smooth structure

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.

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 $X$.

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

This was shown in (Stallings).

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

Many topological spaces have canonical or “obvious” smooth structures. For instance a Cartesian space $\mathbb{R}^n$ 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

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

given by $S^n = \{\vec x \in \mathbb{R}^n | \sum_i (x^i)^2 = 1\}$ and this induces a smooth structure of $\mathbb{S}^n$.

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

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

Revised on May 11, 2013 15:39:22
by WCP?
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