Classifying space for U(n)

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 (\cdot,\cdot) 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 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

: \begin{align} F_n(\mathbf{C}^k) & \longrightarrow \mathbf{S}^{2k-1} \ (e_1,\ldots,e_n) & \longmapsto e_n \end{align}

is a fibre bundle of fibre F**n−1(Ck−1). Thus because \pi_p(\mathbf{S}^{2k-1}) is trivial and because of the long exact sequence of the fibration, we have

: \pi_p(F_n(\mathbf{C}^k))=\pi_p(F_{n-1}(\mathbf{C}^{k-1}))

whenever p\leq 2k-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_n(\mathbf{C}^k)) = \pi_p(F_{n-1}(\mathbf{C}^{k-1})) = \cdots = \pi_p(F_1(\mathbf{C}^{k+1-n})) = \pi_p(\mathbf{S}^{k-n}).

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

: EU(n)={\lim_{\to}};_{k\to\infty}F_n(\mathbf{C}^k)

be the direct limit of all the F**n(Ck) (with the induced topology). Let

: G_n(\mathbf{C}^\infty)={\lim_\to};_{k\to\infty}G_n(\mathbf{C}^k)

be the direct limit of all the G**n(Ck) (with the induced topology).

Proof: Let γ : Sp → EU(n), since Sp is compact, there exists k such that γ(Sp) is included in F**n(Ck). 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(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 ring of \operatorname{BU}(n) with coefficients in the ring \mathbb{Z} of integers is generated by the Chern classes:

: H^*(\operatorname{BU}(n);\mathbb{Z}) =\mathbb{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 that the cohomology of CPk is isomorphic to \mathbb{Z}\lbrack c_1\rbrack/c_1^{k+1}, 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

: \mathbb{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 while the splitting V = (\mathbb{C} u) \oplus u^\perp , trivialized by u, realizes the map B U(n-1) \to B U(n) representing direct sum with \mathbb{C}.

Applying the Gysin sequence, one has a long exact sequence

: H^p ( BU(n) ) \overset{\smile d_{2n} \eta}{\longrightarrow} H^{p+2n} ( BU(n) ) \overset{j^*}{\longrightarrow} H^{p+2n} (BU(n-1)) \overset{\partial}{\longrightarrow} H^{p+1}(BU(n)) \longrightarrow \cdots

where \eta is the fundamental class of the fiber \mathbb{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 , 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}\eta must always be injective. We therefore have short exact sequences split by a ring homomorphism

: 0 \to H^p ( BU(n) ) \overset{\smile d_{2n} \eta}{\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_{2n}] where c_{2n} = d_{2n} \eta. 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 \mathbb{Z}[t,t^{-1}]c_1,...,c_n, and KU_(BU(n)) is the free \mathbb{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. 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) \otimes K_0(X), where \pi_*(K)=\mathbf{Z}[t,t^{-1}], 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,\dots,w_n) \mapsto \frac{1}{n!} \sum_{\sigma \in S_n} f(x_{\sigma(1)}, \dots, 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,\dots,k_n) \in \mathbf{Z}

where

: {n \choose k_1, k_2, \ldots, k_m} = \frac{n!}{k_1!, k_2! \cdots k_m!}

is the multinomial coefficient and k_1,\dots,k_n contains r distinct integers, repeated n_1,\dots,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\rightarrow\infty}\operatorname{U}(n); : \operatorname{BU} :=\lim_{n\rightarrow\infty}\operatorname{BU}(n).

\operatorname{BU} is indeed the classifying space of \operatorname{U}.

Notes

References

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References

1. Lawson & Michelson 90, Theorem B.10
2. Hatcher 02, Theorem 4D.4.
3. R. Bott, L. W. Tu-- ''Differential Forms in Algebraic Topology'', Graduate Texts in Mathematics 82, Springer
4. Adams 1974, p. 49
5. Adams 1974, p. 47