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Milds # Smooth structure

In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold.

## Definition

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

A smooth manifold is a topological manifold $M$ together with a smooth structure on $M.$ ### 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 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

Let $\mu$ and $\nu$ be two maximal atlases on $M.$ The two smooth structures associated to $\mu$ and $\nu$ are said to be equivalent if there is a diffeomorphism $f:M\to M$ such that $\mu \circ f=\nu .$ [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 we only require the transition maps to be $k$ -times continuously differentiable; or strengthened, so that we require the transition maps to be real-analytic. Accordingly, this gives a $C^{k}$ or (real-)analytic structure on the manifold rather than a smooth one. Similarly, we can define a complex structure by requiring the transition maps to be holomorphic.