An embedding of a manifold or subspace that is ambiently isotopic to a smooth or piecewise-linear embedding and is locally flat at every point, meaning each point has a neighborhood homeomorphic to a standard flat model.

Principle

Tameness requires that local neighborhoods be standard (locally flat), so the embedding can be straightened by an ambient isotopy into a smooth or PL representative; it preserves classical local geometric and homological tools.

Demonstration

A standardly embedded k-sphere in Rn (for n>k+1) with a tubular neighborhood diffeomorphic to S^k × R^{n-k} is tame; small perturbations that do not create accumulation of knots remain ambiently isotopic to this smooth model.

Misapplication

Assuming that every locally tame-looking feature guarantees global tameness without checking for accumulation phenomena; or applying smooth tubular-neighborhood theorems in codimensions where local flatness fails.

Consequence

A tame embedding admits standard local invariants, tubular neighborhoods, transversality arguments, and PL or smooth classification methods; computations relying on local flatness and duality hold as expected.

Reversal

The opposite is a wild embedding, where local flatness fails and no ambient isotopy to a smooth or PL model exists; reversing yields loss of tubular neighborhoods and breakdown of classical local invariants.

Boundary

Applies to embeddings that admit ambient isotopies to smooth or PL models; does not include embeddings with local accumulation of pathological linking, nor does it address immersions with self-intersections.

Semantic Tension

Tension arises with embeddings that are piecewise tame except on a small Cantor set of points: locally most points are flat but global wildness persists, forcing a careful distinction between almost-everywhere tameness and genuine global tameness.

Synthesis

A tame embedding is one whose local neighborhoods are standard and which can be straightened by ambient isotopy to a smooth or PL embedding, restoring the usual local geometric and topological toolkit.