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Integral total Pontryagin multiplicativity cannot ignore two-torsion

Statement refuted

Assume AC. The statement refuted is the integral total Pontryagin multiplicativity formula $p (E \oplus F) = p (E) p (F) in H^{*} (B; Z)$ for all real vector bundles $E, F$ over a path-connected CW base. Let $λ \to {R P}^{\infty}$ be the universal real line, put $E = λ \oplus λ$ and let $a = c_{1} (λ_{C})$ be the two-torsion class of Integral powers of the complexified universal real line. Then $p_{1} (E) = - a^{2} \neq 0, p (λ) p (λ) = 1,$ while the left-hand side $p (E)$ has nonzero degree-four component $- a^{2}$ ; hence $p (λ \oplus λ) \neq p (λ) p (λ)$ integrally, and the integral multiplicativity formula fails. This is the witness constructed below.

Facts & Assumptions

Given: AC and the universal real line $λ$ over ${R P}^{\infty}$ .

[A1]

The Axiom of Choice is assumed, exactly as inherited from the two-torsion lemma (The Axiom of Choice).

[F1]

$p_{i} (V) = (- 1)^{i} c_{2 i} (V_{C})$ , with $p_{0} = 1$ and $p_{i} = 0$ whenever $2 i > rank V$ (Pontryagin classes by complexification).

[F2]

One has $2 a = 0$ and $a^{2} \neq 0$ of exact order two (Integral powers of the complexified universal real line). Complexification is formed by real transition matrices acting complex-linearly (Complexification is conjugation invariant), and direct sums use their block diagonal matrices (Whitney sum, tensor, dual, Hom, and exterior-power bundles).

[F3]

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

Counterexample

technique · direct

1.1

The Pontryagin class of the line: $λ$ has real rank one, so by the cutoff in [F1] we have $p_{i} (λ) = 0$ for all $i \geq 1$ , while $p_{0} (λ) = 1$ ; hence $p (λ) = 1$ .

F1given

1.2

The complexification of $E = λ \oplus λ$ : by [F2], the two bundles $E_{C}$ and $λ_{C} \oplus λ_{C}$ have identical complex block diagonal transition matrices and hence a canonical isomorphism, and by multiplicativity [F3] applied to the two lines, $c (E_{C}) = (1 + a)^{2} = 1 + 2 a + a^{2} = 1 + a^{2}$ in $H^{*} ({R P}^{\infty}; Z)$ , because $2 a = 0$ by the two-torsion relation of [F2].

F2F3

2.1

The top component of $E_{C}$ in degree four is $c_{2} (E_{C}) = a^{2}$ , so by [F1] with $i = 1$ , $p_{1} (E) = (- 1)^{1} c_{2} (E_{C}) = - a^{2}$ , which is nonzero by the exact-order-two statement of [F2].

F1F2step 1.2

3.1

The right-hand side of the refuted formula is $p (λ) p (λ) = 1 \cdot 1 = 1$ by step 1.1, whose degree-four component is $0$ ; the left-hand side $p (E)$ has degree-four component $- a^{2} \neq 0$ by step 2.1.

step 1.1step 2.1

4.1

Hence $p (λ \oplus λ) \neq p (λ) p (λ)$ integrally: the degree-four components differ by the nonzero two-torsion class $- a^{2}$ . The witness pair is $(λ, λ)$ ; its Whitney sum is the bundle $E = λ \oplus λ$ used above. The argument uses no additive universal-coefficient computation beyond the nonvanishing $a^{2} \neq 0$ proved on the companion $A$ -page lemma.

F2step 2.1step 3.1

5.1

Boundary cases. After tensoring the integral cohomology ring with $Z [1 / 2]$ , $a \otimes 1 = (2 a) \otimes (1 / 2) = 0$ , so the two sides for this witness both become $1$ ; the trivial bundle case $a = 0$ gives no failure, and the rank cutoffs are used at rank one ( $p (λ) = 1$ ) and rank two ( $p_{1}$ is the first nonzero Pontryagin class). The base ${R P}^{\infty}$ is path connected and the coefficient group $Z$ is nonzero. AC is used only through [A1].

A1F1F2step 1.1step 3.1∎

Source notes

This is the two-torsion obstruction recorded in Miller's Lecture 36 and Hatcher's section 3.2: the odd Chern classes of complexified real bundles are two-torsion, and integrally they contribute cross terms to $p (E \oplus F)$ that are not seen by $p (E) p (F)$ . The companion theorem on the $A$ page states the multiplicativity only away from two.

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Sources

Miller, MIT 18.906 Algebraic Topology II, Lecture 36 (standard reference, not scraped)
Hatcher, Vector Bundles & K-Theory, section 3.2 (standard reference, not scraped)