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An odd-rank Euler class need not vanish in the presence of two-torsion

Statement refuted

The slogan "the Euler class of an oriented odd-rank bundle vanishes" is false. There is an oriented real rank-three bundle over $B = {R P}^{\infty} \times {R P}^{\infty}$ whose integral Euler class is nonzero and of order two. Only the weaker statement $2 e = 0$ is true in general.

Facts & Assumptions

Given: AC and the base $B = {R P}^{\infty} \times {R P}^{\infty}$ with coordinate projections.

[F1]

Real line bundles over a CW complex, and more generally over an admissible base, in particular over $B$ and its factors, correspond bijectively to $H^{1} (-; F_{2})$ through their first Stiefel–Whitney class: the correspondence is a natural bijection and the trivial bundle corresponds to $0$ (The first Stiefel–Whitney class classifies orientability).

[F2]

$H^{*} (B; F_{2}) = F_{2} [a, b]$ where $a, b$ are the pullbacks of the generators of the two factors, and $w_{1}$ of a pullback of the universal line is the corresponding coordinate class (Total Stiefel–Whitney class of a sum of universal lines, Cohomological Kunneth cross product is a ring isomorphism).

[F3]

The defining rank convention gives $w_{i} (L) = 0$ for $i > 1$ for every line bundle, hence $w (L) = 1 + w_{1} (L)$ . Together with naturality and the Whitney product formula this gives $w_{1} (E \oplus F) = w_{1} (E) + w_{1} (F)$ and $w_{3} (L_{1} \oplus L_{2} \oplus L_{3}) = w_{1} (L_{1}) w_{1} (L_{2}) w_{1} (L_{3})$ for line bundles $L_{j}$ (Stiefel–Whitney classes from the projective-bundle relation, Whitney sum formula for Stiefel–Whitney classes, Naturality of Stiefel–Whitney classes).

[F4]

A real bundle with $w_{1} = 0$ is orientable, so it admits an orientation; the equivalence between vanishing first Stiefel–Whitney class and orientability is available over admissible bases (The first Stiefel–Whitney class classifies orientability).

[F5]

For a real bundle with the canonical $F_{2}$ -orientation, $ρ_{2} (e (E, o)) = w_{n} (E)$ for either integral orientation $o$ ; the odd-rank Euler class satisfies $2 e (E, o) = 0$ (The mod-two Euler class is the top Stiefel–Whitney class, The Euler class of an oriented odd-rank bundle is two-torsion, Euler class by zero-section pullback of the Thom class).

[A1]

AC is the Axiom of Choice in the form fixed by The Axiom of Choice.

Counterexample

1.1

The line bundles. By [F1] choose line bundles $L_{a}, L_{b}, L_{a + b}$ over $B$ with $w_{1} (L_{a}) = a$ , $w_{1} (L_{b}) = b$ and $w_{1} (L_{a + b}) = a + b$ ; such bundles exist because the classification bijection is surjective and the three displayed classes lie in $H^{1} (B; F_{2})$ .

F1F2

2.1

The witness is orientable. Let $E = L_{a} \oplus L_{b} \oplus L_{a + b}$ , a real rank-three bundle over $B$ . By [F3] and step 1.1, $w_{1} (E) = w_{1} (L_{a}) + w_{1} (L_{b}) + w_{1} (L_{a + b}) = a + b + (a + b) = 0$ in $F_{2} [a, b]$ . Hence $E$ is orientable by [F4]; fix an orientation $o$ .

F3F4step 1.1

2.2

Its top class does not vanish. Again by [F3], $w_{3} (E) = w_{1} (L_{a}) w_{1} (L_{b}) w_{1} (L_{a + b}) = a b (a + b) \in F_{2} [a, b],$ which is a nonzero polynomial since it is a sum of the two distinct monomials $a^{2} b$ and $a b^{2}$ of degree three. Hence $w_{3} (E) \neq 0$ in $H^{3} (B; F_{2})$ by [F2].

F2F3step 1.1

3.1

The Euler class is nonzero of order two. By [F5] the mod-two reduction of the integral Euler class is the top Stiefel–Whitney class, so $ρ_{2} (e (E, o)) = w_{3} (E) \neq 0$ by step 2.2; in particular $e (E, o) \neq 0$ in $H^{3} (B; Z)$ . Also by [F5] the odd rank three gives $2 e (E, o) = 0$ . Therefore $e (E, o)$ is a nonzero element of order two, and the slogan of the statement refuted is false.

F5step 2.1step 2.2

4.1

Boundary remarks. The construction uses three line bundles of rank one, so the sum has odd rank three and the two-torsion conclusion applies; if the three line classes summed to a nonzero class the bundle would not be orientable and no integral Euler class would be defined. For the base with the second factor replaced by a point the same computation gives $w_{3} = 0$ , so the two factors are both needed for the witness. The orientation $o$ is determined only up to sign, and the statement is sign-independent because both $e \neq 0$ and $2 e = 0$ are invariant under negation. AC is used through the classification of line bundles and the Thom class, as recorded.

F1F4F5A1step 2.1step 3.1∎

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Sources

Allen Hatcher, Vector Bundles & K-Theory (standard reference, not scraped)
Milnor and Stasheff, Characteristic Classes (standard reference, not scraped)