# Diffeomorphism

A diffeomorphism is an infinitely differentiable homeomorphism with an infinitely differentiable inverse. Diffeomorphisms provide the right notion for two smooth manifolds to be "the same": if there exists a diffeomorphism, between two manifolds, then they will have all the same topological properties.

Put another way, a diffeomorphism is a type of function that satisfies the following condition: both the function and its inverse have continuous mixed partial derivatives of all orders in neighborhoods of every possible point.

## Examples

Let $\mathbb R^+$ denote the positive real numbers. The map $\phi : \R \to \R^+$ defined by φ(x) = ex is a diffeomorphism.

The quotient space $\mathbb R/\mathbb Z$ (that is, the real numbers modulo 1) is diffeomorphic to the unit circle S1 in the complex plane. $\phi : \mathbb R/\mathbb Z \to S^1$ defined by $[x] \mapsto e^{2\pi i x}$ provides a diffeomorphism. Note that this map is well-defined: if a and b represent the same class in $\mathbb R/\mathbb Z$, then a = b + n for some integer n. Then φ(a) = eia = eia + 2πni = eib = φ(b).

## Distinct from Homeomorphism

Two topological spaces are said to be homeomorphic if there exists a homeomorphism between them. In this case, the two spaces are topologically the same. Similarly, two smooth manifolds (which carry the structure of topological spaces) are said to be diffeomorphic if there exists a diffeomorphism between them. Since a diffeomorphism is a special kind of homeomorphism, any two smooth manifolds which are diffeomorphic are in fact homeomorphic. One might expect that if two smooth manifolds are homeomorphic (i.e., equivalent as topological spaces) then they are diffeomorphic (i.e., equivalent as manifolds). However, this is not the case! It is possible to give the 7-dimensional sphere S7 the structure of a smooth manifold in such a way that it is not diffeomorphic to S7 with the standard smooth structure, though it is homeomorphic! In fact, there are exactly 28 such exotic structures on the 7-sphere up to diffeomorphism. Such examples are called exotic spheres. While the exotic spheres were the first known example of this phenomenon, it was later shown that even 4-dimension Euclidean space may be endowed with any of an uncountable number of non-diffeomorphic smooth structures.

In some important cases, however, the notions of diffeomorphism and homeomorphism coincide. It is a fundamental theorem of low-dimensional topology that two smooth 3-manifolds are diffeomorphic if and only if they are homeomorphic. Naturally, this must apply to lower dimensional manifolds as well. This is in stark contrast to higher dimensional manifolds - many 4-manifolds admit an infinite number of smooth structures, and many cannot have any smooth structure at all. Thus, there exist infinite families of 4-manifolds which are all homeomorphic, but none of which are diffeomorphic.