# 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.^{[1]}

## 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 and be two
maximal atlases on *M*.
The two smooth structures associated to
and
are said to be equivalent if there is a homeomorphism
such that .

## 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 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.

## See also

## References

- ↑ Callahan, James J. (1974). "Singularities and plane maps".
*Amer. Math. Monthly*.**81**: 211–240. doi:10.2307/2319521.

- Hirsch, Morris (1976).
*Differential Topology*. Springer-Verlag. ISBN 3-540-90148-5. - Lee, John M. (2006).
*Introduction to Smooth Manifolds*. Springer-Verlag. ISBN 978-0-387-95448-6. - Sepanski, Mark R. (2007).
*Compact Lie Groups*. Springer-Verlag. ISBN 978-0-387-30263-8.