Euler number equals self-intersection ($⟨e({\nu }_S),[S]⟩=[S]\cdot [S]$)

For an oriented closed surface smoothly embedded in an oriented four-manifold, a transverse section of the normal bundle gives a homologous small push-off. Its signed zeros are exactly the signed intersections with the original surface. This identifies the evaluation of the Euler class with the self-intersection number.