# Diffeomorphism

A one-to-one continuously-differentiable mapping $f : M \rightarrow N$ of a differentiable manifold $M$( e.g. of a domain in a Euclidean space) into a differentiable manifold $N$ for which the inverse mapping is also continuously differentiable. If $f ( M) = N$, one says that $M$ and $N$ are diffeomorphic. From the point of view of differential topology, diffeomorphic manifolds have the same properties, and one is interested in a classification of manifolds up to a diffeomorphism (this classification is not identical with the coarser classification up to a homeomorphism, except for cases involving small dimensions).
Topological (more exactly, homotopic) properties of the group $\mathop{\rm Diff} M$ of all diffeomorphisms of a manifold $M$ onto itself, in which a topology has been introduced in a suitable manner, have also been studied. They may be unexpectedly complicated (see, e.g., , , , which also contain reviews and references). This problem is connected with a number of important problems in homotopic topology (e.g. with the homotopy groups of spheres). In principle, knowledge of the properties of $\mathop{\rm Diff} M$ would be of assistance in solving these problems, but at the time of writing (1987) the situation seems to be almost the opposite: Advances in the study of $\mathop{\rm Diff} M$ involve the use of the already known features of the problems or, at best, are realized in parallel with the solutions of these problems and by the same methods. As regards the algebraic properties of the group of diffeomorphisms of class $C ^ {r}$( including the case $r = \infty$) of a closed $n$- dimensional manifold, it has been proved that if $r \neq n + 1$, then its connected component of the unit is a simple group, i.e. has no non-trivial normal subgroups (see Normal subgroup; cf. , ; the situation is not clear for $r = n + 1$). As regards a non-closed $n$- dimensional manifold $M$, it has been proved that the group of all diffeomorphisms of class $C ^ {r}$( $r \neq n + 1$) which may be connected with the identity mapping $1 _ {M}$ by way of a continuous family of diffeomorphisms $f _ {t}$( $0 \leq t \leq 1$, $f _ {0} = 1 _ {M}$; $f _ {1} = f$) which does not displace points outside a certain compact set (depending on the family), is simple.