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The Euler number of the tangent bundle is the Euler characteristic

Statement

Assume the Axiom of Choice (The Axiom of Choice).

(i) Let $M$ be a closed oriented smooth $n$ -manifold, $n \geq 1$ . Then the Euler number of the tangent bundle satisfies $⟨ e (T M), [M] ⟩ = χ (M),$ where $e (T M) \in H^{n} (M; Z)$ is the Euler class of Euler class by zero-section pullback of the Thom class and the bracket is the Kronecker evaluation; equivalently the diagonal satisfies $Δ_{M} \cdot Δ_{M} = χ (M)$ in the self-intersection number of The self-intersection number of a complementary-dimensional oriented submanifold.

(ii) For any closed smooth $n$ -manifold with $n \geq 1$ , $⟨ w_{n} (T M), [M] ⟩_{2} \equiv χ (M) (m o d 2),$ with $w_{n}$ the top Stiefel-Whitney class.

Facts & Assumptions

Given: A closed smooth $n$ -manifold $M$ , oriented in part (i).

[F1]

Choose an excellent Morse function $f$ and a Riemannian metric $g$ . The section $s = {grad}_{g} f$ of $T M$ vanishes exactly at $Crit (f)$ and is transverse to the zero section there (the linearization is the nondegenerate Hessian) (The Riemannian gradient is the metric dual of the differential, The Riemannian gradient vanishes exactly at the critical points, Every compact smooth manifold admits an excellent Morse function, Every smooth manifold admits a riemannian metric, Morse functions and excellent Morse functions).

[F2]

The signed zero count of a transverse section of a rank- $n$ oriented bundle over a closed oriented $n$ -manifold equals $⟨ e (E), [M] ⟩$ ; the local sign of a zero of a vector field, read on the zero-section/graph intersection in $T M$ with the horizontal-then-vertical orientation, is the index of the zero (The self-intersection number is the Euler number of the normal bundle, The index of a zero is its zero-section intersection number, Euler class by zero-section pullback of the Thom class).

[F3]

The signed zero count of ${grad}_{g} f$ is $\sum_{p} (- 1)^{ind (p)}$ , and this equals $χ (M)$ ; in particular the Euler number is $χ (M)$ . The diagonal form is the self-intersection statement for $Δ_{M}$ , and over $F_{2}$ the unsigned zero count satisfies $# Crit (f) \equiv ⟨ w_{n} (T M), [M] ⟩_{2}$ , since the canonical mod 2 Euler class is the top Stiefel-Whitney class (A Morse gradient zero contributes $(- 1)^{λ}$ to the index, Morse Euler characteristic identity, The diagonal self-intersection is the Euler number of the tangent bundle, The mod two self-intersection is the top Stiefel-Whitney evaluation, The mod-two Euler class is the top Stiefel–Whitney class).

[F4]

Under the assumed AC, $M$ is paracompact Hausdorff and CGWH with CW homotopy type, and the smooth tangent bundle is numerable (Smooth manifolds have CW homotopy type). These are the base and bundle hypotheses of the cited Thom and Stiefel-Whitney results.

Proof

1.1F1F2F3F4algebra

For (i): by [F1] the section $s = {grad}_{g} f$ is transverse to the zero section with zero set $Crit (f)$ , and by [F2] its signed zero count equals $⟨ e (T M), [M] ⟩$ ; by [F3] that signed count is $\sum_{p} (- 1)^{ind (p)} = χ (M)$ , so $⟨ e (T M), [M] ⟩ = χ (M)$ .

2.1F3step 1.1algebra

The diagonal statement of [F3] identifies $Δ_{M} \cdot Δ_{M}$ with $⟨ e (T M), [M] ⟩$ , so it equals $χ (M)$ as well.

3.1F3F4step 1.1algebra∎

For (ii): the unsigned count $# Crit (f)$ of the transverse section ${grad}_{g} f$ satisfies $# Crit (f) \equiv ⟨ w_{n} (T M), [M] ⟩_{2}$ by [F3], while $# Crit (f) \equiv \sum_{p} (- 1)^{ind (p)} = χ (M) (m o d 2)$ because $(- 1)^{λ} \equiv 1$ ; hence $⟨ w_{n} (T M), [M] ⟩_{2} \equiv χ (M) (m o d 2)$ .

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Sources

John W. Milnor and James D. Stasheff, Characteristic Classes (Princeton University Press, 1974; complete PDF) (standard reference, not scraped)
Victor Guillemin and Alan Pollack, Differential Topology (Prentice-Hall 1974; complete PDF) (standard reference, not scraped)
Ralph L. Cohen, Bundles, Manifolds, and Homotopy (author draft) (standard reference, not scraped)