Smooth structure

 ( Differential Topology ) 

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.

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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 map 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 $M.$

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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 space, then one can find an atlas with only finitely many charts.

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Equivalence of smooth structures If $\backslash mu$ and $\backslash nu$ are two maximal atlases on $M$ the two smooth structures associated to $\backslash mu$ and $\backslash nu$ are said to be equivalent if there is a diffeomorphism $f\; :\; M\; \backslash to\; M$ such that $\backslash mu\; \backslash circ\; f\; =\; \backslash nu.$

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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.

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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.

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

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See also

• Morris Hirsch (1976). 9783540901488, Springer-Verlag. ISBN 9783540901488
• John M. Lee (2025). 9780387954486, Springer-Verlag. . ISBN 9780387954486
• Mark R. Sepanski (2025). 9780387302638, Springer-Verlag. ISBN 9780387302638

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