The self-intersection number is the Euler number of the normal bundle

The finite spanning-section construction in The self-intersection number of a complementary-dimensional oriented submanifold, using Every vector in a fibre extends to a compactly supported smooth section and Parametric transversality, supplies a small section $s$ transverse to the zero section. Smooth bundles on $A$ meet the Thom hypotheses by Smooth manifolds have CW homotopy type. Use a normalized tube and let $A_s=\phi (s(A))$ be its push-off. By [F1], $A\cap A_s=\phi (Z(s))$ with ${\epsilon }_{(A,A_s)}(\phi (x))=\mathrm{sign}\det {\mathrm{\partial }}_{\nu }{s}_x$ for every $x\in Z(s)$, so $A\cdot A=I(A,A_s)={\sum }_{x\in Z(s)}\mathrm{sign}\det {\mathrm{\partial }}_{\nu }{s}_x$ by [F4].

The Euler number of the section. Apply [F2] to the bundle ${\nu }_A\to A$ and the section $s$: with the orientation of $Z(s)$ induced by $A$ and ${\nu }_{Z(s)}\cong {\nu }_A{\mathrm{\mid }}_{Z(s)}$, and with $z=\dim Z(s)=a-a=0$ so that the Koszul factor is $1$, $e({\nu }_A)\cap [A]=(i_{Z(s)}{)}_{\ast }[Z(s)]$. Applying zero-chain augmentation to this cap class (the top-degree cap formula evaluates the cocycle on each simplex) gives $⟨e({\nu }_A),[A]⟩={\sum }_x{\sigma }_x$ with ${\sigma }_x$ the orientation sign of $x\in Z(s)$, where $e({\nu }_A)$ is the class of [F7]. Comparing with step 1.1, it remains to identify ${\sigma }_x$ with $\mathrm{sign}\det {\mathrm{\partial }}_{\nu }{s}_x$: the normal bundle of the zero locus in $A$ is ${\nu }_A{\mathrm{\mid }}_{Z(s)}$ oriented by Normal bundle of the zero locus of a transverse section, and its fibre orientation is exactly the one in which ${\mathrm{\partial }}_{\nu }{s}_x$ is measured in the determinant computation of [F1]; hence the signs agree.

Well-definedness. Every small transverse section in every normalized tube gives the same Euler number by steps 1.1–2.1, so no tube-isotopy theorem is needed. This also proves the signed-zero-count clause for a rank-$r$ oriented bundle over a closed $r$-manifold by applying [F2] and the same augmentation argument. Hence the value depends only on $(A,{\nu }_A)$, and the definition is well posed. The closed-oriented assumption on $A$ and the compactness of $A$ are used; compactness of $M$ is not.