# Extensive Definition

In mathematics, a
diffeomorphism is a kind of 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 smooth.

Given two manifolds M and N, a bijective
map f from M to N is called a diffeomorphism if both

- f:M\to N

and its inverse

- f^:N\to M

are differentiable (if these functions are r
times continuously differentiable, f is called a
C^r-diffeomorphism).

Two manifolds M and N are diffeomorphic (symbol
being usually \simeq) if there is a diffeomorphism f from M to
N.

## Examples

- \mathbb/\mathbb \simeq S^1.

That is, the quotient
group of the real numbers
modulo the integers is again a smooth
manifold, which is diffeomorphic to the 1-sphere, usually known as the
circle. The diffeomorphism is given by

- x\mapsto e^.

This map provides not only a diffeomorphism, but
also an isomorphism
of Lie
groups between the two spaces.

## Local description

Model example: if U and V are two open subsets of \mathbb^n, a differentiable map f from U to V is a diffeomorphism if- it is a bijection,
- its derivative Df is invertible (as the matrix of all \partial f_i/\partial x_j, 1 \leq i,j \leq n), which means the same as having non-zero Jacobian determinant.

- Condition 2 excludes diffeomorphisms going from dimension n to a different dimension k (the matrix Df would not be square hence certainly not invertible).
- A differentiable bijection is not necessarily a diffeomorphism, e.g. f(x)=x^3 is not a diffeomorphism from \mathbb to itself because its derivative vanishes at 0 (and hence its inverse is not differentiable at 0).
- f also happens to be a homeomorphism, as easily follows from the inverse function theorem

Now, f from M to N is called a diffeomorphism if
in
coordinates charts it satisfies the definition above. More
precisely, pick any cover of M by compatible
coordinate charts, and do the same for N. Let \phi and \psi be
charts on M and N respectively, with U being the image of \phi and
V the image of \psi. Then the conditions says that the map \psi f
\phi^ from U to V is a diffeomorphism as in the definition above
(whenever it makes sense). One has to check that for every couple
of charts \phi, \psi of two given
atlases, but once checked, it will be true for any other
compatible chart. Again we see that dimensions have to agree.

## Diffeomorphism group

The diffeomorphism group of a manifold is the group of all its automorphisms (diffeomorphisms to itself). For dimension greater than or equal to one this is a large group. For a connected manifold M the diffeomorphisms act transitively on M: this is true locally because it is true in Euclidean space and then a topological argument shows that given any p and q there is a diffeomorphism taking p to q. That is, all points of M in effect look the same, intrinsically. The same is true for finite configurations of points, so that the diffeomorphism group is k- fold multiply transitive for any integer k ≥ 1, provided the dimension is at least two (it is not true for the case of the circle or real line). This group can be given the structure of an infinite dimensional Lie group, modeled on the space of vector fields on the manifold. In general, this will not be a Banach Lie group, and the exponential map will not be a local diffeomorphism.In 1926, Tibor
Radó asked whether the harmonic
extension of any homeomorphism (or diffeomorphism) of the
unit
circle to the unit disc
yields a diffeomorphism on the open disc. An elegant proof was
provided shortly afterwards by Hellmuth
Kneser and a completely different proof was discovered in 1945
by Gustave
Choquet, apparently unaware that the theorem was already
known.

The (orientation-preserving) diffeomorphism group
of the circle is pathwise connected. This can be seen by noting
that any such diffeomorphism can be lifted to a diffeomorphism f of
the reals satisfying f(x+1) = f(x) +1; this space is convex and
hence path connected. A smooth eventually constant path to the
identity gives a second more elementary way of extending a
diffeomorphism from the circle to the open unit disc (this is a
special case of the Alexander
trick).

The corresponding extension problem for
diffeomorphisms of higher dimensional spheres Sn-1 was much studied
in the 1950s and 1960s, with notable contributions from René Thom,
John
Milnor and Stephen
Smale. An obstruction to such extensions is given by the
finite
Abelian group Γn, the "group
of twisted spheres", defined as the quotient
of the Abelian component
group of the diffeomorphism group by the subgroup of classes
extending to diffeomorphisms of the ball Bn.

For oriented manifolds of dimension >1, the
diffeomorphism group is usually not connected. Its component group
is called the mapping
class group. In dimension 2, i.e. for surfaces, the mapping class
group is a finitely
presented group, generated by Dehn twists
(Dehn,
Lickorish,
Hatcher).
Max Dehn
and
Jakob Nielsen showed that it can be identified with the
outer automorphism group of the fundamental
group of the surface.

William
Thurston refined this analysis by
classifying elements of the mapping class group into three
types: those equivalent to a
periodic diffeomorphism; those equivalent to a diffeomorphism
leaving a simple closed curve invariant; and those equivalent to
pseudo-Anosov
diffeomorphisms. In the case of the torus S¹ x S¹ = R²/Z², the mapping
class group is just the modular
group SL(2,Z) and the classification reduces to the classical
one in terms of
elliptic,
parabolic and
hyperbolic matrices. Thurston accomplished his classification
by observing that the mapping class group acted naturally on a
compactification of
Teichmuller
space; since this enlarged space was homeomorphic to a closed
ball, the
Brouwer fixed point theorem became applicable.

If M is an oriented smooth closed manifold, it
was conjectured by Smale that the identity
component of the group of orientation-preserving
diffeomorphisms is simple. This had first been proved for a product
of circles by Michel
Herman; it was proved in full generality by Thurston.

## Homeomorphism and diffeomorphism

It is easy to find a homeomorphism which is not a diffeomorphism, but it is more difficult to find a pair of homeomorphic manifolds that are not diffeomorphic. In dimensions 1, 2, 3, any pair of homeomorphic smooth manifolds are diffeomorphic. In dimension 4 or greater, examples of homeomorphic but not diffeomorphic pairs have been found. The first such example was constructed by John Milnor in dimension 7, he constructed a smooth 7-dimensional manifold (called now Milnor's sphere) which is homeomorphic to the standard 7-sphere but not diffeomorphic to it. There are in fact 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere (each of them is a fiber bundle over the 4-sphere with fiber the 3-sphere).Much more extreme phenomena occur for 4-manifolds: in
the early 1980s, a combination of results due to Simon
Donaldson and Michael
Freedman led to the discovery of exotic R4s:
there are uncountably many pairwise non-diffeomorphic open subsets
of \mathbb^4 each of which is homeomorphic to \mathbb^4, and also
there are uncountably many pairwise non-diffeomorphic
differentiable manifolds homeomorphic to \mathbb^4 which do not
embed smoothly in \mathbb^4.

## See also

## References

diffeomorphism in German: Diffeomorphismus

diffeomorphism in French: Difféomorphisme

diffeomorphism in Italian: Diffeomorfismo

diffeomorphism in Hebrew: דיפאומורפיזם

diffeomorphism in Dutch: Diffeomorfisme

diffeomorphism in Polish: Dyfeomorfizm

diffeomorphism in Portuguese:
Difeomorfismo

diffeomorphism in Russian: Диффеоморфизм

diffeomorphism in Swedish: Diffeomorfi

diffeomorphism in Ukrainian: Дифеоморфізм

diffeomorphism in Chinese:
微分同胚