View source for Smooth structure

{{Short description|Maximal smooth atlas for a topological manifold}}
In [[Mathematics|mathematics]], a '''smooth structure''' on a [[Manifold|manifold]] allows for an unambiguous notion of smooth function. In particular, a smooth structure allows [[Mathematical analysis|mathematical analysis]] to be performed on the manifold.<ref>{{cite journal|author=Callahan, James J.|title=Singularities and plane maps|journal=Amer. Math. Monthly|volume=81|year=1974|pages=211–240|url=http://www.maa.org/programs/maa-awards/writing-awards/singularities-and-plane-maps|doi=10.2307/2319521}}</ref>

== Definition ==

A smooth structure on a manifold <math>M</math> is a collection of smoothly equivalent smooth atlases. Here, a '''smooth atlas''' for a topological manifold <math>M</math> is an [[Atlas (topology)|atlas]] for <math>M</math> such that each transition function is a smooth map, and two smooth atlases for <math>M</math> are '''smoothly equivalent''' provided their [[Union (set theory)|union]] is again a smooth atlas for <math>M.</math> This gives a natural [[Equivalence relation|equivalence relation]] on the set of smooth atlases.

A smooth manifold is a topological manifold <math>M</math> together with a smooth structure on <math>M.</math>

=== Maximal smooth atlases ===

By taking the union of all [[Atlas (topology)|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|compact]], then one can find an atlas with only finitely many charts.

=== Equivalence of smooth structures ===

If <math>\mu</math> and <math>\nu</math> are two maximal atlases on <math>M</math> the two smooth structures associated to <math>\mu</math> and <math>\nu</math> are said to be equivalent if there is a [[Diffeomorphism|diffeomorphism]] <math>f : M \to M</math> such that <math>\mu \circ f = \nu.</math> {{Citation needed|reason=It's not clear what composing a function with an atlas should mean here. |date=June 2020}}

== Exotic spheres ==

[[Biography:John Milnor|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|exotic sphere]].

== E8 manifold ==

The [[E8 manifold]] is an example of a [[Topological manifold|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 <math>k</math>-times continuously differentiable; or strengthened, so that the transition maps are required to be real-analytic. Accordingly, this gives a <math>C^k</math> or [[Analytic manifold|(real-)analytic structure]] on the manifold rather than a smooth one. Similarly, a [[Complex manifold|complex structure]] can be defined by requiring the transition maps to be holomorphic.

== See also==

* {{annotated link|Atlas (topology)}}

== References ==

{{reflist}}
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* {{cite book | first = Morris | last = Hirsch | year = 1976 | title = Differential Topology | publisher = Springer-Verlag | isbn = 3-540-90148-5}}
* {{cite book | first = John M. | last = Lee | year = 2006 | title = Introduction to Smooth Manifolds | publisher = Springer-Verlag | isbn = 978-0-387-95448-6 | url=https://books.google.com/books/about/Introduction_to_Smooth_Manifolds.html?id=eqfgZtjQceYC}}
* {{cite book | first = Mark R. | last = Sepanski | year = 2007 | title = Compact Lie Groups | publisher = Springer-Verlag | isbn = 978-0-387-30263-8}}
{{refend}}


[[Category:Differential topology]]
[[Category:Structures on manifolds]]

{{Sourceattribution|Smooth structure}}