Vanishing primary and characteristic obstructions do not classify embeddings

Assume AC for the characteristic-class clauses (The Axiom of Choice). Every embedding is an immersion with empty double point set (Smooth embeddings, Immersions, submersions, and constant-rank maps), so, in the $m$-into-$2m$ setting of The primary double point obstruction to removing self-intersections, its defined primary counts vanish and its selected-pair conditions are vacuous; and its normal bundle satisfies the usual rank restrictions on characteristic classes. A rank-$k$ real bundle has $w_i=0$ for $i>k$ and $p_i=0$ for $2i>k$; the cohomological degree $4i$ of $p_i$ need not be at most $k$ (Stiefel–Whitney classes from the projective-bundle relation, Pontryagin classes by complexification). Nevertheless these vanishings do not classify embeddings up to isotopy, and Smale–Hirsch theory for immersions must not be applied to embeddings without additional knotting data.

A proved witness consists of the standard and reflected parametrized embeddings $S^1\hookrightarrow {\mathbb{R}}^2$ in The round circle and its reflection are not isotopic embeddings in the plane. Both have empty double point sets and trivial normal line bundles, hence zero primary unoriented double point count and identical stable characteristic classes, but they are not isotopic. The primary count is within the definition’s $m$-into-$2m$ setting with $m=1$, and is zero because the double point set is empty. Every selected-pair Whitney-circle condition is vacuous. The integral count is not invoked, since $m$ is odd. This witness shows that these primary and characteristic data do not classify embeddings in general. It does not assert that every characteristic class vanishes for every embedding, or that no restricted embedding problem can be classified by such data.

Consequently the disjunction statement of Whitney disjunction removes algebraically cancelling double points is a statement about regularly homotoping a self-transverse immersion, not about isotoping embeddings, and The isotopy extension theorem converts isotopies of embeddings into ambient isotopies only for families that are already given. The cancellation criterion concerns the finite branch-pair count and admissible Whitney circles, with self-transverse endpoints. In the simply connected oriented even-dimensional stable range it supplies a regular homotopy to an embedding after first separating triple images; it does not supply an isotopy between two given embeddings.