An embedding of a manifold or subspace into Euclidean space that is not ambiently isotopic to any smooth or piecewise-linear embedding, producing pathological local topology such as fractal-like neighborhoods or non-locally-flat points.

Principle

An embedding is called wild when its local complement or neighborhood structure cannot be straightened by an ambient isotopy; wildness arises from accumulation of nontrivial knotting or linking at arbitrarily small scales.

Demonstration

Alexander's horned sphere in R3 is a classic example: it is a topological 2-sphere embedded so that its complement is not simply connected, and there is no ambient isotopy to the standard round sphere, hence it is wildly embedded.

Misapplication

Assuming every topological embedding in codimension one is equivalent to a smooth embedding; or using differentiable invariants (like curvature) to detect wildness, which can miss purely topological pathologies.

Consequence

Recognition of a wild embedding warns that local geometric intuition fails: invariants depending on local flatness (tubular neighborhoods, Alexander duality in naive form) may not apply and special topological techniques are required.

Reversal

The inverse concept is a tame embedding, where local neighborhoods are standard and the embedding can be smoothed or PL-approximated; reversing wildness restores local flatness and ambient isotopy to classical models.

Boundary

Concerns embeddings in codimensions where pathological phenomena can occur (notably codimension two or one in low dimensions); excludes embeddings that admit ambient isotopies to smooth or PL models and excludes purely algebraic immersions without topological embedding issues.

Semantic Tension

Competes with the notion of a pathological but tameable embedding: some embeddings appear wild at first glance but admit local untangling by delicate isotopies; the tension is between detectable algebraic invariants and subtle topological obstructions.

Synthesis

A wild embedding is a topological embedding whose neighborhood or complement resists straightening by ambient isotopy, producing genuine local pathological topology that distinguishes it from smooth or PL embeddings.