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Zero Euler class does not in general imply a nowhere-zero section

Statement refuted

The converse of the vanishing criterion fails in general: the implication $an oriented rank- 3 real bundle E \to S^{4} has e (E) = 0 ⟹ E admits a nowhere-zero section$ is false. There is an oriented rank-three real bundle over $S^{4}$ whose Euler class vanishes and which admits no nowhere-zero section.

Facts & Assumptions

Given: AC, the sphere $S^{4}$ with its standard structure, and the covering homomorphism $ρ : S^{3} \to S O (3)$ .

[F1]

For $n \geq 1$ and $k \geq 1$ , orientation-preserving isomorphism classes of oriented rank- $n$ real bundles over $S^{k}$ correspond bijectively to $[S^{k - 1}, SO (n)]$ by clutching; the trivial bundle corresponds to the class of a constant map (Oriented clutching classifies oriented bundles over spheres).

[F2]

Conjugation by the unit quaternions is a continuous surjective two-sheeted covering homomorphism $ρ : S^{3} \to S O (3)$ with kernel ${\pm 1}$ , where $ρ (q) (v) = q v q^{- 1}$ . Its homotopy class generates $π_{3} (S O (3)) ≅ Z$ , and the rank-three bundle $E_{ρ} \to S^{4}$ clutched by $ρ$ is nontrivial (The quaternion double cover generates the third homotopy group of SO(3)).

[F5]

${\tilde{H}}_{k} (S^{4}; Z) = 0$ for $k \neq 4$ and ${\tilde{H}}_{4} (S^{4}; Z) = Z$ ; hence $H^{3} (S^{4}; Z) = 0$ by the universal coefficient sequence (Homology of spheres, Topological universal coefficient short exact sequence for cohomology).

[F6]

If a short exact sequence of numerable bundles over a paracompact Hausdorff base splits, here via a nowhere-zero section spanning a trivial line subbundle and a bundle metric on the quotient, then the middle bundle is the direct sum of the ends (Short exact sequences of numerable vector bundles split).

[F7]

The projection $R \to R / Z ≅ S^{1}$ is the universal covering of the circle, and a map from a simply connected space into $S^{1}$ lifts through it; since $R$ is contractible, every map $S^{3} \to S^{1}$ is nullhomotopic ( $R \to R / Z$ is a universal covering, Lifting criterion for maps from path-connected locally path-connected spaces).

[F8]

The Euler class of an oriented rank-three bundle over $S^{4}$ lies in $H^{3} (S^{4}; Z)$ (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 witness. Via the clutching bijection [F1] with $k = 4$ and $n = 3$ , let $E \to S^{4}$ be the oriented rank-three real bundle clutched by the map $ρ : S^{3} \to SO (3)$ of [F2]. This is the witness; it is an oriented numerable bundle since $S^{4}$ is a CW complex.

F1F2

1.2

The Euler class vanishes. The bundle $E$ has rank three, so $e (E) \in H^{3} (S^{4}; Z)$ by [F8]; this group is zero by [F5]. Hence $e (E) = 0$ .

F5F8

2.1

The witness is nontrivial. Clause 4 of [F2] is exactly the assertion that the clutching construction over the equatorial $S^{3}$ with clutching map $ρ$ produces a nontrivial oriented rank-three bundle $E_{ρ} \to S^{4}$ . The witness of step 1.1 is this bundle, so it is nontrivial. Equivalently, clauses 2 and 3 of [F2] identify $[ρ]$ with a generator of $π_{3} (S O (3)) ≅ Z$ , hence with a nonconstant based homotopy class. No conversion from an unbased nullhomotopy to a based one is used here.

F2step 1.1

3.1

There is no nowhere-zero section. Suppose, for contradiction, that $σ$ is a nowhere-zero section of $E$ . It spans a trivial line subbundle $ε^{1} \subseteq E$ , and a bundle metric on the paracompact Hausdorff base $S^{4}$ splits the resulting sequence, so [F6] gives $E ≅ ε^{1} \oplus F$ with $F$ an oriented rank-two real bundle over $S^{4}$ . By [F1] the bundle $F$ is clutched by a map $S^{3} \to SO (2) ≅ S^{1}$ , which is nullhomotopic by [F7], since every map from the simply connected $S^{3}$ to $S^{1}$ lifts through the contractible universal cover. Hence $F$ is trivial by [F1] and $E ≅ ε^{1} \oplus F ≅ ε^{3}$ is trivial, contradicting step 2.1. Therefore no nowhere-zero section exists.

F1F6F7step 2.1assume-contraA1

4.1

Conclusion. The bundle $E$ of step 1.1 is an oriented rank-three real bundle over $S^{4}$ with $e (E) = 0$ by step 1.2 and no nowhere-zero section by step 3.1. This refutes the displayed implication and completes the counterexample.

step 1.1step 1.2step 3.1discharge-contradiction: the triviality of E forced by a section contradicts its established nontriviality∎

Depends on

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Sources

Allen Hatcher, Vector Bundles & K-Theory (standard reference, not scraped)
Haynes Miller, MIT 18.906 Algebraic Topology II lecture notes (standard reference, not scraped)