Editing Diffeomorphism (section)

===Homotopy types===
* The diffeomorphism group of <math>S^2</math> has the homotopy-type of the subgroup <math>O(3)</math>. This was proven by Steve Smale.<ref>{{cite journal | last1 = Smale | year = 1959 | title = Diffeomorphisms of the 2-sphere | journal = Proc. Amer. Math. Soc. | volume = 10 | issue = 4| pages = 621–626 | doi=10.1090/s0002-9939-1959-0112149-8| doi-access = free }}</ref> 
* The diffeomorphism group of the torus has the homotopy-type of its linear [[automorphism]]s: <math>S^1\times S^1\times\text{GL}(2,\Z)</math>.
* The diffeomorphism groups of orientable surfaces of [[Genus (mathematics)|genus]] <math>g>1</math> have the homotopy-type of their mapping class groups (i.e. the components are contractible).
* The homotopy-type of the diffeomorphism groups of 3-manifolds are fairly well understood via the work of Ivanov, Hatcher, Gabai and Rubinstein, although there are a few outstanding open cases (primarily 3-manifolds with finite [[fundamental group]]s).
* The homotopy-type of diffeomorphism groups of <math>n</math>-manifolds for <math>n>3</math> are poorly understood. For example, it is an open problem whether or not <math>\text{Diff}(S^4)</math> has more than two components. Via Milnor, Kahn and Antonelli, however, it is known that provided <math>n>6</math>, <math>\text{Diff}(S^n)</math> does not have the homotopy-type of a finite [[CW-complex]].