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.

Definition

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
1. it is a bijection,
2. 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.
Remarks:
• 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.

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