Editing Whitney embedding theorem

{{short description|Any smooth real m-dimensional manifold can be smoothly embedded in real 2m-space}}
In [[mathematics]], particularly in [[differential topology]], there are two Whitney embedding theorems, named after [[Hassler Whitney]]:

*The '''strong Whitney embedding theorem''' states that any [[differentiable manifold|smooth]] [[real numbers|real]] {{mvar|m}}-[[dimension (mathematics)|dimensional]] [[manifold]] (required also to be [[Hausdorff space|Hausdorff]] and [[second-countable]]) can be [[smooth map|smoothly]] [[embedding|embedded]] in the [[real coordinate space|real {{math|2''m''}}-space]], {{tmath|\R^{2m},}} if {{math|''m'' > 0}}. This is the best linear bound on the smallest-dimensional Euclidean space that all {{mvar|m}}-dimensional manifolds embed in, as the [[real projective space]]s of dimension {{mvar|m}} cannot be embedded into real {{math|(2''m'' − 1)}}-space if {{mvar|m}} 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 {{mvar|n}}-dimensional manifold to an {{mvar|m}}-dimensional manifold may be approximated by a smooth embedding provided {{math|''m'' > 2''n''}}.  Whitney similarly proved that such a map could be approximated by an [[immersion (mathematics)|immersion]] provided {{math|''m'' > 2''n'' − 1}}. This last result is sometimes called the '''[[Whitney immersion theorem]]'''.

==About the proof==

=== Weak embedding theorem ===
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">{{Cite book |last=Hirsch |first=Morris W. |author-link=Morris Hirsch |title=Differential topology |date=1976 |publisher=[[Springer Publishing|Springer]] |isbn=978-1-4684-9449-5 |series=Graduate texts in mathematics |location=New York Heidelberg Berlin |language=en}}</ref>{{rp|Ch. 1 §3}}<ref>{{Cite book |last=Lee |first=John M. |author-link=John M. Lee |url=https://www.worldcat.org/title/800646950 |title=Introduction to smooth manifolds |date=2013 |publisher=Springer |isbn=978-1-4419-9981-8 |edition=2nd |series=Graduate texts in mathematics |location=New York; London |oclc=800646950}}</ref>{{rp|Ch. 6}}<ref>{{Cite book |last=Prasolov |first=Victor V. |author-link=Prasolov |title=Elements of Combinatorial and Differential Topology |publisher=[[American Mathematical Society]] |year=2006 |isbn=978-1-4704-1153-4 |location=Providence}}</ref>{{rp|Ch. 5 §3}}

=== Strong embedding theorem ===
The general outline of the proof is to start with an immersion {{tmath|f:M \to \R^{2m} }} with [[transversality (mathematics)|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 {{mvar|M}} has boundary, one can remove the self-intersections simply by isotoping {{mvar|M}} into itself (the isotopy being in the domain of {{mvar|f}}), to a submanifold of {{mvar|M}} that does not contain the double-points.  Thus, we are quickly led to the case where {{mvar|M}} 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. [[File:whitneytrickstep1.svg|thumb|350px|right|Introducing double-point.]] Once one has two opposite double points, one constructs a closed loop connecting the two, giving a closed path in {{tmath|\R^{2m}.}}  Since {{tmath|\R^{2m} }} is [[simply connected]], one can assume this path bounds a disc, and provided {{math|2''m'' > 4}} one can further assume (by the '''weak Whitney embedding theorem''') that the disc is embedded in {{tmath|\R^{2m} }} such that it intersects the image of {{mvar|M}} only in its boundary. Whitney then uses the disc to create a [[homotopy|1-parameter family]] of immersions, in effect pushing {{mvar|M}} 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).[[Image:whitneytrickstep2.svg|thumb|450px|right|Cancelling opposite double-points.]] 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 {{tmath|\alpha_m: \R^m \to \R^{2m} }} which are approximately linear outside of the unit ball, but containing a single double point.  For {{math|1=''m'' = 1}} 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 {{math|α}} is considered as a map to {{tmath|\R^3}} 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 {{math|1=β(''t'', 0) = α(''t'')}} and for {{math|''a'' ≠ 0}} then as a function of {{mvar|t}}, {{math|β(''t'', ''a'')}} is an embedding.

For higher dimensions {{mvar|m}}, there are {{math|α<sub>''m''</sub>}} that can be similarly resolved in {{tmath|\R^{2m+1}.}} For an embedding into {{tmath|\R^5,}} 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 {{math|α<sub>''m''</sub>}} is that it is an embedding except for the double-point {{math|1=α<sub>''m''</sub>(1, 0, ... , 0) = α<sub>''m''</sub>(−1, 0, ... , 0)}}. Moreover, for {{math|{{!}}(''t''<sub>1</sub>, ... , ''t<sub>m</sub>''){{!}}}} large, it is approximately the linear embedding {{math|(0, ''t''<sub>1</sub>, 0, ''t''<sub>2</sub>, ... , 0, ''t<sub>m</sub>'')}}.

===Eventual consequences of the Whitney trick===
The Whitney trick was used by [[Stephen Smale]] to prove the [[h-cobordism theorem|''h''-cobordism theorem]]; from which follows the [[Poincaré conjecture]] in dimensions {{math|''m'' ≥ 5}}, and the classification of [[smooth structure]]s 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==
{{see also|History of manifolds and varieties}}
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 results==
Although every {{mvar|n}}-manifold embeds in {{tmath|\R^{2n},}} one can frequently do better.  Let {{math|''e''(''n'')}} denote the smallest integer so that all compact connected {{mvar|n}}-manifolds embed in {{tmath|\R^{e(n)}.}} Whitney's strong embedding theorem states that {{math|''e''(''n'') ≤ 2''n''}}.  For {{math|1=''n'' = 1, 2}} we have {{math|1=''e''(''n'') = 2''n''}}, as the [[circle]] and the [[Klein bottle]] show. More generally, for {{math|1=''n'' = 2<sup>''k''</sup>}} we have {{math|1=''e''(''n'') = 2''n''}}, as the {{math|2<sup>''k''</sup>}}-dimensional [[real projective space]] show. Whitney's result can be improved to {{math|''e''(''n'') ≤ 2''n'' − 1}} unless {{mvar|n}} is a power of 2. This is a result of [[André Haefliger]] and [[Morris Hirsch]] (for {{math|''n'' > 4}}) and [[C. T. C. Wall]] (for {{math|1=''n'' = 3}}); these authors used important preliminary results and particular cases proved by Hirsch, [[William S. Massey]], [[Sergei Novikov (mathematician)|Sergey Novikov]] and [[Vladimir Rokhlin (Soviet mathematician)|Vladimir Rokhlin]].<ref name=skopenkov2>See section 2 of Skopenkov (2008)</ref> At present the function {{mvar|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|{{mvar|n}}-sphere]] always embeds in {{tmath|\R^{n+1} }}&nbsp;– which is the best possible (closed {{mvar|n}}-manifolds cannot embed in {{tmath|\R^n}}). Any compact ''orientable'' surface and any compact surface ''with non-empty boundary'' embeds in {{tmath|\R^3,}} though any ''closed non-orientable'' surface needs {{tmath|\R^4.}}

If {{mvar|N}} is a compact orientable {{mvar|n}}-dimensional  manifold, then {{mvar|N}} embeds in {{tmath|\R^{2n-1} }} (for {{mvar|n}} not a power of 2 the orientability condition is superfluous). For {{mvar|n}} a power of 2 this is a result of [[André Haefliger]] and [[Morris Hirsch]] (for {{math|''n'' > 4}}), and Fuquan Fang (for {{math|1=''n'' = 4}}); 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 {{mvar|N}} is a compact {{mvar|n}}-dimensional [[n-connected|{{mvar|k}}-connected]] manifold, then {{mvar|N}} embeds in {{tmath|\R^{2n-k} }} provided {{math|2''k'' + 3 ≤ ''n''}}.<ref name=skopenkov2/>

==Isotopy versions==
A relatively 'easy' result is to prove that any two embeddings of a 1-manifold into {{tmath|\R^4}} are isotopic (see [[Knot theory#Higher dimensions]]). This is proved using general position, which also allows to show that any two embeddings of an {{mvar|n}}-manifold into {{tmath|\R^{2n+2} }} are isotopic. This result is an isotopy version of the weak Whitney embedding theorem.

Wu proved that for {{math|''n'' ≥ 2}}, any two embeddings of an {{mvar|n}}-manifold into {{tmath|\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, [[André Haefliger|Haefliger]] proved that if {{mvar|N}} is a compact {{mvar|n}}-dimensional {{mvar|k}}-connected manifold, then any two embeddings of {{mvar|N}} into {{tmath|\R^{2n-k+1} }} are isotopic provided {{math|2''k'' + 2 ≤ ''n''}}. The dimension restriction {{math|2''k'' + 2 ≤ ''n''}} is sharp: Haefliger went on to give examples of non-trivially embedded 3-spheres in {{tmath|\R^6}} (and, more generally, {{math|(2''d'' − 1)}}-spheres in {{tmath|\R^{3d} }}). See [http://www.map.mpim-bonn.mpg.de/High_codimension_embeddings:_classification further generalizations].

==See also==
* {{annotated link|Representation theorem}}
* {{annotated link|Whitney immersion theorem}}
* {{annotated link|Nash embedding theorem}}
* {{annotated link|Takens's theorem}}
* {{annotated link|Nonlinear dimensionality reduction}}
* {{annotated link|Universal space}}

==Notes==
{{reflist}}

==References==
*{{citation |title=Collected Papers |first=Hassler |last=Whitney |author-link=Hassler Whitney|editor1-first=James |editor1-last=Eells |editor1-link=James Eells|editor2-first=Domingo |editor2-last=Toledo |year=1992 |location=Boston |publisher=Birkhäuser |isbn=0-8176-3560-2 }}
*{{citation |title=Lectures on the ''h''-cobordism theorem |author-link=John Milnor |first=John |last=Milnor |publisher=Princeton University Press |year=1965 }}
*{{citation |url=https://books.google.com/books?id=JcMwHWSBSB4C |title=Embeddings and Immersions |first=Masahisa |last=Adachi |translator-first=Kiki |translator-last=Hudson |year=1993 |publisher=American Mathematical Society |isbn=0-8218-4612-4 }}
*{{Citation | last1=Skopenkov | first1=Arkadiy | contribution=Embedding and knotting of manifolds in Euclidean spaces | arxiv=math/0604045 | year=2008 | title=Surveys in Contemporary Mathematics|editor1= Nicholas Young |editor2=Yemon Choi|series= London Math. Soc. Lect. Notes. | publisher= [[Cambridge University Press]]|location= Cambridge |volume=347 | issue=2 | pages=248–342| bibcode=2006math......4045S|mr=2388495 }}

==External links==
* [http://www.map.mpim-bonn.mpg.de/Embeddings_in_Euclidean_space:_an_introduction_to_their_classification Classification of embeddings]

{{Manifolds}}

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[[Category:Theorems in differential topology]]