The self-intersection number of a complementary-dimensional oriented submanifold

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M$ be an oriented boundaryless smooth $n$-manifold and let $A^a\subseteq M$ be a compact boundaryless oriented embedded submanifold with $2a=n$. Orient ${\nu }_A=TM{\mathrm{\mid }}_A\mathrm{/}TA$ by the tangent-first rule. Choose a smooth bundle metric and a tubular chart $\phi$ whose normal differential along the zero section is the identity, as constructed in The tubular neighbourhood theorem in a smooth ambient manifold. Compactness of $A$ permits a uniform small disk bundle inside the tube. For any small smooth section $s$ transverse to the zero section, let $A_s=\phi (s(A))$, oriented by the parametrization $\phi \circ s:A\to A_s$. The self-intersection number is $A\cdot A:=I(A,A_s),$ with $A$ first and $A_s$ second.

Every section is an embedding into its total space: projection is a left inverse, its differential is injective, and projection restricted to its graph is its continuous inverse. Composing with the tube therefore gives an embedding. The homotopy $\phi (ts)$ consists of embeddings and joins the inclusion to the push-off. Compactness makes $A_s$ closed and the transverse count finite. Small transverse sections exist: extend local frame vectors to compactly supported sections by Every vector in a fibre extends to a compactly supported smooth section, take finitely many which span every fibre by compactness, and apply Parametric transversality to their parameter-linear sum. The full evaluation is transverse because the parameter directions span every fibre. A good parameter can be chosen arbitrarily small, including the rank-zero case where every section is already transverse.

The same definition using The mod 2 intersection number gives $A{\cdot }_{2}A$ for any boundaryless ambient $M$ and any compact boundaryless embedded $A$ with $2a=n$, using no orientations. The numerical value is independent of the tube and small section: each push-off map is homotopic to the inclusion, and the two-map diagonal construction underlying Intersection number under factor interchange and The mod 2 intersection number is homotopy invariant makes the ordered count invariant under that homotopy. This argument requires no assertion that the tube-germ comparison is an isotopy. Under AC the integral Euler evaluation is proved in The self-intersection number is the Euler number of the normal bundle ↗. The local signs and finite counts themselves require no choice.

Normal differential normalization matters for local signs: a tubular chart merely fixed on the zero section may reverse the integral normal generator. The tangent-first normalized chart makes the local ordered intersection sign agree with the zero sign. When $a=0$, signs are comparisons of the supplied determinant rays, rather than the unsigned determinant of an empty matrix. Compactness of $M$ is unnecessary; compactness of $A$ is essential. Exchanging the factors multiplies the integral count by $(-1{)}^{a^2}=(-1{)}^a$.