In
mathematics, a diffeomorphism is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
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 m ...
s. It is an
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
function that maps one
differentiable 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 ...
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 ad ...
are
differentiable.
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 ...
and
, a
differentiable map is called a diffeomorphism if it is a
bijection and its inverse
is differentiable as well. If these functions are
times
continuously 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 ...
,
is called a
-diffeomorphism.
Two manifolds
and
are diffeomorphic (usually denoted
) if there is a diffeomorphism
from
to
. They are
-diffeomorphic if there is an
times continuously differentiable bijective map between them whose inverse is also
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 ...
of a manifold
and a subset
of a manifold
, a function
is said to be smooth if for all
in
there is a
neighborhood of
and a smooth function
such that the
restrictions agree:
(note that
is an extension of
). The function
is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth.
Local description
; Hadamard-Caccioppoli Theorem
If
,
are
connected open subsets of
such that
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 spa ...
, a
differentiable map
is a diffeomorphism if it is
proper and if the
differential is bijective (and hence a
linear isomorphism) at each point
in
.
; First remark
It is essential for 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 spa ...
for the function 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 square function
:
Then 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
:
Thus, though is bijective at each point, is not invertible because it fails to be injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...
(e.g. ).
; Second remark
Since the differential at a point (for a differentiable function)
:
is a linear map
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 pr ...
, it has a well-defined inverse if and only if 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'' (franchi ...
representation of is the matrix of first-order partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s whose entry in the -th row and -th column is . This so-called Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...
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 coor ...
. Imagine going from dimension to dimension . 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(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 field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
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\to N is called a diffeomorphism if, in
coordinate charts
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an a ...
, it satisfies the definition above. More precisely: Pick any cover of
M by compatible
coordinate charts
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an a ...
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(\phi^(U))\subseteq\psi^(V).
Examples
Since any manifold can be locally parametrised, we can consider some explicit maps from
\R^2 into
\R^2.
* Let
::
f(x,y) = \left (x^2 + y^3, x^2 - y^3 \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, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
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(x,y) = \left (a_0 + a_x + a_y + \cdots, \ b_0 + b_x + b_y + \cdots \right )
: where the
a_ and
b_ are arbitrary
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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(0,0) = \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 ...
as
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
s.
* Let
::
h(x,y) = \left (\sin(x^2 + y^2), \cos(x^2 + y^2) \right ).
: We can calculate the Jacobian matrix:
::
J_h = \begin 2x\cos(x^2 + y^2) & 2y\cos(x^2 + y^2) \\ -2x\sin(x^2+y^2) & -2y\sin(x^2 + y^2) \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 Eucli ...
.
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 ...
, a stress-induced transformation is called a
deformation 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. Suppose that in a chart of the surface,
f(x,y) = (u,v).
The
total differential of ''u'' is
:
du = \frac dx + \frac dy, and similarly for ''v''.
Then the image
(du, dv) = (dx, dy) 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 pr ...
, 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
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 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 used ...
) 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 of all
C^r diffeomorphisms of
M to itself, denoted by
\text^r(M) or, when
r is understood,
\text(M). This is a "large" group, in the sense that—provided
M is not zero-dimensional—it is not
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
.
Topology
The diffeomorphism group has two natural
topologies: ''weak'' and ''strong'' . When the manifold is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
, 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 , \inf ...
. 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 spac ...
on
M, the weak topology is the topology induced by the family of metrics
:
d_K(f,g) = \sup\nolimits_ d(f(x),g(x)) + \sum\nolimits_ \sup\nolimits_ \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^\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
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
. 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
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 with respect ...
. Moreover, the transition maps are smooth, making the diffeomorphism group into a
Fréchet manifold In mathematics, in particular in nonlinear analysis, a Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space.
More precisely, a Fréchet manifold consists of a Haus ...
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 iden ...
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^(x)
so the infinitesimal generators are the vector fields
:
L_ = h^(x)\frac.
Examples
* When
M=G is a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
, there is a natural inclusion of
G in its own diffeomorphism group via left-translation. Let
\text(G) denote the diffeomorphism group of
G, then there is a splitting
\text(G)\simeq G\times\text(G,e), where
\text(G,e) 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 subgrou ...
of
\text(G) that fixes the
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
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, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible ...
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 deform ...
of the subgroup
\text(\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 (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 group ...
. Similarly, if
M is any manifold there is a
group extension 0\to\text_0(M)\to\text(M)\to\Sigma(\pi_0(M)). Here
\text_0(M) is the subgroup of
\text(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(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
The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
transitively
Transitivity or transitive may refer to:
Grammar
* Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects
* Transitive verb, a verb which takes an object
* Transitive case, a grammatical case to mark a ...
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 ...
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
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-mani ...
. In 1945,
Gustave Choquet
Gustave Choquet (; 1 March 1915 – 14 November 2006) was a 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 creati ...
, 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
(x+1)=f(x)+1/math>; this space is convex and hence path-connected. A smooth, eventually constant path to the identity gives a second more elementary way of extending a diffeomorphism from the circle to the open unit disc (a special case of the Alexander trick). Moreover, the diffeomorphism group of the circle has the homotopy-type of the orthogonal group
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. ...
O(2).
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
René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958.
He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became ...
, 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 Un ...
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 facult ...
. 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 com ...
\Gamma_n, the " group of twisted spheres", defined as the quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
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 mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
Mo ...
. 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 generated by Dehn twist
In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold).
Definition
Suppose that ''c'' is a simple closed curve in a closed, orientable surface ''S''. Let ...
s ( Dehn, Lickorish, Hatcher Hatcher is a surname. Notable people with the surname include:
* Allen Hatcher (born 1944), U.S. mathematician
* Anna Granville Hatcher (1905–1978), U.S. linguist
* Edwin Starr (born Charles Edwin Hatcher, 1942–2003), U.S. soul singer
*Chris Ha ...
). 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. H ...
and Jakob Nielsen Jacob or Jakob Nielsen may refer to:
* Jacob Nielsen, Count of Halland (died c. 1309), great grandson of Valdemar II of Denmark
* , Norway (1768-1822)
* Jakob Nielsen (mathematician) (1890–1959), Danish mathematician known for work on automorphi ...
showed that it can be identified with the outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a ...
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 ...
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.
Thursto ...
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 not ...
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 fraction ...
\text(2,\Z) and the classification becomes classical in terms of elliptic, parabolic and 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 ...
matrices. Thurston accomplished his classification by observing that the mapping class group acted naturally on a compactification
Compactification may refer to:
* Compactification (mathematics), making a topological space compact
* Compactification (physics), the "curling up" of extra dimensions in string theory
See also
* Compaction (disambiguation)
{{disambiguation ...
of Teichmüller space
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmülle ...
; as this enlarged space was homeomorphic to a closed ball, the Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simpl ...
became applicable. Smale conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...
d that if M is an oriented 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 comp ...
of the group of orientation-preserving diffeomorphisms is simple. This had first been proved for a product of circles by Michel Herman
Michel may refer to:
* Michel (name), a given name or surname of French origin (and list of people with the name)
* Míchel (nickname), a nickname (a list of people with the nickname, mainly Spanish footballers)
* Míchel (footballer, born 1963), S ...
; it was proved in full generality by Thurston.
Homotopy types
* The diffeomorphism group of S^2 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: S^1\times S^1\times\text(2,\Z).
* The diffeomorphism groups of orientable surfaces of genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...
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, of ...
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(S^4) has more than two components. Via Milnor, Kahn and Antonelli, however, it is known that provided n>6, \text(S^n) does not have the homotopy-type of a finite CW-complex.
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 Un ...
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
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
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 dimens ...
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
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
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.
See also
* 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 "cont ...
such as Arnold's cat map
In mathematics, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name.
Thinking of the torus \mathbb^2 as the quotient space ...
* Diffeo anomaly also known as a gravitational anomaly
In theoretical physics, a gravitational anomaly is an example of a gauge anomaly: it is an effect of quantum mechanics — usually a one-loop diagram—that invalidates the general covariance of a theory of general relativity combined with s ...
, a type anomaly
Anomaly may refer to:
Science
Natural
*Anomaly (natural sciences)
** Atmospheric anomaly
** Geophysical anomaly
Medical
* Congenital anomaly (birth defect), a disorder present at birth
** Physical anomaly, a deformation of an anatomical struct ...
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, q ...
* Diffeology
In mathematics, a diffeology on a set generalizes the concept of smooth charts in a differentiable manifold, declaring what the "smooth parametrizations" in the set are.
The concept was first introduced by Jean-Marie Souriau in the 1980s under ...
, 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 \operato ...
, 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 t ...
* Large diffeomorphism
* Local diffeomorphism
* Superdiffeomorphism
Notes
References
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Mathematical physics