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

${\displaystyle 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 ${\displaystyle \delta _{ij}}$ is the Kronecker delta. The symbol ${\displaystyle (\cdot ,\cdot )}$ 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

${\displaystyle BU(n)=EU(n)/U(n)}$

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

${\displaystyle 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

${\displaystyle BU(1)=PU(H),}$

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

${\displaystyle {\begin{aligned}F_{n}(\mathbf {C} ^{k})&\longrightarrow \mathbf {S} ^{2k-1}\\(e_{1},\ldots ,e_{n})&\longmapsto e_{n}\end{aligned}}}$

is a fibre bundle of fibre F_n−1(C^k−1). Thus because ${\displaystyle \pi _{p}(\mathbf {S} ^{2k-1})}$ is trivial and because of the long exact sequence of the fibration, we have

${\displaystyle \pi _{p}(F_{n}(\mathbf {C} ^{k}))=\pi _{p}(F_{n-1}(\mathbf {C} ^{k-1}))}$

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

${\displaystyle \pi _{p}(F_{n}(\mathbf {C} ^{k}))=\pi _{p}(F_{n-1}(\mathbf {C} ^{k-1}))=\cdots =\pi _{p}(F_{1}(\mathbf {C} ^{k+1-n}))=\pi _{p}(\mathbf {S} ^{k-n}).}$

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

${\displaystyle EU(n)={\lim _{\to }}\;_{k\to \infty }F_{n}(\mathbf {C} ^{k})}$

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

${\displaystyle G_{n}(\mathbf {C} ^{\infty })={\lim _{\to }}\;_{k\to \infty }G_{n}(\mathbf {C} ^{k})}$

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

Lemma: The group ${\displaystyle \pi _{p}(EU(n))}$ 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.${\displaystyle \Box }$

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 ${\displaystyle \mathbf {R} \lbrack c_{1}\rbrack /c_{1}^{k+1}}$, 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

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

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

Applying the Gysin sequence, one has a long exact sequence

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

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

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

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

K-theory of BU(n)

Consider topological complex K-theory as the cohomology theory represented by the spectrum ${\displaystyle KU}$. In this case, ${\displaystyle KU^{*}(BU(n))\cong \mathbb {Z} [t,t^{-1}][[c_{1},...,c_{n}]]}$,[2] and ${\displaystyle KU_{*}(BU(n))}$ is the free ${\displaystyle \mathbb {Z} [t,t^{-1}]}$ module on ${\displaystyle \beta _{0}}$ and ${\displaystyle \beta _{i_{1}}\ldots \beta _{i_{r}}}$ for ${\displaystyle n\geq i_{j}>0}$ and ${\displaystyle r\leq n}$.[3] In this description, the product structure on ${\displaystyle KU_{*}(BU(n))}$ comes from the H-space structure of ${\displaystyle BU}$ 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 ${\displaystyle K_{*}(X)=\pi _{*}(K)\otimes K_{0}(X)}$, where ${\displaystyle \pi _{*}(K)=\mathbf {Z} [t,t^{-1}]}$, 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

${\displaystyle f(w_{1},\dots ,w_{n})\mapsto {\frac {1}{n!}}\sum _{\sigma \in S_{n}}f(x_{\sigma (1)},\dots ,x_{\sigma (n)})}$

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

${\displaystyle {n \choose n_{1},n_{2},\ldots ,n_{r}}f(k_{1},\dots ,k_{n})\in \mathbf {Z} }$

where

${\displaystyle {n \choose k_{1},k_{2},\ldots ,k_{m}}={\frac {n!}{k_{1}!\,k_{2}!\cdots k_{m}!}}}$

is the multinomial coefficient and ${\displaystyle k_{1},\dots ,k_{n}}$ contains r distinct integers, repeated ${\displaystyle n_{1},\dots ,n_{r}}$ times, respectively.

See also

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

Notes

1. R. Bott, L. W. Tu-- Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer
2. Adams 1974, p. 49
3. 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 ${\displaystyle KU^{*}(BU(n))}$ and ${\displaystyle KU_{*}(BU(n))}$.
• 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 ${\displaystyle K_{0}(BG)}$ as a ${\displaystyle K_{0}(K)}$-comodule for any compact, connected Lie group.
• L. Schwartz (1983), "K-théorie et homotopie stable", Thesis, Université de Paris–VII Explicit description of ${\displaystyle K_{0}(BU(n))}$
• 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, doi:10.2307/2001355, JSTOR 2001355