# What is the significance of incompatible coordinate charts for a manifold?

For reference, here is my definition of a "manifold".

A $\,C^\infty$ manifold is a topological manifold together with all the admissible charts of some $C^\infty$ atlas.

When considering the real line, the standard manifold structure given by the atlas $\{f(x): x \mapsto x\}$ is different from the manifold structure given by the atlas $\{g(x): x \mapsto x^{3}\}$ because $f$ and $g$ are not compatible.

What are the significant differences in the manifolds given by these two atlases? Why would I care which selection of admissible charts I am given?

In this case, the manifolds end up being the same, in the sense that there is a diffeomorphism between them. In general, a smooth manifold of dimension less than 4 can be given only one unique differential structure up to diffeomorphism. In dimension 4 and higher, one can endow sets with multiple distinct atlases. In dimension 4, one can endow $\mathbb{R}^4$ with an atlas such that the manifold is homeomorphic but not diffeomorphic to the standard $\mathbb{R}^4$; such manifolds are called "exotic $\mathbb{R}^4$'s".