Now this model must be embedded in so that the actual manifolds and and two points of intersection and correspond to the manifolds and points in the model.
If both and are connected, then the arcs and exist, and if and are simply connected (as they often are in applications), then the arcs are unique up to homotopy. If is simply connected, then the disk can be mapped into . If not, then must be connected by an arc (unique up to homotopy if is simply connected) to a base point which is connected by an arc to a base point . Similarly with arcs to a base point . It follows that then determines an element of by running from to to to and back to . Now if and both represent the same element of , then we can still map a disk into . (This is important in proving the -cobordism theorem.)
Once is mapped into , we can embed it if the dimension of , , is five or more. Furthermore, if each of and is three or more, then the embedding of can be chosen to miss and except along its boundary.
Now that is embedded missing and , it remains to find the embedding of the normal bundle of . The normal -bundle to (in fact, ) in can be split along as the normal -bundle to in direct sum
the orthogonal -bundle. That splitting extends across . The only problem remaining is that this -plane bundle may not coincide with the -plane bundle which is the normal bundle to in .
The problem reduces to an arc of -planes in which we want to isotope to the trivial arc, relative to the endpoints.
Note that the trivial arc, as in the model, corresponds to and having opposite signs, so this is necessary.
Now, this is possible because the fundamental group of the Stiefel manifold of -planes in is trivial when
(see
[e3],
p. 202).
For more details, see the excellent description in
[e4].
Whitney developed the Whitney trick in order to embed in
[e1].
For , this is easy. In higher dimensions, only immerses in (by general position), so for each double point, Whitney introduces in local fashion another double point of opposite sign (some thought is needed if 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 -cobordism theorem
[e2].