Template:Short description 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 coordinate space|real Template:Math-space]], Template:Tmath if Template:Math. 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 Template:Math-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 Template:Math. Whitney similarly proved that such a map could be approximated by an immersion provided Template:Math. This last result is sometimes called the Whitney immersion theorem.
About the proofEdit
Weak embedding theoremEdit
The weak Whitney embedding is proved through a projection argument.
When the manifold is compact, one can first use a covering by finitely many local charts and then reduce the dimension with suitable projections.<ref name="Hirsch">Template:Cite book</ref>Template:Rp<ref>Template:Cite book</ref>Template:Rp<ref>Template:Cite book</ref>Template:Rp
Strong embedding theoremEdit
The general outline of the proof is to start with an immersion Template:Tmath 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 Template:Tmath Since Template:Tmath is simply connected, one can assume this path bounds a disc, and provided Template:Math one can further assume (by the weak Whitney embedding theorem) that the disc is embedded in Template:Tmath 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 immersions Template:Tmath which are approximately linear outside of the unit ball, but containing a single double point. For Template:Math such an immersion is given by
- <math>\begin{cases}
\alpha : \R^1 \to \R^2 \\ \alpha(t)=\left(\frac{1}{1+t^2},\ t - \frac{2t}{1+t^2}\right) \end{cases}</math>
Notice that if Template:Math is considered as a map to Template:Tmath like so:
- <math>\alpha(t) = \left( \frac{1}{1+t^2},\ t - \frac{2t}{1+t^2},0\right)</math>
then the double point can be resolved to an embedding:
- <math>\beta(t,a) = \left(\frac{1}{(1+t^2)(1+a^2)},\ t - \frac{2t}{(1+t^2)(1+a^2)},\ \frac{ta}{(1+t^2)(1+a^2)}\right).</math>
Notice Template:Math and for Template:Math then as a function of Template:Mvar, Template:Math is an embedding.
For higher dimensions Template:Mvar, there are Template:Math that can be similarly resolved in Template:Tmath For an embedding into Template:Tmath for example, define
- <math>\alpha_2(t_1,t_2) = \left(\beta(t_1,t_2),\ t_2\right) = \left(\frac{1}{(1+t_1^2)(1+t_2^2)},\ t_1 - \frac{2t_1}{(1+t_1^2)(1+t_2^2)},\ \frac{t_1t_2}{(1+t_1^2)(1+t_2^2)},\ t_2 \right).</math>
This process ultimately leads one to the definition:
- <math>\alpha_m(t_1,t_2,\cdots,t_m) = \left(\frac{1}{u},t_1 - \frac{2t_1}{u}, \frac{t_1t_2}{u}, t_2, \frac{t_1t_3}{u}, t_3, \cdots, \frac{t_1t_m}{u}, t_m \right),</math>
where
- <math>u=(1+t_1^2)(1+t_2^2)\cdots(1+t_m^2).</math>
The key properties of Template:Math is that it is an embedding except for the double-point Template:Math. Moreover, for Template:Math large, it is approximately the linear embedding Template:Math.
Eventual consequences of the Whitney trickEdit
The Whitney trick was used by Stephen Smale to prove the h-cobordism theorem; from which follows the Poincaré conjecture in dimensions Template:Math, 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.
HistoryEdit
Template:See also 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 manifolds extrinsically defined as submanifolds of Euclidean space. See also the history of manifolds and varieties for context.
Sharper resultsEdit
Although every Template:Mvar-manifold embeds in Template:Tmath one can frequently do better. Let Template:Math denote the smallest integer so that all compact connected Template:Mvar-manifolds embed in Template:Tmath Whitney's strong embedding theorem states that Template:Math. For Template:Math we have Template:Math, as the circle and the Klein bottle show. More generally, for Template:Math we have Template:Math, as the Template:Math-dimensional real projective space show. Whitney's result can be improved to Template:Math unless Template:Mvar is a power of 2. This is a result of André Haefliger and Morris Hirsch (for Template:Math) and C. T. C. Wall (for Template:Math); these authors used important preliminary results and particular cases proved by Hirsch, William S. Massey, Sergey Novikov and Vladimir Rokhlin.<ref name=skopenkov2>See section 2 of Skopenkov (2008)</ref> At present the function Template:Mvar is not known in closed-form for all integers (compare to the Whitney immersion theorem, where the analogous number is known).
Restrictions on manifoldsEdit
One can strengthen the results by putting additional restrictions on the manifold. For example, the [[n-sphere|Template:Mvar-sphere]] always embeds in Template:Tmath – which is the best possible (closed Template:Mvar-manifolds cannot embed in Template:Tmath). Any compact orientable surface and any compact surface with non-empty boundary embeds in Template:Tmath though any closed non-orientable surface needs Template:Tmath
If Template:Mvar is a compact orientable Template:Mvar-dimensional manifold, then Template:Mvar embeds in Template:Tmath (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 André Haefliger and Morris Hirsch (for Template:Math), and Fuquan Fang (for Template:Math); these authors used important preliminary results proved by Jacques Boéchat and Haefliger, Simon Donaldson, Hirsch and William S. Massey.<ref name=skopenkov2/> Haefliger proved that if Template:Mvar is a compact Template:Mvar-dimensional [[n-connected|Template:Mvar-connected]] manifold, then Template:Mvar embeds in Template:Tmath provided Template:Math.<ref name=skopenkov2/>
Isotopy versionsEdit
A relatively 'easy' result is to prove that any two embeddings of a 1-manifold into Template:Tmath are isotopic (see Knot theory#Higher dimensions). This is proved using general position, which also allows to show that any two embeddings of an Template:Mvar-manifold into Template:Tmath are isotopic. This result is an isotopy version of the weak Whitney embedding theorem.
Wu proved that for Template:Math, any two embeddings of an Template:Mvar-manifold into Template:Tmath 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 Template:Tmath are isotopic provided Template:Math. The dimension restriction Template:Math is sharp: Haefliger went on to give examples of non-trivially embedded 3-spheres in Template:Tmath (and, more generally, Template:Math-spheres in Template:Tmath). See further generalizations.
See alsoEdit
- Template:Annotated link
- Template:Annotated link
- Template:Annotated link
- Template:Annotated link
- Template:Annotated link
- Template:Annotated link