Diffeomorphism Guide, Meaning , Facts, Information and Description

In mathematics, a diffeomorphism is a kind 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

.

Table of contents

1 Local description 2 Diffeomorphism group 3 Homeomorphism and diffeomorphism

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.

Diffeomorphism group

The diffeomorphism group of a manifold is the group of all its self-diffeomorphisms. For dimension ≥ 1 this is a large group (too big to be a Lie group). For a connected manifold M the diffeomorphisms act transitivelyly 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 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: 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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