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Integral cohomology of BU(n)

Statement

Assume AC and $n \geq 1$ . Let $E U (n) \to B U (n)$ be the universal principal $U (n)$ -bundle, and let $E = E U (n) \times_{U (n)} C^{n}$ be its associated universal rank- $n$ complex vector bundle, the tautological bundle over the Grassmannian model $B U (n) = {Gr}_{n} (C^{\infty})$ , and let $q : B T^{n} = E U (n) / T^{n} \to B U (n)$ be the universal flag bundle of The universal complex flag bundle is BT-n. Then restriction along $q$ identifies $H^{*} (B U (n); Z) ≅ Z [c_{1}, \dots, c_{n}],$ the polynomial ring on the Chern classes of the universal bundle, and sends $c_{i}$ to the $i$ -th elementary symmetric polynomial $e_{i} (t_{1}, \dots, t_{n})$ in the coordinate Chern roots of $B T^{n}$ .

Facts & Assumptions

[A1]

The Axiom of Choice is assumed, exactly as inherited from the splitting and classifying-space suppliers (The Axiom of Choice).

[F1]

The flag projection $q$ splits $q^{*} E = L_{1} \oplus \dots \oplus L_{n}$ into the tautological lines and $q^{*} : H^{*} (B U (n); Z) \to H^{*} (B T^{n}; Z)$ is injective (Complex splitting principle with integral injective pullback).

[F2]

Chern classes are natural and multiplicative over Whitney sums, with $c (L) = 1 + c_{1} (L)$ for a line (Naturality, normalization, and Whitney sum for Chern classes).

[F3]

$H^{*} (B T^{n}; Z) = Z [t_{1}, \dots, t_{n}]$ with $t_{i} = c_{1} (L_{i})$ , the image of $q^{*}$ is contained in the symmetric invariants; its quotient model has $q$ as the projection to the Grassmannian (The universal complex flag bundle is BT-n).

[F4]

Over every commutative ring the symmetric polynomials in $n$ variables are the polynomials in the elementary symmetric functions $e_{1}, \dots, e_{n}$ , with $R [T_{1}, \dots, T_{n}] \to R [x_{1}, \dots, x_{n}]^{Σ_{n}}$ an isomorphism (Fundamental theorem of symmetric polynomials: unique expression as a polynomial in $e_{1}, \dots, e_{n}$ ).

[F5]

The Grassmannian ${Gr}_{n} (C^{\infty})$ is the chosen model of $B U (n)$ and classifies numerable rank- $n$ complex bundles (Real and complex vector bundles are classified by stable Grassmannians).

[F6]

The tautological lines $L_{i}$ over the flag bundle are complex line bundles with $q^{*} E = L_{1} \oplus \dots \oplus L_{n}$ (Complex flag bundle and Chern roots).

[F7]

Homotopic maps give equal cohomology pullbacks, hence a homotopy equivalence gives a ring isomorphism (Homotopic maps induce equal maps in singular cohomology).

[F8]

First Chern classes of lines are their complex-oriented Euler classes on the allowed CW-type bases (Chern classes from the projective-bundle relation); those Euler classes are natural for oriented pullbacks (Naturality, orientation sign, and Whitney product for Euler classes).

[F9]

The Stiefel frame projection has the tautological bundle as its associated standard vector bundle (Stiefel spaces, Grassmannians, and tautological bundles).

Proof

technique · direct

Given: AC, the universal bundle $E \to B U (n)$ and its flag bundle $q : B T^{n} \to B U (n)$ .

1.1

Put $Q = B T^{n}$ in its flag-quotient model. By [F3] and its explicit product comparison there is a homotopy equivalence $g : P = ({C P}^{\infty})^{n} \to Q$ , with $P$ a path-connected CW complex. Pull the splitting of [F1] back along $g$ . Naturality and Whitney multiplication in [F2] apply on the actual CW base $P$ and give $(q g)^{*} c (E) = \prod_{i} (1 + c_{1} (g^{*} L_{i}))$ . By [F8], $c_{1} (g^{*} L_{i}) = g^{*} c_{1} (L_{i}) = g^{*} t_{i}$ , with the fixed complex orientation. Since $g^{*}$ is an isomorphism by [F7], the equality descends to $q^{*} c (E) = \prod_{i} (1 + t_{i})$ on $Q$ , hence $q^{*} c_{i} (E) = e_{i} (t_{1}, \dots, t_{n})$ . This does not apply the CW-base Whitney interface directly on a space known only to have CW type.

F1F2F3F6F7F8

1.2

By [F3] the image of $q^{*}$ is contained in the symmetric invariants of $Z [t_{1}, \dots, t_{n}]$ , and by [F4] those invariants are exactly $Z [e_{1}, \dots, e_{n}]$ .

F3F4

2.1

By step 1.1 the image of $q^{*}$ contains $Z [e_{1}, \dots, e_{n}]$ , since it contains the images of the classes $c_{i}$ ; with step 1.2 this forces the image of $q^{*}$ to be exactly the invariant subring $Z [e_{1}, \dots, e_{n}]$ .

F4step 1.1step 1.2

3.1

Let $φ : Z [C_{1}, \dots, C_{n}] \to H^{*} (B U (n); Z)$ be the ring map $C_{i} \mapsto c_{i}$ and let $s : Z [C_{1}, \dots, C_{n}] \to Z [t_{1}, \dots, t_{n}]^{Σ_{n}}$ be the substitution $C_{i} \mapsto e_{i}$ of [F4]. Then $q^{*} \circ φ = s$ , which is an isomorphism by [F4]; injectivity of $q^{*}$ from [F1] makes $φ$ injective, and step 2.1 makes $φ$ surjective. Hence $φ$ is an isomorphism, which is the assertion.

F1F4step 2.1

4.1

Boundary cases. For $n = 1$ the statement reads $H^{*} (B U (1); Z) = Z [c_{1}]$ with $c_{1} = e_{1} = t_{1}$ , which is the ring $Z [u]$ of ${C P}^{\infty}$ ; the flag bundle is $q = i d$ and [F1] is trivial. The rank-zero case is excluded by $n \geq 1$ ; the coefficient ring $Z$ is a PID and the polynomial rings considered are free, so the fundamental theorem applies verbatim. The vector bundle is the associated tautological bundle by [F9] and is universal by [F5], and AC is used only through [A1] in the splitting, classifying-space, Euler and metric suppliers.

A1F1F4F5F9step 3.1∎

Source notes

Miller's Lecture 35 and Hatcher's section 3.1 prove $H^{*} (B U (n); Z) = Z [c_{1}, \dots, c_{n}]$ exactly by the symmetric-polynomial argument used here: the flag pullback is injective, the image lies in the invariants because the Weyl group permutes the roots, and the elementary symmetric functions generate the invariant ring. The proof above avoids any finite-index or Gysin shortcut.

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Sources

Miller, MIT 18.906 Algebraic Topology II, Lecture 35 (standard reference, not scraped)
Hatcher, Vector Bundles & K-Theory, section 3.1 (standard reference, not scraped)