Editing Diffeomorphism (section)

== Diffeomorphism group ==

Let <math>M</math> be a differentiable manifold that is [[second-countable]] and [[Hausdorff space|Hausdorff]]. The '''diffeomorphism group''' of <math>M</math> is the [[group (mathematics)|group]] of all <math>C^r</math> diffeomorphisms of <math>M</math> to itself, denoted by <math>\text{Diff}^r(M)</math> or, when <math>r</math> is understood, <math>\text{Diff}(M)</math>. This is a "large" group, in the sense that—provided <math>M</math> is not zero-dimensional—it is not [[locally compact]].

===Topology===
The diffeomorphism group has two natural [[Topological space|topologies]]: ''weak'' and ''strong'' {{harv|Hirsch|1997}}. When the manifold is [[Compact space|compact]], these two topologies agree.  The weak topology is always [[Metrizable space|metrizable]].  When the manifold is not compact, the strong topology captures the behavior of functions "at infinity" and is not metrizable.  It is, however, still [[Baire space|Baire]].

Fixing a [[Riemannian metric]] on <math>M</math>, the weak topology is the topology induced by the family of metrics
: <math>d_K(f,g) = \sup\nolimits_{x\in K} d(f(x),g(x)) + \sum\nolimits_{1\le p\le r} \sup\nolimits_{x\in K} \left \|D^pf(x) - D^pg(x) \right \|</math>
as <math>K</math> varies over compact subsets of <math>M</math>.  Indeed, since <math>M</math> is <math>\sigma</math>-compact, there is a sequence of compact subsets <math>K_n</math> whose [[Union (set theory)|union]] is <math>M</math>.  Then:
: <math>d(f,g) = \sum\nolimits_n 2^{-n}\frac{d_{K_n}(f,g)}{1+d_{K_n}(f,g)}.</math>

The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of <math>C^r</math> vector fields {{harv|Leslie|1967}}. Over a compact subset of <math>M</math>, this follows by fixing a Riemannian metric on <math>M</math> and using the [[Exponential map (Riemannian geometry)|exponential map]] for that metric. If <math>r</math> is finite and the manifold is compact, the space of vector fields is a [[Banach space]]. Moreover, the transition maps from one chart of this atlas to another are smooth, making the diffeomorphism group into a [[Banach manifold]] with smooth right translations; left translations and inversion are only continuous. If <math>r=\infty</math>,  the space of vector fields is a [[Fréchet space]]. Moreover, the transition maps are smooth, making the diffeomorphism group into a [[Fréchet manifold]] and even into a [[Convenient vector space#Regular Lie groups|regular Fréchet Lie group]]. If the manifold is <math>\sigma</math>-compact and not compact the full diffeomorphism group is not locally contractible for any of the two topologies. One has to restrict the group by controlling the deviation from the identity near infinity to obtain a diffeomorphism group which is a manifold; see {{harv|Michor|Mumford|2013}}.

===Lie algebra===
The [[Lie algebra]] of the diffeomorphism group of <math>M</math> consists of all [[vector field]]s on <math>M</math> equipped with the [[Lie bracket of vector fields]].  Somewhat formally, this is seen by making a small change to the coordinate <math>x</math> at each point in space:
: <math>x^{\mu} \mapsto x^{\mu} + \varepsilon h^{\mu}(x)</math>
so the infinitesimal generators are the vector fields
: <math> L_{h} = h^{\mu}(x)\frac{\partial}{\partial x^\mu}.</math>

===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.

===Transitivity===
For a connected manifold <math>M</math>, the diffeomorphism group [[Group action (mathematics)|acts]] [[Group_action#Remarkable properties of actions|transitively]] on <math>M</math>. More generally, the diffeomorphism group acts transitively on the [[Configuration space (physics)|configuration space]] <math>C_k M</math>.  If <math>M</math> is at least two-dimensional, the diffeomorphism group acts transitively on the configuration space <math>F_k M</math> and the action on <math>M</math> is [[Group action (mathematics)#Remarkable properties of actions|multiply transitive]] {{harv|Banyaga|1997|p=29}}.

===Extensions of diffeomorphisms===
In 1926, [[Tibor Radó]] asked whether the [[Poisson integral|harmonic extension]] of any homeomorphism or diffeomorphism of the unit circle to the [[unit disc]] yields a diffeomorphism on the open disc. An elegant proof was provided shortly afterwards by [[Hellmuth Kneser]]. In 1945, [[Gustave Choquet]], apparently unaware of this result, produced a completely different proof.

The (orientation-preserving) diffeomorphism group of the circle is pathwise connected. This can be seen by noting that any such diffeomorphism can be lifted to a diffeomorphism <math>f</math> of the reals satisfying <math>[f(x+1)=f(x)+1]</math>; this space is convex and hence path-connected. A smooth, eventually constant path to the identity gives a second more elementary way of extending a diffeomorphism from the circle to the open unit disc (a special case of the [[Alexander trick]]). Moreover, the diffeomorphism group of the circle has the homotopy-type of the [[orthogonal group]] <math>O(2)</math>.

The corresponding extension problem for diffeomorphisms of higher-dimensional spheres <math>S^{n-1}</math> was much studied in the 1950s and 1960s, with notable contributions from [[René Thom]], [[John Milnor]] and [[Stephen Smale]]. An obstruction to such extensions is given by the finite [[abelian group]] <math>\Gamma_n</math>, the "[[Exotic sphere#Twisted spheres|group of twisted spheres]]", defined as the [[Quotient group|quotient]] of the abelian [[component group]] of the diffeomorphism group by the subgroup of classes extending to diffeomorphisms of the ball <math>B^n</math>.

===Connectedness===
For manifolds, the diffeomorphism group is usually not connected. Its component group is called the [[mapping class group]]. In dimension 2 (i.e. [[Surface (topology)|surface]]s), the mapping class group is a [[finitely presented group]] generated by [[Dehn twist]]s; this has been proved by [[Max Dehn]], [[W. B. R. Lickorish]], and [[Allen Hatcher]]).{{Citation needed|date=December 2009}} Max Dehn and [[Jakob Nielsen (mathematician)|Jakob Nielsen]] showed that it can be identified with the [[outer automorphism group]] of the [[fundamental group]] of the surface.

[[William Thurston]] refined this analysis by [[Nielsen-Thurston classification|classifying elements of the mapping class group]] into three types: those equivalent to a [[Periodic function#Periodic mapping|periodic]] diffeomorphism; those equivalent to a diffeomorphism leaving a simple closed curve invariant; and those equivalent to [[Pseudo-Anosov map|pseudo-Anosov diffeomorphisms]]. In the case of the [[torus]] <math>S^1\times S^1=\R^2/\Z^2</math>, the mapping class group is simply the [[modular group]] <math>\text{SL}(2,\Z)</math> and the classification becomes classical in terms of [[Möbius transformation#Elliptic transforms|elliptic]], [[Möbius transformation#Parabolic transforms|parabolic]] and [[Möbius transformation#Hyperbolic transforms|hyperbolic]] matrices. Thurston accomplished his classification by observing that the mapping class group acted naturally on a [[Compactification (mathematics)|compactification]] of [[Teichmüller space]]; as this enlarged space was homeomorphic to a closed ball, the [[Brouwer fixed-point theorem]] became applicable. Smale [[conjecture]]d that if <math>M</math> is an [[Orientability#Orientability_of_manifolds|oriented]] smooth closed manifold, the [[identity component]] of the group of orientation-preserving diffeomorphisms is [[Simple group|simple]]. This had first been proved for a product of circles by [[Michel Herman]]; it was proved in full generality by Thurston.

===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]].