Editing Whitney embedding theorem (section)

=== 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>'')}}.