Differentiable structure

A differentiable structure on a topological manifold is a maximal smooth atlas — a collection of charts whose transition maps are all smooth (infinitely differentiable). It is what transforms a bare topological space into a geometric object on which calculus can be performed. The existence of multiple incompatible differentiable structures on the same topological manifold — most famously on the 7-sphere, giving rise to exotic spheres — revealed that smoothness is not determined by topology alone. A map between manifolds that preserves the differentiable structure is a diffeomorphism.