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In mathematics, a diffeomorphism is an isomorphism of
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in i ...
.

# Definition

Given two
manifolds In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...
$M$ and $N$, a
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in i ...
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Altho ...
$f \colon M \rightarrow N$ is called a diffeomorphism if it is a bijection and its inverse $f^ \colon N \rightarrow M$ 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 (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, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
$X$ of a manifold $M$ and a subset $Y$ of a manifold $N$, a function $f:X\to Y$ is said to be smooth if for all $p$ in $X$ there is a neighborhood $U\subset M$ of $p$ and a smooth function $g:U\to N$ 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 Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
open subsets of $\R^n$ such that $V$ is
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
, a
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in i ...
map $f:U\to V$ is a diffeomorphism if it is
proper Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...
and if the differential $Df_x:\R^n\to\R^n$ is bijective (and hence a linear isomorphism) at each point $x$ in $U$.
; First remark It is essential for $V$ to be
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
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 Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
square function : $\begin f : \R^2 \setminus \ \to \R^2 \setminus \ \\ \left(x,y\right)\mapsto\left(x^2-y^2,2xy\right). \end$ Then $f$ is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
and it satisfies : $\det Df_x = 4\left(x^2+y^2\right) \neq 0.$ Thus, though $Df_x$ is bijective at each point, $f$ is not invertible because it fails to be injective (e.g. $f\left(1,0\right)=\left(1,0\right)=f\left(-1,0\right)$).
; Second remark Since the differential at a point (for a differentiable function) : $Df_x : T_xU \to T_V$ is a linear map, it has a well-defined inverse if and only if $Df_x$ is a bijection. The
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
representation of $Df_x$ is the $n\times n$ matrix of first-order partial derivatives whose entry in the $i$-th row and $j$-th column is $\partial f_i / \partial x_j$. This so-called Jacobian matrix is often used for explicit computations.
; Third remark Diffeomorphisms are necessarily between manifolds of the same
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
. Imagine $f$ going from dimension $n$ to dimension $k$. If
; 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. $f\left(x\right)=x^3$, 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 In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in i ...
; 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 ...
. Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism.
$f:M\to N$ 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 $\phi$ and $\psi$ be charts on, respectively, $M$ and $N$, with $U$ and $V$ as, respectively, the images of $\phi$ and $\psi$. The map $\psi f\phi^:U\to V$ is then a diffeomorphism as in the definition above, whenever $f\left(\phi^\left(U\right)\right)\subseteq\psi^\left(V\right)$.

# Examples

Since any manifold can be locally parametrised, we can consider some explicit maps from $\R^2$ into $\R^2$. * 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 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 $f\left(x,y\right)=f\left(-x,y\right)$, 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 numbers, 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 spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ar ...
as polynomials. * 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, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
.

## Surface deformations

In
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects re ...
, 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 (physics), such changes considered and analyzed as displacements of continuum bodies. * Def ...
and may be described by a diffeomorphism. A diffeomorphism $f:U\to V$ between two
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
s $U$ and $V$ has a Jacobian matrix $Df$ that is an
invertible matrix In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplicati ...
. In fact, it is required that for $p$ in $U$, there is a neighborhood of $p$ in which the Jacobian $Df$ stays
non-singular In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In cas ...
. Suppose that in a chart of the surface, $f\left(x,y\right) = \left(u,v\right).$ The total differential 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, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
, 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 In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
) 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. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
and Hausdorff. The diffeomorphism group of $M$ is the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of all $C^r$ diffeomorphisms of $M$ to itself, denoted by $\text^r\left(M\right)$ or, when $r$ is understood, $\text\left(M\right)$. 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 In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
. 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 , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
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 $\sigma$-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 $C^r$ 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 In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
of the diffeomorphism group of $M$ consists of all vector fields 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 fields ''X'' and ''Y'' on a smooth ...
. 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, there is a natural inclusion of $G$ in its own diffeomorphism group via left-translation. Let $\text\left(G\right)$ denote the diffeomorphism group of $G$, then there is a splitting $\text\left(G\right)\simeq G\times\text\left(G,e\right)$, where $\text\left(G,e\right)$ is the
subgroup In group theory, a branch of mathematics, given a 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, ''H'' is a subgro ...
of $\text\left(G\right)$ 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 In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deforma ...
of the subgroup $\text\left(\R^n,0\right)$ of diffeomorphisms fixing the origin under the map $f\left(x\right)\to f\left(tx\right)/t, t\in\left(0,1\right]$. 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, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
. Similarly, if $M$ is any manifold there is a
group extension In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\overs ...
$0\to\text_0\left(M\right)\to\text\left(M\right)\to\Sigma\left(\pi_0\left(M\right)\right)$. Here $\text_0\left(M\right)$ is the subgroup of $\text\left(M\right)$ that preserves all the components of $M$, and $\Sigma\left(\pi_0\left(M\right)\right)$ is the permutation group of the set $\pi_0\left(M\right)$ (the components of $M$). Moreover, the image of the map $\text\left(M\right)\to\Sigma\left(\pi_0\left(M\right)\right)$ is the bijections of $\pi_0\left(M\right)$ 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 Tibor Radó (June 2, 1895 – December 29, 1965) was a Hungarian mathematician who moved to the United States after World War I. Biography Radó was born in Budapest and between 1913 and 1915 attended the Polytechnic Institute, studying civ ...
asked whether the harmonic extension of any homeomorphism or diffeomorphism of the unit circle to the
unit disc In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...
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 for 3-manifold ...
. 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
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. T ...
$O\left(2\right)$. The corresponding extension problem for diffeomorphisms of higher-dimensional spheres $S^$ was much studied in the 1950s and 1960s, with notable contributions from René Thom,
John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
and
Stephen Smale Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics faculty ...
. An obstruction to such extensions is given by the finite
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
$\Gamma_n$, 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.
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
s), the mapping class group is a
finitely presented group In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
generated by Dehn twists ( Dehn, Lickorish, Hatcher).
Max Dehn Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. ...
and Jakob Nielsen showed that it can be identified with the outer automorphism group of the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, ...
of the surface.
William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. 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 In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does no ...
$S^1\times S^1=\R^2/\Z^2$, the mapping class group is simply the
modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractiona ...
$\text\left(2,\Z\right)$ 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 In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
smooth closed manifold, the
identity component In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element. In point set topology, the identity compon ...
of the group of orientation-preserving diffeomorphisms is
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
. 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 $O\left(3\right)$. This was proven by Steve Smale. * The diffeomorphism group of the torus has the homotopy-type of its linear automorphisms: $S^1\times S^1\times\text\left(2,\Z\right)$. * 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 group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, ...
s). * 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 $\text\left(S^4\right)$ has more than two components. Via Milnor, Kahn and Antonelli, however, it is known that provided $n>6$, $\text\left(S^n\right)$ does not have the homotopy-type of a finite
CW-complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
.

# Homeomorphism and diffeomorphism

Since every diffeomorphism is a homeomorphism, every diffeomorphic manifolds are homeomorphic, but the converse is not true. While it is easy to find homeomorphisms that are non-diffeomorphisms, it is more 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 known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
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 In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dime ...
as the fiber). More unusual phenomena occur for
4-manifold In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. ...
s. In the early 1980s, a combination of results due to
Simon Donaldson Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. H ...
and Michael Freedman led to the discovery of
exotic Exotic may refer to: Mathematics and physics * Exotic R4, a differentiable 4-manifold, homeomorphic but not diffeomorphic to the Euclidean space R4 *Exotic sphere, a differentiable ''n''-manifold, homeomorphic but not diffeomorphic to the ordinar ...
$\R^4$: there are uncountably many pairwise non-diffeomorphic open subsets of $\R^4$ each of which is homeomorphic to $\R^4$, and also there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic to $\R^4$ that do not embed smoothly in $\R^4$.

*
Anosov diffeomorphism In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contr ...
such as Arnold's cat map * Diffeo anomaly also known as a gravitational anomaly, a type anomaly in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
* Diffeology, smooth parameterizations on a set, which makes a diffeological space *
Diffeomorphometry Diffeomorphometry is the metric study of imagery, shape and form in the discipline of computational anatomy (CA) in medical imaging. The study of images in computational anatomy rely on high-dimensional diffeomorphism groups \varphi \in \operat ...
, metric study of shape and form in computational anatomy *
Étale morphism In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy th ...
*
Large diffeomorphism In mathematics and theoretical physics, a large diffeomorphism is an equivalence class of diffeomorphisms under the equivalence relation where diffeomorphisms that can be continuously connected to each other are in the same equivalence class. For ...
* Local diffeomorphism * Superdiffeomorphism

# References

* * * * * * * * * * * * {{Manifolds Mathematical physics