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The diagonal self-intersection is the Euler number of the tangent bundle

Statement

Assume AC. Let $M$ be a closed oriented smooth $n$ -manifold and give $M \times M$ the product orientation. Then the diagonal $Δ_{M} \subseteq M \times M$ , oriented by the transport of the orientation of $M$ along $x \mapsto (x, x)$ , is a closed oriented embedded $n$ -submanifold with $2 \dim Δ_{M} = \dim (M \times M)$ and $Δ_{M} \cdot Δ_{M} = ⟨ e (T M), [M] ⟩ \in Z,$ where $e (T M)$ is the Euler class of the tangent bundle and the self-intersection number is that of The self-intersection number of a complementary-dimensional oriented submanifold. This is the geometric form of the evaluation of the Euler class of the tangent bundle and the bridge to the Euler characteristic in the later Euler/index pair.

Facts & Assumptions

Given: The closed oriented smooth $n$ -manifold $M$ , the product $M \times M$ with its product orientation, and the diagonal oriented by transport from $M$ .

[F1]

The diagonal $Δ_{M} = {(p, p) : p \in M} \subseteq M \times M$ is an embedded submanifold of dimension $\dim M$ (The diagonal is an embedded submanifold).

[F2]

$M \times M$ carries the canonical product smooth structure, so it is a closed orientable $2 n$ -manifold with the product orientation when $M$ is oriented (Products of smooth manifolds have a canonical product smooth structure).

[F3]

The product orientation is defined by the ordered determinant isomorphism $\det (V \oplus W) ≅ \det V \otimes \det W$ , tensoring the selected rays (Product orientations).

[F4]

The normal bundle of the diagonal is canonically $T M$ , oriented so that a positive tangent basis of $Δ_{M}$ followed by a positive normal basis is positive in $M \times M$ , and under this convention the canonical isomorphism is orientation-preserving when $Δ_{M}$ carries the orientation transported from $M$ (The normal bundle of the diagonal is canonically the tangent bundle).

[F5]

The self-intersection number satisfies $A \cdot A = ⟨ e (ν_{A}), [A] ⟩$ for a closed oriented $A$ with $2 \dim A = \dim M$ (The self-intersection number is the Euler number of the normal bundle).

Proof

technique · apply the self-intersection/Euler-number theorem to the diagonal using the normal-bundle identification

1.1F1F2F3given

The diagonal is a closed embedded submanifold of dimension $n$ with $2 n = \dim (M \times M)$ [F1], and the orientation transported from $M$ along $x \mapsto (x, x)$ makes it a closed oriented submanifold of the closed oriented manifold $M \times M$ [F2]; the product orientation is the ordered tensor product of the two copies of the orientation of $M$ [F3].

2.1F4F5step 1.1algebra∎

By [F4] the normal bundle of the diagonal is canonically $T M$ with the induced orientation, and the identification is orientation-preserving. Applying [F5] to $Δ_{M} \subseteq M \times M$ gives $Δ_{M} \cdot Δ_{M} = ⟨ e (ν_{Δ_{M}}), [Δ_{M}] ⟩ = ⟨ e (T M), [M] ⟩$ under the identification $Δ_{M} ≅ M$ .

Depends on

Used by

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61 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ralph L. Cohen, Bundles, Manifolds, and Homotopy (author draft bookR4) (standard reference, not scraped)
Victor Guillemin and Alan Pollack, Differential Topology (Prentice-Hall, 1974; complete 236-page PDF) (standard reference, not scraped)
Eleny-Nicoleta Ionel (notes by Andrew Lin), Stanford Math 215B Differential Topology, Winter 2023 (complete 63-page lecture notes) (standard reference, not scraped)