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

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 {\mathcal {H}}\right\}.

Here, H is 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 {\mathcal {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({\mathcal {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{aligned}F_{n}(\mathbf {C} ^{k})&\longrightarrow \mathbf {S} ^{2k-1}\\(e_{1},\ldots ,e_{n})&\longmapsto e_{n}\end{aligned}}

is a fibre bundle of fibre F_n−1(C^k−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(C^k) (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(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 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^[1] that the cohomology of CP^k is isomorphic to $\mathbf {R} \lbrack c_{1}\rbrack /c_{1}^{k+1}$ , 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

$\mathbb {S} ^{2n-1}\to BU(n-1)\to BU(n)$

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

Applying the Gysin Sequence, one has a long exact sequence

$H^{p}(BU(n)){\overset {\smile d_{2n}\eta }{\to }}H^{p+2n}(BU(n)){\overset {j^{*}}{\to }}H^{p+2n}(BU(n-1)){\overset {\partial }{\to }}H^{p+1}(BU(n))\dots$

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 $p<-1$ , 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, $\smile d_{2n}\eta$ must always be injective. We therefore have short exact sequences split by a ring homomorphism

$0\to H^{p}(BU(n)){\overset {\smile d_{2n}\eta }{\to }}H^{p+2n}(BU(n)){\overset {j^{*}}{\to }}H^{p+2n}(BU(n-1))\to 0$

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)

Let us consider topological complex K-theory as the cohomology theory represented by the spectrum $KU$ . In this case, $KU^{*}(BU(n))\cong \mathbb {Z} [t,t^{-1}][[c_{1},...,c_{n}]]$ ,^[2] 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 $r\leq n$ .^[3] 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.

Notes

↑ R. Bott, L. W. Tu-- Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer
↑ Adams 1974, p. 49
↑ Adams 1974, p. 47

References

J. F. Adams (1974), Stable Homotopy and Generalised Homology, University Of Chicago Press, ISBN 0-226-00524-0 Contains calculation of $KU^{*}(BU(n))$ and $KU_{*}(BU(n))$ .
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 $K_{0}(BG)$ as a $K_{0}(K)$ -comodule for any compact, connected Lie group.
L. Schwartz (1983), "K-théorie et homotopie stable", Thesis, Université de Paris–VII Explicit description of $K_{0}(BU(n))$
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, JSTOR 2001355, doi:10.2307/2001355

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