nLab
smooth structure

Idea

A differential structure on a topological space $X$ is the extra structure of a differential manifold on $X$. A smooth structure on $X$ is the extra structure of a smooth manifold.

Definition

For $k\in \mathbb{N}$ a $C^k$-differential structure on a topological space $X$ is a manifold $\hat{X}$ whose charts have transition functions that have continuous derivatives to degree $k$, such that $X$ is the topological space underlying $\hat{X}$.

A smooth structure on $X$ is a smooth manifold $\hat{X}$ (transition functions have derivatives to all degrees) with underlying topological space $X$.

Properties

Theorem

For $n\in \mathbb{N}$ a natural number with $n\ne 4$, there is a unique (up to isomorphism) smooth structure on the Cartesian space ${ℝ}^n$.

This was shown in (Stallings).

Theorem

In $d=4$ the analog of this statement is false. One says that on ${ℝ}^4$ there exist exotic smooth structures.

Exotic smooth structures

Many topological spaces have canonical or “obvious” smooth structures. For instance a Cartesian space ${ℝ}^n$ has the evident smooth structure induced from the fact that it can be covered by a single chart – itself.

From this example, various topological spaces inherit a canonical smooth structure by embedding. For instance the $n$-sphere may naturally be thought of as the collection of points

$$S^n\hookrightarrow {ℝ}^n$$S^n \hookrightarrow \mathbb{R}^n

given by $S^n=\{\stackrel{\rightharpoonup }{x}\in {ℝ}^n\mid {\sum }_i(x^i{)}^2=1\}$ and this induces a smooth structure of ${𝕊}^n$.

But there may be other, non-equivalent smooth structures than these canonical ones. These are called exotic smooth structures. See there for more details.

References

• John Stallings, The piecewise linear structure of Euclidean space , Proc. Cambridge Philos. Soc. 58 (1962), 481-488. (pdf)