [t,t ^{-1} ] module on \beta _{0} and \beta _{i _{1}} \ldots\beta _{i _{r}} for n\geq i _{j} > 0 and r\leq n .^[4] In this description, the product structure on KU _{*(BU(n))} comes from the H-space structure of BU 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)} = \pi _{*(K)} ⊗ K _{0(X)} , where \pi ^{-1} _{*(K) = Z [t,t} ] , 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

f(w _{1,...,w} _{n)} \mapsto (1)/(n!) \sum _{\sigma \in S _{n}} f(x _{\sigma(1)} , ... , x _{\sigma(n)} )

and the image can be identified as the symmetric polynomials satisfying the integrality condition that

{n \choose n _{1,} n _{2,} \ldots, n _{r}f(k} _{1,...,k} _{n)} \in Z

where

{n \choose k _{1,} k _{2,} \ldots, k _{m}} = (n!)/(k _{1!} k _{2!} … k _{m!})

is the multinomial coefficient and

k _{1,...,k} _{n} contains r distinct integers, repeated n _{1,...,n} _{r} times, respectively.

Infinite classifying space

The canonical inclusions

\operatorname{U}(n)\hookrightarrow\operatorname{U}(n +1 ) induce canonical inclusions \operatorname{BU}(n)\hookrightarrow\operatorname{BU}(n +1 ) on their respective classifying spaces. Their respective colimits are denoted as:\operatorname{U} : = \lim _{n → infty} \operatorname{U}(n); \operatorname{BU} : = \lim _{n → infty} \operatorname{BU}(n). \operatorname{BU} is indeed the classifying space of \operatorname{U} .

See also

• Classifying space for O(n)
• Classifying space for SO(n)
• Classifying space for SU(n)
• Topological K-theory
• Atiyah–Jänich theorem

Notes

1. Hatcher 02, Theorem 4D.4.
2. R. Bott, L. W. Tu-- Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer
3. Adams 1974, p. 49
4. Adams 1974, p. 47

References

• Contains calculation of

KU ^{*(BU(n))} and KU _{*(BU(n))} .

• Contains a description of

K _{0(BG)} as a K _{0(K)} -comodule for any compact, connected Lie group.

• Explicit description of

K _{0(BU(n))}

• • Book: Hatcher, Allen. Algebraic topology. Cambridge University Press. Cambridge. 2002. en. 0-521-79160-X.
• Book: Mitchell, Stephen. Universal principal bundles and classifying spaces. August 2001.

External links

• classifying space on nLab
• BU(n) on nLab