User:MWinter4/Whitney trick

In geometric topology, the Whitney trick is a technique used to remove a pair of intersections between two orientable differentiable submanifolds (or a pair self-intersections of a single submanifold). It was initially introduced by Hassler Whitney for proving the Whitney embedding theorem, though has found many applications since.

The Whitney trick can be used to remove a pair of transversal points of intersection of opposite signs. To apply it one also needs to provide a particular embedded 2-dimensional disc called the Whitney disc. Intuitively, one gets rid of the intersections by pulling one submanifold over the other one guided by the Whitney disc. This disc can always be found if the ambient space has dimension at least five and the submanifolds are of codimension at least two. This is a main reason for why geometric topology in higher dimensions is of an essentially different nature, and why dimension three and four are generally seen as the most challenging.

Setting

Let $M,N\subset {\mathbb {R}}^{d}$ be orientable differentiable submanifold of dimension $m$ and $n$ respectively. If $m+n<d$, then intersections between these submanifolds can be removed by perturbing them ever so slightly. Liekwise, if $m+n=d$ then then by perturbing them one can arrange that they intersect transversally in isolated points. Since the submanifolds are orientable, each point of intersection can be assigned a signs denoting the orientation. The Whitney trick is used to remove a pair of intersection points of opposite sign.

Let $x^{+},x^{-}\in M\cap N$ be intersection points of opposite sign. To apply the Whitney trick to remove both intersections one has to provide a Whitney disc $W\subset {\mathbb {R}}^{d}$, an embedded 2-dimensional disc with for which

$\gamma _{M}:=W\cap M,\quad \gamma _{N}:=W\cap N$

are curves in $M$ and $N$ respectively whose end points are $x^{+}$ and $x^{-}$, and that do not pass through any other intersection points.

If $d\geq 5$ and $m,n\leq d-3$ then one can construct a Whitney disc from any embedded disc and remove self-intersections and intersections with $M$ and $N$ by perturbation. In these circumstances the Whitney trick therefore applies without further assumptions. In dimension $d\leq 4$ or if the submanifolds are of codimension $\leq 2$, constructing the Whitney disc can be rather involved.

Perfoming the Whitney trick

The Whitney trick is performed by showing that the setting is locally diffeomorphic to a model configuration in which the intersections can be removed easily. For this, let $B\subset {\mathbb {R}}^{d}$ be an open neighborhood of the Whitney disc $W$ whose closure is diffeomorphic to ball. One shows that $(B,B\cap M,B\cap N)$ is diffeomorphic to $({\mathbb {R}}^{d},M',N')$, where

$M':=U\times {\mathbb {R}}^{m-1}\times \{0\}^{n-1}\;\;$ and $\;\;N':=I\times \{0\}^{m-1}\times {\mathbb {R}}^{n-1}$

with $U,I\subset {\mathbb {R}}^{2}$ being curves in a configuration as shown in the figure. In this model configuration a simple translation of $M'$ will remove the intersections.

Comments

The Whitney trick as presented here only applies in the smooth category as the Whitney disc is sufficiently nice. The same trick cannot be used to remove self-intersections in the PL category as the Whitney disc might be internally knotted.

In dimension $d=4$ the Whitney disc might intersect itself, and due to missing codimension, the self-intersections cannot be removed by a perturbation. One can then attempt to remove self-intersections using the Whitney trick on the disc itself, which requires another Whitney disc. This new Whitney disc might again intersect itself. One can then apply the trick recursively and potentially infinitely many times. This leads to the notions of a Casson tower. It was shown by Friedmann that this structure is itself homeomorphic to a disc and can be used to perform the Whitney trick.