#### by Rob Kirby

The
Whitney trick is a method for removing points of intersection between two submanifolds.
It can be seen in its most elementary form in
Figure 1,
in which it is obvious that the two points of intersection can be removed by an isotopy (a 1-parameter family of embeddings) of the arc labeled __\( P^p \)__
which pulls the arc across the disk __\( D \)__.
(Note that __\( x \)__ and __\( y \)__ have opposite signs.)
More generally the Whitney trick is used to remove a pair of intersections, __\( x \)__ and __\( y \)__, between two manifolds __\( P^p \)__ and __\( Q^q \)__ which are embedded in an ambient manifold __\( M^{p+q} \)__. To see how this is done, we first construct a model, then show how to embed it in __\( M \)__ (if possible), and then sketch some applications of the
Whitney trick.

The model is merely a *stabilization* of the example in
Figure 1.
We cross the plane in which __\( D \)__ is embedded with __\( \mathbb R^{(p-1) + (q-1)} \)__ so that the ambient space is just __\( \mathbb R^{p+q} \)__, and then we cross the curve which includes __\( \alpha \)__ by __\( \mathbb R^{p-1} \)__ to get an __\( p \)__-dimensional manifold __\( P \)__, and similarly cross with __\( \mathbb R^{q-1} \)__ to get an __\( q \)__-manifold __\( Q \)__. These two manifolds still meet in two points __\( x \)__ and __\( y \)__, which are connected in __\( P \)__ by the original arc __\( \alpha \)__ and in __\( Q \)__ by the original arc __\( \beta \)__. Note that the two arcs still bound a 2-dimensional disk __\( D \)__, and that __\( D \)__ lies inside a larger open disk __\( \Delta \)__ in the plane. Also note that __\( \Delta \)__ has a normal __\( (p-1)+(q-1) \)__-plane bundle which splits as the direct sum
(also called “Whitney sum”)
of a __\( (p-1) \)__-plane bundle which coincides along __\( \alpha \)__ with the normal bundle of __\( \alpha \)__ in __\( P \)__, and an __\( (q-1) \)__-plane bundle which coincides along __\( \beta \)__ with the normal bundle of __\( \beta \)__ in __\( Q \)__.

The plane isotopy described in
Figure 1
easily extends to an isotopy taking place in the plane crossed with the __\( p-1 \)__ coordinates of __\( P \)__, as drawn for __\( p=2 \)__ in
Figure 2
[e4];
nothing happens with the other __\( q-1 \)__ coordinates.

Now this model must be embedded in __\( M^{p+q} \)__ so that the actual manifolds __\( P \)__ and __\( Q \)__ and two points of intersection __\( x \)__ and __\( y \)__ correspond to the manifolds and points in the model.

If both __\( P \)__ and __\( Q \)__ are connected, then the arcs __\( \alpha \)__ and __\( \beta \)__ exist, and if __\( P \)__ and __\( Q \)__ are simply connected (as they often are in applications), then the arcs are unique up to homotopy. If __\( M \)__ is simply connected, then the disk __\( D \)__ can be mapped into __\( M \)__. If not, then __\( x \)__ must be connected by an arc (unique up to homotopy if __\( P \)__ is simply connected) to a base point __\( x_0 \in P \)__ which is connected by an arc to a base point __\( z \in M \)__. Similarly with arcs to a base point __\( y_0 \in Q \)__. It follows that __\( x \)__ then determines an element of __\( \pi_1(M) \)__ by running from __\( z \)__ to __\( x_0 \)__ to __\( x \)__ to __\( y_0 \)__ and back to __\( z \)__. Now if __\( x \)__ and __\( y \)__ both represent the same element of __\( \pi_1(M) \)__, then we can still map a disk __\( D \)__ into __\( M \)__. (This is important in proving the __\( s \)__-cobordism theorem.)

Once __\( D \)__ is mapped into __\( M \)__, we can embed it if the dimension of __\( M \)__, __\( p+q \)__, is five or more. Furthermore, if each of __\( p \)__ and __\( q \)__ is three or more, then the embedding of __\( D \)__ can be chosen to miss __\( P \)__ and __\( Q \)__ except along its boundary.

Now that __\( D \)__ is embedded missing __\( P \)__ and __\( Q \)__, it remains to find the embedding of the normal bundle of __\( D \)__. The normal __\( (p+q-2) \)__-bundle to __\( D \)__ (in fact, __\( \Delta \)__) in __\( M \)__ can be split along __\( \alpha \)__ as the normal __\( (p-1) \)__-bundle to __\( \alpha \)__ in __\( P \)__ direct sum
the orthogonal __\( (q-1) \)__-bundle. That splitting extends across __\( \Delta \)__. The only problem remaining is that this __\( (p-1) \)__-plane bundle may not coincide with the __\( (p-1) \)__-plane bundle which is the normal bundle to __\( \beta \)__ in __\( Q \)__.

The problem reduces to an arc of __\( (p-1) \)__-planes in __\( \mathbb R^{(p-1) + (q-1)} \)__ which we want to isotope to the trivial arc, relative to the endpoints.
Note that the trivial arc, as in the model, corresponds to __\( x \)__ and __\( y \)__ having opposite signs, so this is necessary.
Now, this is possible because the fundamental group of the Stiefel manifold of __\( (p-1) \)__-planes in __\( \mathbb R^{p+q-2} \)__ is trivial when __\( p > 2 \)__
(see
[e3],
p. 202).
For more details, see the excellent description in
[e4].

Whitney developed the Whitney trick in order to embed __\( P^p \)__ in __\( \mathbb R^{2p} \)__
[e1].
For __\( p=2 \)__, this is easy. In higher dimensions, __\( P \)__ only immerses in __\( \mathbb R^{2p} \)__ (by general position), so for each double point, Whitney introduces in local fashion another double point of opposite sign (some thought is needed if __\( P \)__ is non-orientable), and then uses the Whitney trick to remove both points of intersection.

A later, and crucial, use of the Whitney trick is in Smale’s proof of the __\( h \)__-cobordism theorem
[e2].