Classifying space for U(n)

Short description: Exact homotopy case

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

[math]\displaystyle{ EU(n)=\left \{e_1,\ldots,e_n \ : \ (e_i,e_j)=\delta_{ij}, e_i\in H \right \}. }[/math]

Here, H denotes an infinite-dimensional complex Hilbert space, the e_i are vectors in H, and [math]\displaystyle{ \delta_{ij} }[/math] is the Kronecker delta. The symbol [math]\displaystyle{ (\cdot,\cdot) }[/math] 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

[math]\displaystyle{ BU(n)=EU(n)/U(n) }[/math]

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

[math]\displaystyle{ BU(n) = \{ V \subset H \ : \ \dim V = n \} }[/math]

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

[math]\displaystyle{ BU(1)= PU(H), }[/math]

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

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

is a fibre bundle of fibre F_n−1(C^k−1). Thus because [math]\displaystyle{ \pi_p(\mathbf{S}^{2k-1}) }[/math] is trivial and because of the long exact sequence of the fibration, we have

[math]\displaystyle{ \pi_p(F_n(\mathbf{C}^k))=\pi_p(F_{n-1}(\mathbf{C}^{k-1})) }[/math]

whenever [math]\displaystyle{ p\leq 2k-2 }[/math]. By taking k big enough, precisely for [math]\displaystyle{ k\gt \tfrac{1}{2}p+n-1 }[/math], we can repeat the process and get

[math]\displaystyle{ \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}). }[/math]

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

[math]\displaystyle{ EU(n)={\lim_{\to}}\;_{k\to\infty}F_n(\mathbf{C}^k) }[/math]

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

[math]\displaystyle{ G_n(\mathbf{C}^\infty)={\lim_\to}\;_{k\to\infty}G_n(\mathbf{C}^k) }[/math]

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

Lemma: The group [math]\displaystyle{ \pi_p(EU(n)) }[/math] 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.[math]\displaystyle{ \Box }[/math]

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 of the classifying space H*(BU(n)) is a ring of polynomials in n variables c₁, ..., c_n where c_p is of degree 2p.

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 CP^k is isomorphic to [math]\displaystyle{ \mathbf{R}\lbrack c_1\rbrack/c_1^{k+1} }[/math], 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

[math]\displaystyle{ \mathbb{S}^{2n-1} \to B U(n-1) \to B U(n) }[/math]

Concretely, a point of the total space [math]\displaystyle{ BU(n-1) }[/math] is given by a point of the base space [math]\displaystyle{ BU(n) }[/math] classifying a complex vector space [math]\displaystyle{ V }[/math], together with a unit vector [math]\displaystyle{ u }[/math] in [math]\displaystyle{ V }[/math]; together they classify [math]\displaystyle{ u^\perp \lt V }[/math] while the splitting [math]\displaystyle{ V = (\mathbb{C} u) \oplus u^\perp }[/math], trivialized by [math]\displaystyle{ u }[/math], realizes the map [math]\displaystyle{ B U(n-1) \to B U(n) }[/math] representing direct sum with [math]\displaystyle{ \mathbb{C}. }[/math]

Applying the Gysin sequence, one has a long exact sequence

[math]\displaystyle{ 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 }[/math]

where [math]\displaystyle{ \eta }[/math] is the fundamental class of the fiber [math]\displaystyle{ \mathbb{S}^{2n-1} }[/math]. By properties of the Gysin Sequence, [math]\displaystyle{ j^* }[/math] is a multiplicative homomorphism; by induction, [math]\displaystyle{ H^*BU(n-1) }[/math] is generated by elements with [math]\displaystyle{ p \lt -1 }[/math], where [math]\displaystyle{ \partial }[/math] must be zero, and hence where [math]\displaystyle{ j^* }[/math] must be surjective. It follows that [math]\displaystyle{ j^* }[/math] 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, [math]\displaystyle{ \smile d_{2n}\eta }[/math] must always be injective. We therefore have short exact sequences split by a ring homomorphism

[math]\displaystyle{ 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 }[/math]

Thus we conclude [math]\displaystyle{ H^*(BU(n)) = H^*(BU(n-1))[c_{2n}] }[/math] where [math]\displaystyle{ c_{2n} = d_{2n} \eta }[/math]. This completes the induction.

K-theory of BU(n)

Consider topological complex K-theory as the cohomology theory represented by the spectrum [math]\displaystyle{ KU }[/math]. In this case, [math]\displaystyle{ KU^*(BU(n))\cong \mathbb{Z}[t,t^{-1}]c 1,...,c n }[/math], and [math]\displaystyle{ KU_*(BU(n)) }[/math] is the free [math]\displaystyle{ \mathbb{Z}[t,t^{-1}] }[/math] module on [math]\displaystyle{ \beta_0 }[/math] and [math]\displaystyle{ \beta_{i_1}\ldots\beta_{i_r} }[/math] for [math]\displaystyle{ n\geq i_j \gt 0 }[/math] and [math]\displaystyle{ r\leq n }[/math]. In this description, the product structure on [math]\displaystyle{ KU_*(BU(n)) }[/math] comes from the H-space structure of [math]\displaystyle{ BU }[/math] 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 [math]\displaystyle{ K_*(X) = \pi_*(K) \otimes K_0(X) }[/math], where [math]\displaystyle{ \pi_*(K)=\mathbf{Z}[t,t^{-1}] }[/math], 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

[math]\displaystyle{ f(w_1,\dots,w_n) \mapsto \frac{1}{n!} \sum_{\sigma \in S_n} f(x_{\sigma(1)}, \dots, x_{\sigma(n)}) }[/math]

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

[math]\displaystyle{ {n \choose n_1, n_2, \ldots, n_r}f(k_1,\dots,k_n) \in \mathbf{Z} }[/math]

where

[math]\displaystyle{ {n \choose k_1, k_2, \ldots, k_m} = \frac{n!}{k_1!\, k_2! \cdots k_m!} }[/math]

is the multinomial coefficient and [math]\displaystyle{ k_1,\dots,k_n }[/math] contains r distinct integers, repeated [math]\displaystyle{ n_1,\dots,n_r }[/math] times, respectively.

See also

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

References

• J. F. Adams (1974), Stable Homotopy and Generalised Homology, University Of Chicago Press, ISBN 0-226-00524-0  Contains calculation of [math]\displaystyle{ KU^*(BU(n)) }[/math] and [math]\displaystyle{ KU_*(BU(n)) }[/math].
• 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 [math]\displaystyle{ K_0(BG) }[/math] as a [math]\displaystyle{ K_0(K) }[/math]-comodule for any compact, connected Lie group.
• L. Schwartz (1983), "K-théorie et homotopie stable", Thesis (Université de Paris–VII)  Explicit description of [math]\displaystyle{ K_0(BU(n)) }[/math]
• 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