Why would surgery theory require $5$ dimensions?

Question

In the Wikipedia page for geometric topology it says "The Whitney trick requires $2+1$ dimensions, hence surgery theory requires $5$ dimensions". I am having trouble with understanding why surgical theory would have to require five dimensions and why it would not work with less dimensions.

Answer 1

It's a complicated subject, but the main idea is that you need to turn collections of spheres $S^n$ in $2n$-dimensions into disjoint embeddings in order to get surgery to work. The main tool for turning a sphere into an embedded sphere is the Whitney trick. If a sphere intersects another sphere in two algebraically cancelling points, form a circle by connecting the two intersection points by arcs on each of the two spheres. If this is null-homotopic, you can fill it in with a disk. In dimensions higher than $4$, this disk is generically embedded, so can be used to guide a Whitney move that removes the two canceling intersections. However, in dimension $4$, the Whitney disk generically intersects both itself and the spheres.

This introduction to an old paper of mine should also help give context. Even though the Whitney move fails in this dimension, we looked at iterated towers of Whitney moves, which did lead to interesting information.