The primary double point obstruction to removing self-intersections

Let $f:M^m\looparrowright X^{2m}$ be a self-transverse immersion of a closed manifold. Use the ordered locus ${\mathrm{\Delta }}_2(f)$, unordered branch-pair set $D(f)$, and collision image $\mathrm{\Sigma }(f)$ of Self-transverse immersions and the double point locus. The branch-pair set is finite: local injectivity gives an open neighbourhood of the diagonal in compact $M\times M$ containing no off-diagonal coincidence, so ${\mathrm{\Delta }}_2(f)$ is a closed subset of its compact complement; it is discrete directly: in a common target chart, the difference map $(u,v)\mapsto \chi (f(u))-\chi (f(v))$ has invertible derivative at each coincident pair because the two tangent images are complementary, so The smooth inverse function theorem on manifolds isolates that pair. Compact discreteness makes the locus finite. This argument uses Every immersion is locally an embedding and Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right. It also applies whenever finiteness of $D(f)$ is given directly.

The primary double point data are:

1. The unoriented count $\bar{I}(f)=\mathrm{\#}D(f)<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2\in {\mathbb{Z}}_2$, counting unordered branch pairs, including distinct pairs over a triple image.
2. If $M$ and $X$ are oriented and $m$ is even, the integral count $I(f)={\sum }_{d\in D(f)}\epsilon (d)\in \mathbb{Z}\mathrm{.}$ Each sign is the local sign of that branch pair and is independent of its ordering, by The local oriented intersection sign. For odd $m$ the pair sign changes under interchange, so an ordering must be fixed if signs are used; the unoriented count remains defined. If there are no triple or higher-multiplicity images, the branch-pair count is also the image-point count, with the corresponding signs when defined. With higher multiplicities there is no such term-by-term identification, although numerical counts can coincide (a triple image contributes three pairs, which is one modulo two).
3. For a chosen pair of genuine double points $p,q$ with chosen joining source arcs, the group obstruction is the class of the resulting Whitney circle $\gamma =\alpha \ast \beta$ in ${\pi }_1(X,p)$, where the image paths $\alpha$ and $\beta$ run from $p$ to $q$ and back. A base whisker $\lambda$ from $x_0$ to $p$ and compatible label paths are part of the data. In a normalized convention put $g(p)=1$ and $g(q)=[\lambda \ast \gamma \ast \bar{\lambda }]\in {\pi }_1(X,x_0)$. Cancelling $\bar{\lambda }\ast \lambda$ gives the inverse change of base point, so $g(q)=g(p)$ exactly when $[\gamma ]=1$; multiplying both labels on the left by any fixed label preserves this criterion by group cancellation (Based loops and the fundamental group, Loop classes form the group ${\pi }_1(X,x_0)$ under concatenation). This is the chosen-path label convention illustrated by The fundamental-group label controls contractibility of the Whitney circle, proof step 2.2; the calculation uses only paths, not globally embedded closed sheets. No independence from unrelated choices of joining paths is asserted. There is no construction of a nontrivial label by a codimension-at-least-three meridian. If $X$ is simply connected every such circle class is trivial.

An embedding has $D(f)=\mathrm{\Sigma }(f)=\mathrm{\varnothing }$, so both defined counts vanish. Counts alone do not classify embeddings up to isotopy. The disjunction criterion on this page uses admissible pairs and, in the simply connected oriented even-dimensional case, the vanishing integral branch-pair count (the disjunction proposition below). No general invariance statement is asserted here. In the Euclidean even-dimensional oriented setting, the later normal push-off argument identifies twice this integral count with minus the normal Euler number; this definition does not consume that later result. All labelled choices are finite data; no Axiom of Choice is needed by this definition.