Inmathematics, particularly indifferential topology, there are two Whitney embedding theorems, named afterHassler Whitney:

• Thestrong Whitney embedding theorem states that anysmoothrealm-dimensionalmanifold (required also to beHausdorff andsecond-countable) can besmoothlyembedded in thereal2m-space,⁠${\displaystyle {\mathbb{R}}^{2m},}$⁠ ifm > 0. This is the best linear bound on the smallest-dimensional Euclidean space that allm-dimensional manifolds embed in, as thereal projective spaces of dimensionm cannot be embedded into real(2m − 1)-space ifm is apower of two (as can be seen from acharacteristic class argument, also due to Whitney).
• Theweak Whitney embedding theorem states that any continuous function from ann-dimensional manifold to anm-dimensional manifold may be approximated by a smooth embedding providedm > 2n. Whitney similarly proved that such a map could be approximated by animmersion providedm > 2n − 1. This last result is sometimes called theWhitney immersion theorem.

The weak Whitney embedding is proved through a projection argument.

When the manifold iscompact, one can first use a covering by finitely many local charts and then reduce the dimension with suitable projections.^{[1]: Ch. 1 §3 [2]: Ch. 6 [3]: Ch. 5 §3}

The general outline of the proof is to start with an immersion⁠${\displaystyle f:M\to {\mathbb{R}}^{2m}}$⁠ withtransverse self-intersections. These are known to exist from Whitney's earlier work onthe 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. IfM has boundary, one can remove the self-intersections simply by isotopingM into itself (the isotopy being in the domain off), to a submanifold ofM that does not contain the double-points. Thus, we are quickly led to the case whereM 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${\displaystyle {\mathbb{R}}^{2m}.}$⁠ Since⁠${\displaystyle {\mathbb{R}}^{2m}}$⁠ issimply connected, one can assume this path bounds a disc, and provided2m > 4 one can further assume (by theweak Whitney embedding theorem) that the disc is embedded in⁠${\displaystyle {\mathbb{R}}^{2m}}$⁠ such that it intersects the image ofM only in its boundary. Whitney then uses the disc to create a1-parameter family of immersions, in effect pushingM 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 eliminatingopposite sign double-points by pushing the manifold along a disc is called theWhitney Trick.

To introduce a local double point, Whitney created immersions⁠${\displaystyle {\alpha }_m:{\mathbb{R}}^m\to {\mathbb{R}}^{2m}}$⁠ which are approximately linear outside of the unit ball, but containing a single double point. Form = 1 such an immersion is given by

${\displaystyle \{\begin{array}{l}\alpha :{\mathbb{R}}^1\to {\mathbb{R}}^2\\ \alpha (t)=\left(\frac{1}{1+t^2},t-\frac{2t}{1+t^2}\right)\end{array}}$

Notice that ifα is considered as a map to⁠${\displaystyle {\mathbb{R}}^3}$⁠ like so:

${\displaystyle \alpha (t)=\left(\frac{1}{1+t^2},t-\frac{2t}{1+t^2},0\right)}$

then the double point can be resolved to an embedding:

${\displaystyle \beta (t,a)=\left(\frac{1}{(1+t^2)(1+a^2)},t-\frac{2t}{(1+t^2)(1+a^2)},\frac{ta}{(1+t^2)(1+a^2)}\right).}$

Noticeβ(t, 0) = α(t) and fora ≠ 0 then as a function oft,β(t, a) is an embedding.

For higher dimensionsm, there areα_m that can be similarly resolved in⁠${\displaystyle {\mathbb{R}}^{2m+1}.}$⁠ For an embedding into⁠${\displaystyle {\mathbb{R}}^5,}$⁠ for example, define

${\displaystyle {\alpha }_2(t_1,t_2)=\left(\beta (t_1,t_2),t_2\right)=\left(\frac{1}{(1+t_1^2)(1+t_2^2)},t_1-\frac{2t_1}{(1+t_1^2)(1+t_2^2)},\frac{t_1{t}_2}{(1+t_1^2)(1+t_2^2)},t_2\right).}$

This process ultimately leads one to the definition:

${\displaystyle {\alpha }_m(t_1,t_2,\cdots ,t_m)=\left(\frac{1}{u},t_1-\frac{2t_1}{u},\frac{t_1{t}_2}{u},t_2,\frac{t_1{t}_3}{u},t_3,\cdots ,\frac{t_1{t}_m}{u},t_m\right),}$

where

${\displaystyle u=(1+t_1^2)(1+t_2^2)\cdots (1+t_m^2).}$

The key properties ofα_m is that it is an embedding except for the double-pointα_m(1, 0, ... , 0) = α_m(−1, 0, ... , 0). Moreover, for|(t₁, ... , t_m)| large, it is approximately the linear embedding(0, t₁, 0, t₂, ... , 0, t_m).

The Whitney trick was used byStephen Smale to prove theh-cobordism theorem; from which follows thePoincaré conjecture in dimensionsm ≥ 5, and the classification ofsmooth structures on discs (also in dimensions 5 and up). This provides the foundation forsurgery 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.

The occasion of the proof byHassler Whitney of the embedding theorem for smooth manifolds is said (rather surprisingly) to have been the first complete exposition of themanifold 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 thehistory of manifolds and varieties for context.

Although everyn-manifold embeds in⁠${\displaystyle {\mathbb{R}}^{2n},}$⁠ one can frequently do better. Lete(n) denote the smallest integer so that all compact connectedn-manifolds embed in⁠${\displaystyle {\mathbb{R}}^{e(n)}.}$⁠ Whitney's strong embedding theorem states thate(n) ≤ 2n. Forn = 1, 2 we havee(n) = 2n, as thecircle and theKlein bottle show. More generally, forn = 2^k we havee(n) = 2n, as the2^k-dimensionalreal projective space show. Whitney's result can be improved toe(n) ≤ 2n − 1 unlessn is a power of 2. This is a result ofAndré Haefliger andMorris Hirsch (forn > 4) andC. T. C. Wall (forn = 3); these authors used important preliminary results and particular cases proved by Hirsch,William S. Massey,Sergey Novikov andVladimir Rokhlin.^[4] At present the functione is not known in closed-form for all integers (compare to theWhitney immersion theorem, where the analogous number is known).

One can strengthen the results by putting additional restrictions on the manifold. For example, then-sphere always embeds in⁠${\displaystyle {\mathbb{R}}^{n+1}}$⁠ – which is the best possible (closedn-manifolds cannot embed in⁠${\displaystyle {\mathbb{R}}^n}$⁠). Any compactorientable surface and any compact surfacewith non-empty boundary embeds in⁠${\displaystyle {\mathbb{R}}^3,}$⁠ though anyclosed non-orientable surface needs⁠${\displaystyle {\mathbb{R}}^4.}$⁠

IfN is a compact orientablen-dimensional manifold, thenN embeds in⁠${\displaystyle {\mathbb{R}}^{2n-1}}$⁠ (forn not a power of 2 the orientability condition is superfluous). Forn a power of 2 this is a result ofAndré Haefliger andMorris Hirsch (forn > 4), and Fuquan Fang (forn = 4); these authors used important preliminary results proved by Jacques Boéchat and Haefliger,Simon Donaldson, Hirsch andWilliam S. Massey.^[4] Haefliger proved that ifN is a compactn-dimensionalk-connected manifold, thenN embeds in⁠${\displaystyle {\mathbb{R}}^{2n-k}}$⁠ provided2k + 3 ≤n.^[4]

A relatively 'easy' result is to prove that any two embeddings of a 1-manifold into⁠${\displaystyle {\mathbb{R}}^4}$⁠ are isotopic (seeKnot theory#Higher dimensions). This is proved using general position, which also allows to show that any two embeddings of ann-manifold into⁠${\displaystyle {\mathbb{R}}^{2n+2}}$⁠ are isotopic. This result is an isotopy version of the weak Whitney embedding theorem.

Wu proved that forn ≥ 2, any two embeddings of ann-manifold into⁠${\displaystyle {\mathbb{R}}^{2n+1}}$⁠ 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 ifN is a compactn-dimensionalk-connected manifold, then any two embeddings ofN into⁠${\displaystyle {\mathbb{R}}^{2n-k+1}}$⁠ are isotopic provided2k + 2 ≤n. The dimension restriction2k + 2 ≤n is sharp: Haefliger went on to give examples of non-trivially embedded 3-spheres in⁠${\displaystyle {\mathbb{R}}^6}$⁠ (and, more generally,(2d − 1)-spheres in⁠${\displaystyle {\mathbb{R}}^{3d}}$⁠). Seefurther generalizationsArchived 2016-09-30 at theWayback Machine.

• Representation theorem – Proof that every structure with certain properties is isomorphic to another structure
• Whitney immersion theorem – On immersions of smooth m-dimensional manifolds in 2m-space and (2m-1) space
• Nash embedding theorem – Every Riemannian manifold can be isometrically embedded into some Euclidean spacePages displaying short descriptions of redirect targets
• Takens's theorem – Conditions under which a chaotic system can be reconstructed by observation
• Nonlinear dimensionality reduction – Projection of data onto lower-dimensional manifolds
• Universal space

• Whitney, Hassler (1992),Eells, James; Toledo, Domingo (eds.),Collected Papers, Boston: Birkhäuser,ISBN 0-8176-3560-2
• Milnor, John (1965),Lectures on theh-cobordism theorem, Princeton University Press
• Adachi, Masahisa (1993),Embeddings and Immersions, translated by Hudson, Kiki, American Mathematical Society,ISBN 0-8218-4612-4
• Skopenkov, Arkadiy (2008), "Embedding and knotting of manifolds in Euclidean spaces", in Nicholas Young; Yemon Choi (eds.),Surveys in Contemporary Mathematics, London Math. Soc. Lect. Notes., vol. 347, Cambridge:Cambridge University Press, pp. 248–342,arXiv:math/0604045,Bibcode:2006math......4045S,MR 2388495

• Classification of embeddingsArchived 2017-12-22 at theWayback Machine

• Theorems in differential topology

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