Diffeomorphism
From Academic Kids

In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. The mathematical definition is as follows. Given two differentiable manifolds M and N, a bijective map <math>f<math> from M to N is called a diffeomorphism if both <math>f:M\to N<math> and its inverse <math>f^{1}:N\to M<math> are smooth.
Two manifolds M and N are diffeomorphic (symbol being usually <math>\simeq<math>) if there is a diffeomorphism <math>f<math> from M to N. For example
 <math>\mathbb{R}/\mathbb{Z} \simeq S^1.<math>
That is, the quotient group of the real numbers modulo the integers is again a smooth manifold, which is diffeomorphic to the 1sphere, usually known as the circle. The diffeomorphism is given by
 <math>x\mapsto e^{ix}.<math>
This map provides not only a diffeomorphism, but also an isomorphism of Lie groups between the two spaces.
Contents 
Local description
Model example: if <math>U<math> and <math>V<math> are two open subsets of <math>\mathbb{R}^n<math>, a differentiable map <math>f<math> from <math>U<math> to <math>V<math> is a diffeomorphism if
 it is a bijection,
 its differential <math>df<math> is invertible (as the matrix of all <math>\partial f_i/\partial x_j<math>, <math>1 \leq i,j \leq n<math>).
Remarks:
 Condition 2 excludes diffeomorphisms going from dimension <math>n<math> to a different dimension <math>k<math> (the matrix of <math>df<math> would not be square hence certainly not invertible).
 A differentiable bijection is not necessarily a diffeomorphism, e.g. <math>f(x)=x^3<math> is not a diffeomorphism from <math>\mathbb{R}<math> to itself because its derivative vanishes at 0.
 <math>f<math> also happens to be a homeomorphism.
Now, <math>f<math> 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 <math>\phi<math> and <math>\psi<math> be charts on M and N respectively, with <math>U<math> being the image of <math>\phi<math> and <math>V<math> the image of <math>\psi<math>. Then the conditions says that the map <math>\psi f \phi^{1}<math> from <math>U<math> to <math>V<math> is a diffeomorphism as in the definition above (whenever it makes sense). One has to check that for every couple of charts <math>\phi<math>, <math>\psi<math> 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 selfdiffeomorphisms. For dimension ≥ 1 this is a large group (too big to be a Lie 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).
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 7dimensional manifold (called now Milnor's sphere) which is homeomorphic to the standard 7sphere but not diffeomorphic to it. There are in fact 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7sphere (each of them is a fiber bundle over the 4sphere with fiber the 3sphere).
Much more extreme phenomena occur: in the early 1980s, a combination of results due to Fields Medal winners Simon Donaldson and Michael Freedman led to the discoveries that there are uncountably many pairwise nondiffeomorphic open subsets of <math>\mathbb{R}^4<math> each of which is homeomorphic to <math>\mathbb{R}^4<math>, and also that there are uncountably many pairwise nondiffeomorphic differentiable manifolds homeomorphic to <math>\mathbb{R}^4<math> which do not embed smoothly in <math>\mathbb{R}^4<math>.