The Diffeomorphism reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Diffeomorphism

In mathematics, the idea of a diffeomorphism is to be able to have a notion of isomorphism of smooth manifolds. Here is definition

Given two differentiable manifolds M and N, a bijective map from M to N is called a diffeomorphism if both and its inverse are smooth.

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

.

Local description

Model example: if and are two open subsets of , a differentiable map from to is a diffeomorphism if

1. it is a bijection,
2. its differential is invertible (as the matrix of all , ).

Remarks:

• Condition 2 excludes diffeomorphisms going from dimension to a different dimension (the matrix of would not be square hence certainly not invertible).
• A differentiable bijection is not necessarily a diffeomorphism, e.g. is not a diffeomorphism from to itself because its derivative vanishes at 0.
• also happens to be a homeomorphism.

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

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 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: 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 non-diffeomorphic open subsets of each of which is homeomorphic to , and also that there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to which do not embed smoothly in .

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