Editing Diffeomorphism (section)

===Examples===
* When <math>M=G</math> is a [[Lie group]], there is a natural inclusion of <math>G</math> in its own diffeomorphism group via left-translation. Let <math>\text{Diff}(G)</math> denote the diffeomorphism group of <math>G</math>, then there is a splitting <math>\text{Diff}(G)\simeq G\times\text{Diff}(G,e)</math>, where <math>\text{Diff}(G,e)</math> is the [[subgroup]] of <math>\text{Diff}(G)</math> that fixes the [[identity element]] of the group.
* The diffeomorphism group of Euclidean space <math>\R^n</math> consists of two components, consisting of the orientation-preserving and orientation-reversing diffeomorphisms. In fact, the [[general linear group]] is a [[deformation retract]] of the subgroup <math>\text{Diff}(\R^n,0)</math> of diffeomorphisms fixing the origin under the map <math>f(x)\to f(tx)/t, t\in(0,1]</math>.  In particular, the general linear group is also a deformation retract of the full diffeomorphism group.
* For a finite [[Set (mathematics)|set]] of points, the diffeomorphism group is simply the [[symmetric group]]. Similarly, if <math>M</math> is any manifold there is a [[group extension]] <math>0\to\text{Diff}_0(M)\to\text{Diff}(M)\to\Sigma(\pi_0(M))</math>. Here <math>\text{Diff}_0(M)</math> is the subgroup of <math>\text{Diff}(M)</math> that preserves all the components of <math>M</math>, and <math>\Sigma(\pi_0(M))</math> is the permutation group of the set <math>\pi_0(M)</math> (the components of <math>M</math>). Moreover, the image of the map <math>\text{Diff}(M)\to\Sigma(\pi_0(M))</math> is the bijections of <math>\pi_0(M)</math> that preserve diffeomorphism classes.