Certainly one of the most useful things about flat Euclidean space, non-exotic R^n, is that it has a set of global smooth coordinates. But what are coordinates? When we talk about “manifolds”, we are talking about (topological) spaces that locally look like Euclidean space. For example, the surface of the Earth is an oblate 2d-spheroid, it can be approximated by a smooth manifold. But if you “zoom in” enough, as we are well aware, it looks like you can approximate it by a plane. The approximation depends on the curvature of the surface, more on that at a later date. (In fact, you cannot zoom in close enough on a sphere so as not to detect the curvature. This is Gauss’ celebrated “Remarkable Theorem”: https://en.wikipedia.org/wiki/Theorema_Egregium.)

Coordinates are, loosely, a way of setting up a local system of “graph paper” on a manifold so that one can do calculations the same way one does in regular, old, Euclidean space in calculus classes. Coordinates are a way of identifying the where and the how of points in the space and their relations to nearby points. To understand exotic manifolds, it’s necessary to think about coordinates as functions from the Euclidean approximation to the manifold itself.

If M is a 1-dimensional manifold, then there is a coordinate function x that associates to a real number p an element x(p) in M and upon which the function x:R^1 →U is a smooth correspondence on a neighborhood U of x(p). That mouthful is to say “locally M looks like R^1 with coordinate x(p)”.

Certainly an example is in order. Let M be the real line R^1. Then, a coordinate on M is x(p)=p. In other words, the real line is a manifold and you can put coordinates on it that match the numbers. It is this set of coordinates that provides R^1 with its standard smooth structure. Another coordinate on R^1 is x(p) = p^3. Readers with some calculus background will notice that the inverse function p(x) = x^(1/3) is not differentiable, or smooth, at the origin; it has a vertical rate of change:

So, if one wishes to define x(p) = p^3 as a coordinate function on the smooth manifold R^1, then technically you have a function (the reverse correspondence x^(1/3)) that is differentiable in this “new” R^1 that is not differentiable in the “old” R^1. This would indicate a difference in smooth structure.

To a physicist, this is not the kind of change in smooth structure that leads to exotic physics. Rather, what one should be concerned with is what are called isotopy classes of smooth functions. This is a technical definition that we have encountered before when we saw that the three sphere S^3 has no exotic smooth structure. What’s important here is

For n=1,2,3,5,6,…, R^n has exactly one smooth structure, up to isotopy.

What this means is that R^4 is the only Euclidean space where coordinate functions can act different from the way one would expect from calculus. Indeed, there are uncountably infinite non-equivalent smooth structures on R^4. These exotic Euclidean spaces will come in two sizes: large and small, and will often involve “wild” inclusions of 3D submanifolds in them, similar to the Witten trick for spheres.

As we move forward, we will discuss large and small exotic R^4s, and try to wrap our minds around how “exotic-ness” is a true, global property of a manifold.