nLab
smooth structure

Contents

Idea

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

Definition

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

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

Properties

Theorem

For nn \in \mathbb{N} a natural number with n4n \neq 4, there is a unique (up to isomorphism) smooth structure on the Cartesian space n\mathbb{R}^n.

This was shown in (Stallings).

Theorem

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

Exotic smooth structures

Many topological spaces have canonical or “obvious” smooth structures. For instance a Cartesian space n\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 nn-sphere may naturally be thought of as the collection of points

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

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

References

  • 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? (213.219.155.26)