Diffeomorphism
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Modular groups

Infinite dimensional Lie group

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In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.
Definition
Given two manifolds M and N, a differentiable map f : M → N is called a diffeomorphism if it is a bijection and its inverse f^{−1} : N → M is differentiable as well. If these functions are r times continuously differentiable, f is called a C^{r}diffeomorphism.
Two manifolds M and N are diffeomorphic (symbol usually being ≃) if there is a diffeomorphism f from M to N. They are C^{r} diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable.
Diffeomorphisms of subsets of manifolds
Given a subset X of a manifold M and a subset Y of a manifold N, a function f : X → Y is said to be smooth if for all p in X there is a neighborhood U ⊂ M of p and a smooth function g : U → N such that the restrictions agree (note that g is an extension of f). f is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth.
Local description
 Model example
If U, V are connected open subsets of R^{n} such that V is simply connected, a differentiable map f : U → V is a diffeomorphism if it is proper and if the differential Df_{x} : R^{n} → R^{n} is bijective at each point x in U.
 First remark
It is essential for V to be simply connected for the function f to be globally invertible (under the sole condition that its derivative is a bijective map at each point). For example, consider the "realification" of the complex square function
Then f is surjective and it satisfies
Thus, though Df_{x} is bijective at each point, f is not invertible because it fails to be injective (e.g. f(1,0) = (1,0) = f(−1,0).
 Second remark
Since the differential at a point (for a differentiable function)
is a linear map, it has a welldefined inverse if and only if Df_{x} is a bijection. The matrix representation of Df_{x} is the n × n matrix of firstorder partial derivatives whose entry in the ith row and jth column is . This socalled Jacobian matrix is often used for explicit computations.
 Third remark
Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine f going from dimension n to dimension k. If n < k then Df_{x} could never be surjective; and if n > k then Df_{x} could never be injective. In both cases, therefore, Df_{x} fails to be a bijection.
 Fourth remark
If Df_{x} is a bijection at x then f is said to be a local diffeomorphism (since, by continuity, Df_{y} will also be bijective for all y sufficiently close to x).
 Fifth remark
Given a smooth map from dimension n to dimension k, if Df (or, locally, Df_{x}) is surjective, f is said to be a submersion (or, locally, a "local submersion"); and if Df (or, locally, Df_{x}) is injective, f is said to be an immersion (or, locally, a "local immersion").
 Sixth remark
A differentiable bijection is not necessarily a diffeomorphism. f(x) = x^{3}, for example, is not a diffeomorphism from R to itself because its derivative vanishes at 0 (and hence its inverse is not differentiable at 0). This is an example of a homeomorphism that is not a diffeomorphism.
 Seventh remark
When f is a map between differentiable manifolds, a diffeomorphic f is a stronger condition than a homeomorphic f. For a diffeomorphism, f and its inverse need to be differentiable; for a homeomorphism, f and its inverse need only be continuous. Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism.
f : M → N is called a diffeomorphism if, in coordinate charts, it satisfies the definition above. More precisely: Pick any cover of M by compatible coordinate charts and do the same for N. Let φ and ψ be charts on, respectively, M and N, with U and V as, respectively, the images of φ and ψ. The map ψfφ^{−1} : U → V is then a diffeomorphism as in the definition above, whenever f(φ^{−1}(U)) ⊂ ψ^{−1}(V).
Examples
Since any manifold can be locally parametrised, we can consider some explicit maps from R^{2} into R^{2}.
 Let
 We can calculate the Jacobian matrix:
 The Jacobian matrix has zero determinant if, and only if xy = 0. We see that f could only be a diffeomorphism away from the xaxis and the yaxis. However, f is not bijective since f(x,y)=f(x,y), and thus it cannot be a diffeomorphism.
 Let
 where the and are arbitrary real numbers, and the omitted terms are of degree at least two in x and y. We can calculate the Jacobian matrix at 0:
 We see that g is a local diffeomorphism at 0 if, and only if,
 i.e. the linear terms in the components of g are linearly independent as polynomials.
 Let
 We can calculate the Jacobian matrix:
 The Jacobian matrix has zero determinant everywhere! In fact we see that the image of h is the unit circle.
Surface deformations
In mechanics, a stressinduced transformation is called a deformation and may be described by a diffeomorphism. A diffeomorphism f : U → V between two surfaces U and V has a Jacobian matrix Df that is an invertible matrix. In fact, it is required that for p in U, there is a neighborhood of p in which the Jacobian Df stays nonsingular. Since the Jacobian is a 2 × 2 real matrix, Df can be read as one of three types of complex number: ordinary complex, split complex number, or dual number. Suppose that in a chart of the surface,
The total differential of u is
 , and similarly for v.
Then the image is a linear transformation, fixing the origin, and expressible as the action of a complex number of a particular type. When (dx, dy ) is also interpreted as that type of complex number, the action is of complex multiplication in the appropriate complex number plane. As such, there is a type of angle (Euclidean, hyperbolic, or slope) that is preserved in such a multiplication. Due to Df being invertible, the type of complex number is uniform over the surface.
Consequently, a surface deformation or diffeomorphism of surfaces has the conformal property of preserving (the appropriate type of) angles.
Diffeomorphism group
Let M be a differentiable manifold that is secondcountable and Hausdorff. The diffeomorphism group of M is the group of all C^{r} diffeomorphisms of M to itself, denoted by Diff^{r}(M) or, when r is understood, Diff(M). This is a "large" group, in the sense that – provided M is not zerodimensional – it is not locally compact.
Topology
The diffeomorphism group has two natural topologies: weak and strong (Hirsch 1997). 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
as K varies over compact subsets of M. Indeed, since M is σcompact, there is a sequence of compact subsets K_{n} whose union is M. Then:
The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of C^{r} vector fields (Leslie 1967). 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 = ∞, 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 σcompact 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 (Michor & Mumford 2013).
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:
so the infinitesimal generators are the vector fields
Examples
 When M = G is a Lie group, there is a natural inclusion of G in its own diffeomorphism group via lefttranslation. Let Diff(G) denote the diffeomorphism group of G, then there is a splitting Diff(G) ≃ G × Diff(G, e), where Diff(G, e) is the subgroup of 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 subgroup Diff(R^{n}, 0) of diffeomorphisms fixing the origin under the map f(x) ↦ f(tx)/t, t ∈& (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 → Diff_{0}(M) → Diff(M) → Σ(π_{0}(M)). Here Diff_{0}(M)is the subgroup of Diff(M) that preserves all the components of M, and Σ(π_{0}(M)) is the permutation group of the set π_{0}(M) (the components of M). Moreover, the image of the map Diff(M) → Σ(π_{0}(M)) is the bijections of π_{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 twodimensional, the diffeomorphism group acts transitively on the configuration space F_{k}M and the action on M is multiply transitive (Banyaga 1997, p. 29).
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 (orientationpreserving) 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 pathconnected. 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 homotopytype of the orthogonal group O(2).
The corresponding extension problem for diffeomorphisms of higherdimensional 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 Γ_{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 (Dehn, Lickorish, 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 pseudoAnosov diffeomorphisms. In the case of the torus S^{1} × S^{1} = R^{2}/Z^{2}, the mapping class group is simply the modular group 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 fixedpoint theorem became applicable.
Smale conjectured that if M is an oriented smooth closed manifold, the identity component of the group of orientationpreserving 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 homotopytype of the subgroup O(3). This was proved by Steve Smale.^{[1]}
 The diffeomorphism group of the torus has the homotopytype of its linear automorphisms: S^{1} × S^{1} × GL(2, Z).
 The diffeomorphism groups of orientable surfaces of genus g > 1 have the homotopytype of their mapping class groups (i.e. the components are contractible).
 The homotopytype of the diffeomorphism groups of 3manifolds are fairly wellunderstood via the work of Ivanov, Hatcher, Gabai and Rubinstein, although there are a few outstanding open cases (primarily 3manifolds with finite fundamental groups).
 The homotopytype of diffeomorphism groups of nmanifolds for n > 3 are poorly understood. For example, it is an open problem whether or not Diff(S^{4}) has more than two components. Via Milnor, Kahn and Antonelli, however, it is known that provided n > 6, Diff(S^{n}) does not have the homotopytype of a finite CWcomplex.
Homeomorphism and diffeomorphism
Unlike nondiffeomorphic homeomorphisms, it is relatively difficult to find a pair of homeomorphic manifolds that are not diffeomorphic. In dimensions 1, 2, 3, any pair of homeomorphic smooth manifolds are diffeomorphic. In dimension 4 or greater, examples of homeomorphic but not diffeomorphic pairs have been found. The first such example was constructed by John Milnor in dimension 7. He constructed a smooth 7dimensional manifold (called now Milnor's sphere) that is homeomorphic to the standard 7sphere but not diffeomorphic to it. There are, in fact, 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7sphere (each of them is the total space of a fiber bundle over the 4sphere with the 3sphere as the fiber).
More unusual phenomena occur for 4manifolds. In the early 1980s, a combination of results due to Simon Donaldson and Michael Freedman led to the discovery of exotic R^{4}s: there are uncountably many pairwise nondiffeomorphic open subsets of R^{4} each of which is homeomorphic to R^{4}, and also there are uncountably many pairwise nondiffeomorphic differentiable manifolds homeomorphic to R^{4} that do not embed smoothly in R^{4}.
See also
Notes
 ↑ Smale, "Diffeomorphisms of the 2sphere", Proc. Amer. Math. Soc. 10 (1959), pp. 621–626.
References
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 Banyaga, Augustin (1997), The structure of classical diffeomorphism groups, Mathematics and its Applications, 400, Kluwer Academic, ISBN 0792344758
 Duren, Peter L. (2004), Harmonic Mappings in the Plane, Cambridge Mathematical Tracts, 156, Cambridge University Press, ISBN 0521641217
 Hazewinkel, Michiel, ed. (2001), "Diffeomorphism", Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 Hirsch, Morris (1997), Differential Topology, Berlin, New York: SpringerVerlag, ISBN 9780387901480
 Kriegl, Andreas; Michor, Peter (1997), The convenient setting of global analysis, Mathematical Surveys and Monographs, 53, American Mathematical Society, ISBN 0821807803
 Leslie, J. A. (1967), "On a differential structure for the group of diffeomorphisms", Topology, 6 (2): 263–271, doi:10.1016/00409383(67)900389, ISSN 00409383, MR 0210147
 Michor, Peter W.; Mumford, David (2013), "A zoo of diffeomorphism groups on R^{n}.", Annals of Global Analysis and Geometry, 44 (4): 529–540, doi:10.1007/s1045501393802 (arXiv:1211.5704)
 Milnor, John W. (2007), Collected Works Vol. III, Differential Topology, American Mathematical Society, ISBN 0821842307
 Omori, Hideki (1997), Infinitedimensional Lie groups, Translations of Mathematical Monographs, 158, American Mathematical Society, ISBN 0821845756
 Kneser, Hellmuth (1926), "Lösung der Aufgabe 41.", Jahresbericht der Deutschen MathematikerVereinigung (in German), 35 (2): 123