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The zero locus of a transverse section represents the Euler dual

Statement

Assume AC. Let $E \to M$ be a smooth $R$ -oriented rank- $r$ real vector bundle over a closed $R$ -oriented smooth $n$ -manifold, with $R = Z$ or $F_{2}$ . Let $s$ be a smooth section transverse to the zero section. Its zero locus $Z$ is a closed embedded submanifold of dimension $n - r$ when nonempty, and $ν_{Z} ≅ E ∣_{Z}$ . Orient its normal bundle by this isomorphism and orient $Z$ in tangent-first order. Then $e (E) \cap [M] = (- 1)^{r (n - r)} (i_{Z})_{*} [Z] \in H_{n - r} (M; R) .$ In particular $(i_{Z})_{*} [Z] \neq 0$ is equivalent to $e (E) \neq 0$ , and either implies $Z \neq \emptyset$ . When $r = n$ , the Euler number $⟨ e (E), [M] ⟩$ is the signed zero count; a nonzero count forces a zero. On a disconnected base the total count may cancel even when the zero-cycle class is nonzero. For $r \neq n$ , no evaluation of $e (E)$ on the $n$ -dimensional fundamental class is asserted. Over $F_{2}$ orientations are canonical and the sign disappears. The zero section defines the Euler class by absolute pullback, and, when $M \neq \emptyset$ , is transverse to itself exactly in rank zero; for $M = \emptyset$ transversality is vacuous.

Facts & Assumptions

Given: AC and $M, E, s, Z$ with the orientations of the statement.

[F1]

The vertical differential induces $ν_{Z} ≅ E ∣_{Z}$ , and this isomorphism defines the tangent-first induced orientation (Normal bundle of the zero locus of a transverse section).

[F2]

The absolute image $\overset{ˉ}{α}$ of the tangent-first normal Thom class satisfies $\overset{ˉ}{α} \cap [M] = (- 1)^{r z} (i_{Z})_{*} [Z]$ (The normal Thom class realizes the Poincare dual of a closed submanifold).

[F3]

The relative pullback of the Thom class through the punctured-bundle pair is the normal Thom class, and its absolute image is $e (E)$ (Pullback of the Thom class along a transverse section computes the Euler class).

[F4]

Cap duality is an isomorphism; degree-top cap followed by zero-chain augmentation is Kronecker evaluation (Poincaré duality for oriented topological manifolds, Cap product with cohomology written first, Kronecker evaluation pairing).

[F5]

Smooth manifolds are admissible bases and their bundles are numerable under AC. The Euler class of a rank-zero bundle is its supplied orientation unit (Smooth manifolds have CW homotopy type, Euler class by zero-section pullback of the Thom class).

Proof

technique · identify the relative Thom pullback and then forget supports before applying normal Thom duality

1.1F1F3F5given

By [F1], $Z$ has codimension $r$ and normal bundle $E ∣_{Z}$ with the prescribed orientation. By [F5] every bundle and base used here meets the Thom hypotheses. Let $α \in H^{r} (M, M ∖ Z; R)$ be its normal Thom extension and $\overset{ˉ}{α}$ its absolute image. By [F3], $\overset{ˉ}{α} = e (E)$ .

2.1F2F3F4step 1.1algebra

Apply [F2] with $z = n - r$ to obtain $e (E) \cap [M] = (- 1)^{r (n - r)} (i_{Z})_{*} [Z]$ . If $Z$ is empty its relative group is zero and [F3] gives $e (E) = 0$ , including $r > n$ , when transversality forces the zero locus to be empty. Cap duality [F4] gives the stated equivalence of nonzero classes.

3.1F1F4F5step 2.1algebra∎

If $r = n$ , each zero is nondegenerate, and its point orientation is the sign of $D^{v} s : T_{x} M \to E_{x}$ in the supplied orientations. The cap formula of [F4] followed by augmentation therefore gives $⟨ e (E), [M] ⟩ = \sum_{x \in Z} sgn (D^{v} s_{x})$ . If $r = 0$ , $Z = M$ has dimension $n$ ; its induced orientation is the ambient orientation multiplied by the supplied rank-zero orientation unit $o$ (over $Z$ , $o^{- 1} = o$ ). Thus $[Z] = o [M]$ and the formula reads $e (0_{M}, o) \cap [M] = o [M]$ , as [F5] requires. For the zero section the vertical derivative is zero at every base point, so it is surjective exactly when $r = 0$ if the base is nonempty; there are no points to check when the base is empty. Empty manifolds and the canonical mod-two orientations satisfy the same formulas. AC is inherited from the stated suppliers.

Depends on

Used by

Dependency tree · two levels

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Sources

Eleny-Nicoleta Ionel (notes by Andrew Lin), Stanford Math 215B Differential Topology, Winter 2023 (complete 63-page lecture notes) (standard reference, not scraped)
Ralph L. Cohen, Bundles, Manifolds, and Homotopy (author draft bookR4) (standard reference, not scraped)
John W. Milnor and James D. Stasheff, Characteristic Classes (Princeton University Press, 1974; complete PDF) (standard reference, not scraped)