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Smooth structure

## Summary

In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows mathematical analysis to be performed on the manifold.[1]

## Definition

A smooth structure on a manifold ${\displaystyle M}$  is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold ${\displaystyle M}$  is an atlas for ${\displaystyle M}$  such that each transition function is a smooth map, and two smooth atlases for ${\displaystyle M}$  are smoothly equivalent provided their union is again a smooth atlas for ${\displaystyle M.}$  This gives a natural equivalence relation on the set of smooth atlases.

A smooth manifold is a topological manifold ${\displaystyle M}$  together with a smooth structure on ${\displaystyle M.}$

### Maximal smooth atlases

By taking the union of all atlases belonging to a smooth structure, we obtain a maximal smooth atlas. This atlas contains every chart that is compatible with the smooth structure. There is a natural one-to-one correspondence between smooth structures and maximal smooth atlases. Thus, we may regard a smooth structure as a maximal smooth atlas and vice versa.

In general, computations with the maximal atlas of a manifold are rather unwieldy. For most applications, it suffices to choose a smaller atlas. For example, if the manifold is compact, then one can find an atlas with only finitely many charts.

### Equivalence of smooth structures

If ${\displaystyle \mu }$  and ${\displaystyle \nu }$  are two maximal atlases on ${\displaystyle M}$  the two smooth structures associated to ${\displaystyle \mu }$  and ${\displaystyle \nu }$  are said to be equivalent if there is a diffeomorphism ${\displaystyle f:M\to M}$  such that ${\displaystyle \mu \circ f=\nu .}$  [citation needed]

## Exotic spheres

John Milnor showed in 1956 that the 7-dimensional sphere admits a smooth structure that is not equivalent to the standard smooth structure. A sphere equipped with a nonstandard smooth structure is called an exotic sphere.

## E8 manifold

The E8 manifold is an example of a topological manifold that does not admit a smooth structure. This essentially demonstrates that Rokhlin's theorem holds only for smooth structures, and not topological manifolds in general.

## Related structures

The smoothness requirements on the transition functions can be weakened, so that the transition maps are only required to be ${\displaystyle k}$ -times continuously differentiable; or strengthened, so that the transition maps are required to be real-analytic. Accordingly, this gives a ${\displaystyle C^{k}}$  or (real-)analytic structure on the manifold rather than a smooth one. Similarly, a complex structure can be defined by requiring the transition maps to be holomorphic.