TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a diffeomorphism is an
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... of
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
s. It is an invertible
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
that maps one
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
to another such that both the function and its
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
are
smooth Smooth may refer to: Mathematics * Smooth function is a smooth function with compact support. In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuo ... . # Definition

Given two
manifolds The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of suc ...
$M$ and $N$, a
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...
map A map is a symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...
$f \colon M \rightarrow N$ is called a diffeomorphism if it is a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... and its inverse $f^ \colon N \rightarrow M$ is differentiable as well. If these functions are $r$ times
continuously differentiable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, $f$ is called a $C^r$-diffeomorphism. Two manifolds $M$ and $N$ are diffeomorphic (usually denoted $M \simeq N$) 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... ''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 A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
''U'' ⊆ ''M'' of ''p'' and a smooth function ''g'' : ''U'' → ''N'' such that the restrictions agree: $g_ = f_$ (note that ''g'' is an extension of ''f''). The function ''f'' is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth.

# Local description

; Hadamard-Caccioppoli Theorem If ''U'', ''V'' are connected open subsets of R''n'' such that ''V'' is
simply connected In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
, a
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ... map ''f'' : ''U'' → ''V'' is a diffeomorphism if it is proper and if the differential ''Dfx'' : R''n'' → R''n'' is bijective (and hence a
linear isomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
) at each point ''x'' in ''U''.
; First remark It is essential for ''V'' to be
simply connected In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
for the function ''f'' to be globally invertible (under the sole condition that its derivative be a bijective map at each point). For example, consider the "realification" of the
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ... square function : $\begin f : \mathbf^2 \setminus \ \to \mathbf^2 \setminus \ \\ \left(x,y\right)\mapsto\left(x^2-y^2,2xy\right). \end$ Then ''f'' is
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and it satisfies : $\det Df_x = 4\left(x^2+y^2\right) \neq 0.$ Thus, though ''Dfx'' is bijective at each point, ''f'' is not invertible because it fails to be
injective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
(e.g. ''f''(1, 0) = (1, 0) = ''f''(−1, 0)).
; Second remark Since the differential at a point (for a differentiable function) : $Df_x : T_xU \to T_V$ is a
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... , it has a well-defined inverse if and only if ''Dfx'' is a bijection. The
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
representation of ''Dfx'' is the ''n'' × ''n'' matrix of first-order
partial derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... s whose entry in the ''i''-th row and ''j''-th column is $\partial f_i / \partial x_j$. This so-called
Jacobian matrix In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Produ ...
is often used for explicit computations.
; Third remark Diffeomorphisms are necessarily between manifolds of the same
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ... . Imagine ''f'' going from dimension ''n'' to dimension ''k''. If ''n'' < ''k'' then ''Dfx'' could never be surjective, and if ''n'' > ''k'' then ''Dfx'' could never be injective. In both cases, therefore, ''Dfx'' fails to be a bijection.
; Fourth remark If ''Dfx'' is a bijection at ''x'' then ''f'' is said to be a local diffeomorphism (since, by continuity, ''Dfy'' 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, ''Dfx'') is surjective, ''f'' is said to be a submersion (or, locally, a "local submersion"); and if ''Df'' (or, locally, ''Dfx'') 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 In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
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 In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...
; for a homeomorphism, ''f'' and its inverse need only be
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
. 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 ψ''f''φ−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 R2 into R2. * Let :: $f\left(x,y\right) = \left \left(x^2 + y^3, x^2 - y^3 \right \right).$ : We can calculate the Jacobian matrix: :: $J_f = \begin 2x & 3y^2 \\ 2x & -3y^2 \end .$ : The Jacobian matrix has zero
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... if and only if ''xy'' = 0. We see that ''f'' could only be a diffeomorphism away from the ''x''-axis and the ''y''-axis. However, ''f'' is not bijective since ''f''(''x'', ''y'') = ''f''(-''x'', ''y''), and thus it cannot be a diffeomorphism. * Let :: $g\left(x,y\right) = \left \left(a_0 + a_x + a_y + \cdots, \ b_0 + b_x + b_y + \cdots \right \right)$ : where the $a_$ and $b_$ are arbitrary
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, and the omitted terms are of degree at least two in ''x'' and ''y''. We can calculate the Jacobian matrix at 0: :: $J_g\left(0,0\right) = \begin a_ & a_ \\ b_ & b_ \end.$ : We see that ''g'' is a local diffeomorphism at 0 if, and only if, :: $a_b_ - a_b_ \neq 0,$ : i.e. the linear terms in the components of ''g'' are
linearly independent In the theory of vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...
as
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... s. * Let :: $h\left(x,y\right) = \left \left(\sin\left(x^2 + y^2\right), \cos\left(x^2 + y^2\right) \right \right).$ : We can calculate the Jacobian matrix: :: $J_h = \begin 2x\cos\left(x^2 + y^2\right) & 2y\cos\left(x^2 + y^2\right) \\ -2x\sin\left(x^2+y^2\right) & -2y\sin\left(x^2 + y^2\right) \end .$ : The Jacobian matrix has zero determinant everywhere! In fact we see that the image of ''h'' is the
unit circle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ... .

## Surface deformations

In
mechanics Mechanics (Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximat ... , a stress-induced transformation is called a
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (mechanics), such changes considered and analyzed as displacements of continuum bodies. * Defo ...
and may be described by a diffeomorphism. A diffeomorphism ''f'' : ''U'' → ''V'' between two
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
s ''U'' and ''V'' has a Jacobian matrix ''Df'' that is an
invertible matrixIn linear algebra, an ''n''-by-''n'' square matrix is called invertible (also nonsingular or nondegenerate), if there exists an ''n''-by-''n'' square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the ''n''-by-''n'' identit ...
. In fact, it is required that for ''p'' in ''U'', there is a
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
of ''p'' in which the Jacobian ''Df'' stays
non-singular In the mathematical field of algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic tech ...
. Suppose that in a chart of the surface, $f\left(x,y\right) = \left(u,v\right).$ The
total differential In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. ...
of ''u'' is :$du = \frac dx + \frac dy$, and similarly for ''v''. Then the image $\left(du, dv\right) = \left(dx, dy\right) Df$ is a
linear transformation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, 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 Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. It is the name of: Geometry *Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry a ... ,
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ... , or
slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ... ) 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
second-countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base (topology), base. More explicitly, a topological space T is second-countable if there exists some countable ...
and
Hausdorff . The diffeomorphism group of ''M'' is the
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
of all ''Cr'' 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 zero-dimensional—it is not
locally compact In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
.

## Topology

The diffeomorphism group has two natural
topologies s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are p ...
: ''weak'' and ''strong'' . When the manifold is
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
, these two topologies agree. The weak topology is always
metrizable In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
. 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 In differential geometry, a Riemannian manifold or Riemannian space is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ''g'p'' on the tangent space ''T'p'M'' at each poin ...
on ''M'', the weak topology is the topology induced by the family of metrics : $d_K\left(f,g\right) = \sup\nolimits_ d\left(f\left(x\right),g\left(x\right)\right) + \sum\nolimits_ \sup\nolimits_ \left \, D^pf\left(x\right) - D^pg\left(x\right) \right \,$ as ''K'' varies over compact subsets of ''M''. Indeed, since ''M'' is σ-compact, there is a sequence of compact subsets ''K''''n'' whose union is ''M''. Then: : $d\left(f,g\right) = \sum\nolimits_n 2^\frac.$ The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of ''Cr'' 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. 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 In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are Complete space, complete with ...
. 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 σ-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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of the diffeomorphism group of ''M'' consists of all
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ... s on ''M'' equipped with the
Lie bracket of vector fields In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector field In vector calculus and physic ...
. Somewhat formally, this is seen by making a small change to the coordinate $x$ at each point in space: : $x^ \mapsto x^ + \varepsilon h^\left(x\right)$ so the infinitesimal generators are the vector fields : $L_ = h^\left(x\right)\frac.$

## Examples

* When ''M'' = ''G'' is a
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, there is a natural inclusion of ''G'' in its own diffeomorphism group via left-translation. 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 In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...
of Diff(''G'') that fixes the
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
is a
deformation retract In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
of the 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 (mathematics), set of points, the diffeomorphism group is simply the
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
. Similarly, if ''M'' is any manifold there is a
group extension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
0 → Diff0(''M'') → Diff(''M'') → Σ(π0(''M'')). Here Diff0(''M'') is the subgroup of Diff(''M'') that preserves all the components of ''M'', and Σ(π0(''M'')) is the permutation group of the set π0(''M'') (the components of ''M''). Moreover, the image of the map Diff(''M'') → Σ(π0(''M'')) is the bijections of π0(''M'') that preserve diffeomorphism classes.

## Transitivity

For a connected manifold ''M'', the diffeomorphism group
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum), often referred to simply as Acts, or formally the Book of Acts, is the fifth book of the New Testament The New Te ...
transitively on ''M''. More generally, the diffeomorphism group acts transitively on the configuration space ''CkM''. If ''M'' is at least two-dimensional, the diffeomorphism group acts transitively on the configuration space ''FkM'' 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 An open Euclidean unit disk In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathemati ... yields a diffeomorphism on the open disc. An elegant proof was provided shortly afterwards by
Hellmuth Kneser Hellmuth Kneser (16 April 1898 – 23 August 1973) was a Baltic German mathematician, who made notable contributions to group theory and topology. His most famous result may be his theorem on the existence of a prime decomposition (3-manifold), pr ...
. In 1945,
Gustave Choquet Gustave Choquet (; 1 March 1915 – 14 November 2006) was a France, French mathematician. Choquet was born in Solesmes, Nord. His contributions include work in functional analysis, potential theory, topology and measure theory. He is known for ... , 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 trickAlexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after James Waddell Alexander II, J. W. Alexander. Statement Two homeomorphisms of the ''n''-dimensional ball (mathematics), ball D^n which agree o ...
). Moreover, the diffeomorphism group of the circle has the homotopy-type of the
orthogonal group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic comm ...
,
John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic ... and
Stephen Smale Stephen Smale (born July 15, 1930) is an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theor ...
. An obstruction to such extensions is given by the finite
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
Γ''n'', the " group of twisted spheres", defined as the
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, �έχνη ''tiké échne' ...
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.
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
s), the mapping class group is a finitely presented group generated by Dehn twists (Max Dehn, Dehn, W. B. R. Lickorish, Lickorish, Allen Hatcher, Hatcher). 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 S1 × S1 = R2/Z2, the mapping class group is simply the modular group SL(2, Z) 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 conjectured that if ''M'' 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 S2 has the homotopy-type of the subgroup O(3). This was proven by Steve Smale. * The diffeomorphism group of the torus has the homotopy-type of its linear automorphisms: S1 × S1 × 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 Diff(S4) 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 homotopy-type of a finite CW-complex.

# Homeomorphism and diffeomorphism

Unlike non-diffeomorphic homeomorphisms, it is relatively difficult to find a pair of homeomorphic manifolds that are not diffeomorphic. In dimensions 1, 2 and 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 John Willard Milnor (born February 20, 1931) is an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic ... in dimension 7. He constructed a smooth 7-dimensional manifold (called now Milnor's sphere) that is homeomorphic to the standard 7-sphere but not diffeomorphic to it. There are, in fact, 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere (each of them is the total space of a fiber bundle over the 4-sphere with the 3-sphere as the fiber). More unusual phenomena occur for 4-manifolds. In the early 1980s, a combination of results due to Simon Donaldson and Michael Freedman led to the discovery of exotic R4, exotic R4s: there are Uncountable set, uncountably many pairwise non-diffeomorphic open subsets of R4 each of which is homeomorphic to R4, and also there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to R4 that do not Embedding#Differential_topology, embed smoothly in R4.