Whitney embedding and smooth approximation

For a compact smooth n-manifold without boundary, with n≥1, an easy Whitney bound gives an embedding into real (2n+1)-space. The strong Whitney theorem improves existence to real 2n-space. We use the compact version here so that an injective immersion automatically has the required homeomorphism onto its image. These theorems establish the existence of some embedding, not that every formula or projection into the named space is an embedding. The stronger bound is not a lower bound.

A circle embeds in the plane and a standard torus embeds in three-space, although the strong theorem's general surface guarantee is four-space. Thus the theorem can supply a dimension larger than a particular manifold needs. Noncompact manifolds have embedding theorems too, but their proofs and properness controls require additional global care. Compactness in the version stated here is an explicit hypothesis, rather than a silent consequence of the word manifold.

To detect double points of a map into real q-space, compare pairs of distinct source points using the difference of their images. The pair space has dimension 2n. A transverse zero set of that difference map has expected dimension 2n−q. At q=2n+1 the expected dimension is negative, so transversality rules out double points. A separate derivative-rank argument ensures immersion. Turning this sketch into a proof requires controlling behavior near the diagonal and arranging all needed general-position conditions.

At q=2n the same dimension count predicts isolated double points, rather than an empty set. Therefore genericity alone does not prove the strong Whitney theorem. Removing those remaining intersections involves additional geometric arguments, with low-dimensional cases handled appropriately. In particular, do not turn an existence result for some embedding into a claim that embeddings are dense near every prescribed map at this critical dimension. A transverse planar crossing of an immersed circle illustrates a stable local obstruction to such a naive claim.

Smooth approximation permits a continuous map from a compact smooth manifold without boundary to another smooth manifold without boundary to be approximated arbitrarily closely by smooth maps, and a sufficiently close approximation can be chosen homotopic to the original. Euclidean smoothing with a partition of unity, followed by a suitable local retraction when the target is embedded, explains the construction's shape. A C⁰ approximation controls values, not derivatives. Derivative approximation needs corresponding regularity and a specified Cʳ topology.

Immersion and embedding impose additional rank and separation requirements. One useful explicit trick is to retain a coordinate lost by a projection. A planar figure-eight may become an embedded spatial curve when a small third coordinate distinguishes its two crossing parameters. The new coordinate can be arbitrarily small in magnitude without being zero. This example illustrates why changing the ambient dimension can make an embedding problem easier, while preserving a particular planar target can leave the crossing unavoidable under small derivative-level changes.