# Whitney embedding theorem

In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:

- The
**strong Whitney embedding theorem**states that any smooth real Template:Mvar-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real 2*m*-space (**R**^{2m}), if*m*> 0. This is the best linear bound on the smallest-dimensional Euclidean space that all Template:Mvar-dimensional manifolds embed in, as the real projective spaces of dimension Template:Mvar cannot be embedded into real (2*m*− 1)-space if Template:Mvar is a power of two (as can be seen from a characteristic class argument, also due to Whitney).

- The
**weak Whitney embedding theorem**states that any continuous function from an Template:Mvar-dimensional manifold to an Template:Mvar-dimensional manifold may be approximated by a smooth embedding provided*m*> 2*n*. Whitney similarly proved that such a map could be approximated by an immersion provided*m*> 2*n*− 1. This last result is sometimes called the**weak Whitney immersion theorem**.

## A little about the proof

The general outline of the proof is to start with an immersion *f* : *M* → **R**^{2m} with transverse self-intersections. These are known to exist from Whitney's earlier work on **the weak immersion theorem**. Transversality of the double points follows from a general-position argument. The idea is to then somehow remove all the self-intersections. If Template:Mvar has boundary, one can remove the self-intersections simply by isotoping Template:Mvar into itself (the isotopy being in the domain of Template:Mvar), to a submanifold of Template:Mvar that does not contain the double-points. Thus, we are quickly led to the case where Template:Mvar has no boundary. Sometimes it is impossible to remove the double-points via an isotopy—consider for example the figure-8 immersion of the circle in the plane. In this case, one needs to introduce a local double point.

Once one has two opposite double points, one constructs a closed loop connecting the two, giving a closed path in **R**^{2m}. Since **R**^{2m} is simply connected, one can assume this path bounds a disc, and provided 2*m* > 4 one can further assume (by the **weak Whitney embedding theorem**) that the disc is embedded in **R**^{2m} such that it intersects the image of Template:Mvar only in its boundary. Whitney then uses the disc to create a 1-parameter family of immersions, in effect pushing Template:Mvar across the disc, removing the two double points in the process. In the case of the figure-8 immersion with its introduced double-point, the push across move is quite simple (pictured).

This process of eliminating **opposite sign** double-points by pushing the manifold along a disc is called the **Whitney Trick**.

To introduce a local double point, Whitney created a family of immersions α_{m} : **R**^{m} → **R**^{2m} which are approximately linear outside of the unit ball, but containing a single double point. For *m* = 1 such an immersion is defined as

Notice that if α_{1} is considered as a map to **R**^{3} i.e.:

then the double point can be resolved to an embedding:

Notice β_{1}(*t*_{1}, 0) = α_{1}(*t*_{1}) and for *a* ≠ 0 then as a function of *t*_{1}, β_{1}(*t*_{1}, *a*) is an embedding. Define

α_{2} can similarly be resolved in **R**^{5}, this process ultimately leads one to the definition:

where

The key properties of α_{m} is that it is an embedding except for the double-point α_{m}(1, 0, … , 0) = α_{m}(−1, 0, … , 0). Moreover, for |(*t*_{1}, … , *t _{m}*)| large, it is approximately the linear embedding (0,

*t*

_{1}, 0,

*t*

_{2}, … , 0,

*t*).

_{m}### Eventual consequences of the Whitney trick

The Whitney trick was used by Steve Smale to prove the *h*-cobordism theorem; from which follows the Poincaré conjecture in dimensions *m* ≥ 5, and the classification of smooth structures on discs (also in dimensions 5 and up). This provides the foundation for surgery theory, which classifies manifolds in dimension 5 and above.

Given two oriented submanifolds of complementary dimensions in a simply connected manifold of dimension ≥ 5, one can apply an isotopy to one of the submanifolds so that all the points of intersection have the same sign.

## History

{{#invoke:see also|seealso}}
The occasion of the proof by Hassler Whitney of the embedding theorem for smooth manifolds is said (rather surprisingly) to have been the first complete exposition of the *manifold concept* precisely because it brought together and unified the differing concepts of manifolds at the time: no longer was there any confusion as to whether abstract manifolds, intrinsically defined via charts, were any more or less general than manifold extrinsically defined as submanifolds of Euclidean space. See also the history of manifolds and varieties for context.

## Sharper results

Although every Template:Mvar-manifold embeds in **R**^{2n}, one can frequently do better. Let *e*(*n*) denote the smallest integer so that all compact connected Template:Mvar-manifolds embed in **R**^{e(n)}. Whitney's strong embedding theorem states that *e*(*n*) ≤ 2*n*. For *n* = 1, 2 we have *e*(*n*) = 2*n*, as the circle and the Klein bottle show. More generally, for *n* = 2^{k} we have *e*(*n*) = 2*n*, as the 2^{k}-dimensional real projective space show. Whitney's result can be improved to *e*(*n*) ≤ 2*n* − 1 unless Template:Mvar is a power of 2. This is a result of Haefliger–Hirsch (*n* > 4) and C.T.C. Wall (*n* = 3); these authors used important preliminary results and particular cases proved by M. Hirsch, W. Massey, S. Novikov and V. Rokhlin.^{[1]} At present the function *e* is not known in closed-form for all integers (compare to the Whitney immersion theorem, where the analogous number is known).

### Restrictions on manifolds

One can strengthen the results by putting additional restrictions on the manifold. For example, the [[n-sphere|Template:Mvar-sphere]] always embeds in **R**^{n + 1} – which is the best possible (closed Template:Mvar-manifolds cannot embed in **R**^{n}). Any compact *orientable* surface and any compact surface *with non-empty boundary* embeds in **R**^{3}, though any *closed non-orientable* surface needs **R**^{4}.

If Template:Mvar is a compact orientable Template:Mvar-dimensional manifold, then Template:Mvar embeds in **R**^{2n − 1} (for Template:Mvar not a power of 2 the orientability condition is superfluous). For Template:Mvar a power of 2 this is a result of A. Haefliger–M. Hirsch (*n* > 4) and F. Fang (*n* = 4); these authors used important preliminary results proved by J. Bo'echat-A. Haefliger, S. Donaldson, M. Hirsch and W. Massey.^{[1]} Haefliger proved that if Template:Mvar is a compact Template:Mvar-dimensional [[n-connected|Template:Mvar-connected]] manifold, then Template:Mvar embeds in **R**^{2n − k} provided 2*k* + 3 ≤ *n*.^{[1]}

## Isotopy versions

A relatively 'easy' result is to prove that any two embeddings of a 1-manifold into **R**^{4} are isotopic. This is proved using general position, which also allows to show that any two embeddings of an Template:Mvar-manifold into **R**^{2n + 2} are isotopic. This result is an isotopy version of the weak Whitney embedding theorem.

Wu proved that for *n* ≥ 2, any two embeddings of an Template:Mvar-manifold into **R**^{2n + 1} are isotopic. This result is an isotopy version of the strong Whitney embedding theorem.

As an isotopy version of his embedding result, Haefliger proved that if Template:Mvar is a compact Template:Mvar-dimensional Template:Mvar-connected manifold, then any two embeddings of Template:Mvar into **R**^{2n − k + 1} are isotopic provided 2*k* + 2 ≤ *n*. The dimension restriction 2*k* + 2 ≤ *n* is sharp: Haefliger went on to give examples of non-trivially embedded 3-spheres in **R**^{6} (and, more generally, (2*d* − 1)-spheres in **R**^{3d}). See further generalizations.

## See also

## References

- ↑
^{1.0}^{1.1}^{1.2}See section 2 of Skopenkov (2008)

**Hassler Whitney; collected papers.**Hassler Whitney, James Eells, Domingo Toledo. Nelson Thornes, 1992**Lectures on the**John Milnor. Princeton University Press. 1965*h*-cobordism theorem.- Embeddings and immersions, by Masahisa Adachi, translated by Kiki Hudson
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## External links

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