In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a smooth structure on a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
allows for an unambiguous notion of
smooth function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
. In particular, a smooth structure allows one to perform
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
on the manifold.
Definition
A smooth structure on a manifold
is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold
is an
atlas
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth.
Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geographic ...
for
such that each
transition function is a
smooth map
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it ...
, and two smooth atlases for
are smoothly equivalent provided their
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
is again a smooth atlas for
This gives a natural
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
on the set of smooth atlases.
A
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
is a topological manifold
together with a smooth structure on
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
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
, then one can find an atlas with only finitely many charts.
Equivalence of smooth structures
Let
and
be two maximal atlases on
The two smooth structures associated to
and
are said to be equivalent if there is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
such that
Exotic spheres
John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
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
In mathematics, the ''E''8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the ''E''8 lattice.
History
The E_8 manifold was discovered by Michael Freedman in 1982. Rokhlin's theorem shows that ...
is an example of a
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
that does not admit a smooth structure. This essentially demonstrates that
Rokhlin's theorem
In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, closed 4-manifold ''M'' has a spin structure (or, equivalently, the second Stiefel–Whitney class w_2(M) vanishes), then the signature of its interse ...
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
-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
*
*
*
{{Manifolds
Differential topology
Structures on manifolds