Let M be a differentiable manifold that is
second-countable and
Hausdorff. The
diffeomorphism group of M is the
group of all C^r diffeomorphisms of M to itself, denoted by \text{Diff}^r(M) or, when r is understood, \text{Diff}(M). This is a "large" group, in the sense that—provided M is not zero-dimensional—it is not
locally compact.
Topology The diffeomorphism group has two natural
topologies:
weak and
strong . When the manifold is
compact, these two topologies agree. The weak topology is always
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. Fixing a
Riemannian metric on M, the weak topology is the topology induced by the family of metrics : 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 \| as K varies over compact subsets of M. Indeed, since M is \sigma-compact, there is a sequence of compact subsets K_n whose
union is M. Then: : d(f,g) = \sum\nolimits_n 2^{-n}\frac{d_{K_n}(f,g)}{1+d_{K_n}(f,g)}. The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of C^r vector fields . Over a compact subset of M, this follows by fixing a Riemannian metric on M and using the
exponential map for that metric. If r 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 r=\infty, 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
regular Fréchet Lie group. If the manifold is \sigma-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 .
Lie algebra The
Lie algebra of the diffeomorphism group of M consists of all
vector fields on M equipped with the
Lie bracket of vector fields. Somewhat formally, this is seen by making a small change to the coordinate x at each point in space: : x^{\mu} \mapsto x^{\mu} + \varepsilon h^{\mu}(x) so the infinitesimal generators are the vector fields : L_{h} = h^{\mu}(x)\frac{\partial}{\partial x^\mu}.
Examples • When M=G is a
Lie group, there is a natural inclusion of G in its own diffeomorphism group via left-translation. Let \text{Diff}(G) denote the diffeomorphism group of G, then there is a splitting \text{Diff}(G)\simeq G\times\text{Diff}(G,e), where \text{Diff}(G,e) is the
subgroup of \text{Diff}(G) that fixes the
identity element of the group. • The diffeomorphism group of Euclidean space \R^n 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 \text{Diff}(\R^n,0) of diffeomorphisms fixing the origin under the map f(x)\to f(tx)/t, t\in(0,1]. In particular, the general linear group is also a deformation retract of the full diffeomorphism group. • For a finite
set of points, the diffeomorphism group is simply the
symmetric group. Similarly, if M is any manifold there is a
group extension 0\to\text{Diff}_0(M)\to\text{Diff}(M)\to\Sigma(\pi_0(M)). Here \text{Diff}_0(M) is the subgroup of \text{Diff}(M) that preserves all the components of M, and \Sigma(\pi_0(M)) is the permutation group of the set \pi_0(M) (the components of M). Moreover, the image of the map \text{Diff}(M)\to\Sigma(\pi_0(M)) is the bijections of \pi_0(M) that preserve diffeomorphism classes.
Transitivity For a connected manifold M, the diffeomorphism group
acts transitively on M. More generally, the diffeomorphism group acts transitively on the
configuration space C_k M. If M is at least two-dimensional, the diffeomorphism group acts transitively on the configuration space F_k M and the action on M is
multiply transitive .
Extensions of diffeomorphisms In 1926,
Tibor Radó asked whether the
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 f of the reals satisfying [f(x+1)=f(x)+1]; 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 O(2). The corresponding extension problem for diffeomorphisms of higher-dimensional spheres S^{n-1} 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 \Gamma_n, the "
group of twisted spheres", defined as the
quotient of the abelian
component group of the diffeomorphism group by the subgroup of classes extending to diffeomorphisms of the ball B^n.
Connectedness For manifolds, the diffeomorphism group is usually not connected. Its component group is called the
mapping class group. In dimension 2 (i.e.
surfaces), the mapping class group is a
finitely presented group generated by
Dehn twists; this has been proved by
Max Dehn,
W. B. R. Lickorish, and
Allen Hatcher). Max Dehn and
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
classifying elements of the mapping class group into three types: those equivalent to a
periodic diffeomorphism; those equivalent to a diffeomorphism leaving a simple closed curve invariant; and those equivalent to
pseudo-Anosov diffeomorphisms. In the case of the
torus S^1\times S^1=\R^2/\Z^2, the mapping class group is simply the
modular group \text{SL}(2,\Z) and the classification becomes classical in terms of
elliptic,
parabolic and
hyperbolic matrices. Thurston accomplished his classification by observing that the mapping class group acted naturally on a
compactification of
Teichmüller space; as this enlarged space was homeomorphic to a closed ball, the
Brouwer fixed-point theorem became applicable. Smale
conjectured that if M is an
oriented smooth closed manifold, the
identity component of the group of orientation-preserving diffeomorphisms is
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 S^2 has the homotopy-type of the subgroup \mathrm{O}(3). This was proven by Stephen Smale. • The diffeomorphism group of the torus has the homotopy-type of its linear
automorphisms: S^1\times S^1\times\text{GL}(2,\Z). • The diffeomorphism groups of orientable surfaces of
genus g>1 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 groups). • The homotopy-type of diffeomorphism groups of n-manifolds for n>3 are poorly understood. For example, it is an open problem whether or not \mathrm{Diff}(S^4) has more than two components. Via Milnor, Kahn and Antonelli, however, it is known that provided n>6, \mathrm{Diff}(S^n) does not have the homotopy-type of a finite
CW-complex. == Homeomorphism and diffeomorphism ==