In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a topological manifold is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
that locally resembles
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
''n''-
dimensional
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 coordi ...
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All
manifold
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 n ...
s are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g.
differentiable manifolds are topological manifolds equipped with a
differential structure In mathematics, an ''n''-dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for dif ...
). Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.
However, not every topological manifold can be endowed with a particular additional structure. For example, the
E8 manifold
In mathematics, the ''E''8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the ''E''8 lattice.
History
The E_8 manifold was discovered by Michael Freedman in 1982. Rokhlin's theorem shows that ...
is a topological manifold which cannot be endowed with a differentiable structure.
Formal definition
A
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' is called locally Euclidean if there is a non-negative
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''n'' such that every point in ''X'' has a
neighborhood which is
homeomorphic to
real ''n''-space R
''n''.
A topological manifold is a locally Euclidean
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
. It is common to place additional requirements on topological manifolds. In particular, many authors define them to be
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
or
second-countable.
In the remainder of this article a ''manifold'' will mean a topological manifold. An ''n-manifold'' will mean a topological manifold such that every point has a neighborhood homeomorphic to R
''n''.
Examples
''n''-Manifolds
* The
real coordinate space R
''n'' is an ''n''-manifold.
* Any
discrete space is a 0-dimensional manifold.
* A
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
is a
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 ...
1-manifold.
* A
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 tou ...
and a
Klein bottle
In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...
are compact 2-manifolds (or
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 ...
s).
* The
''n''-dimensional sphere ''S''
''n'' is a compact ''n''-manifold.
* The
''n''-dimensional torus T
''n'' (the product of ''n'' circles) is a compact ''n''-manifold.
Projective manifolds
*
Projective spaces over the
reals,
complexes, or
quaternions are compact manifolds.
**
Real projective space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.
Basic properties Construction
A ...
RP
''n'' is a ''n''-dimensional manifold.
**
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
CP
''n'' is a 2''n''-dimensional manifold.
**
Quaternionic projective space HP
''n'' is a 4''n''-dimensional manifold.
* Manifolds related to projective space include
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective ...
s,
flag manifold In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smo ...
s, and
Stiefel manifolds.
Other manifolds
*
Differentiable manifolds are a class of topological manifolds equipped with a
differential structure In mathematics, an ''n''-dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for dif ...
.
*
Lens space
A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions.
In the 3-manifold case, a lens space can be visualized ...
s are a class of differentiable manifolds that are
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 ...
s of odd-dimensional spheres.
*
Lie groups are a class of differentiable manifolds equipped with a compatible
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 ...
structure.
* The
E8 manifold
In mathematics, the ''E''8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the ''E''8 lattice.
History
The E_8 manifold was discovered by Michael Freedman in 1982. Rokhlin's theorem shows that ...
is a topological manifold which cannot be given a differentiable structure.
Properties
The property of being locally Euclidean is preserved by
local homeomorphism
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.
If f : X \to Y is a local homeomorphism, X is said to be an ...
s. That is, if ''X'' is locally Euclidean of dimension ''n'' and ''f'' : ''Y'' → ''X'' is a local homeomorphism, then ''Y'' is locally Euclidean of dimension ''n''. In particular, being locally Euclidean is a
topological property
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spa ...
.
Manifolds inherit many of the local properties of Euclidean space. In particular, they are
locally compact,
locally connected
In topology and other branches of mathematics, a topological space ''X'' is
locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
Background
Throughout the history of topology, connectedness ...
,
first countable
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...
,
locally contractible, and
locally metrizable. Being locally compact Hausdorff spaces, manifolds are necessarily
Tychonoff space
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
s.
Adding the Hausdorff condition can make several properties become equivalent for a manifold. As an example, we can show that for a Hausdorff manifold, the notions of
σ-compactness and second-countability are the same. Indeed, a
Hausdorff manifold is a locally compact Hausdorff space, hence it is (completely) regular. Assume such a space X is σ-compact. Then it is Lindelöf, and because Lindelöf + regular implies paracompact, X is metrizable. But in a metrizable space, second-countability coincides with being Lindelöf, so X is second-countable. Conversely, if X is a Hausdorff second-countable manifold, it must be σ-compact.
A manifold need not be connected, but every manifold ''M'' is a
disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
of connected manifolds. These are just the
connected components of ''M'', which are
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
s since manifolds are locally-connected. Being locally path connected, a manifold is path-connected
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
it is connected. It follows that the path-components are the same as the components.
The Hausdorff axiom
The Hausdorff property is not a local one; so even though Euclidean space is Hausdorff, a locally Euclidean space need
not be. It is true, however, that every locally Euclidean space is
T1.
An example of a non-Hausdorff locally Euclidean space is the
line with two origins In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff. ...
. This space is created by replacing the origin of the real line with ''two'' points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This space is not Hausdorff because the two origins cannot be separated.
Compactness and countability axioms
A manifold is
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) ...
if and only if it is
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
. Since metrizability is such a desirable property for a topological space, it is common to add paracompactness to the definition of a manifold. In any case, non-paracompact manifolds are generally regarded as
pathological
Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in th ...
. An example of a non-paracompact manifold is given by the
long line Long line or longline may refer to:
*'' Long Line'', an album by Peter Wolf
* Long line (topology), or Alexandroff line, a topological space
*Long line (telecommunications), a transmission line in a long-distance communications network
*Longline fi ...
. Paracompact manifolds have all the topological properties of metric spaces. In particular, they are
perfectly normal Hausdorff space
In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. T ...
s.
Manifolds are also commonly required to be
second-countable. This is precisely the condition required to ensure that the manifold
embeds in some finite-dimensional Euclidean space. For any manifold the properties of being second-countable,
Lindelöf, and
σ-compact are all equivalent.
Every second-countable manifold is paracompact, but not vice versa. However, the converse is nearly true: a paracompact manifold is second-countable if and only if it has a
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
number of
connected components. In particular, a connected manifold is paracompact if and only if it is second-countable.
Every second-countable manifold is
separable and paracompact. Moreover, if a manifold is separable and paracompact then it is also second-countable.
Every
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 ...
manifold is second-countable and paracompact.
Dimensionality
By
invariance of domain
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space \R^n.
It states:
:If U is an open subset of \R^n and f : U \rarr \R^n is an injective continuous map, then V := f(U) is open in \R^n and f is a homeomorph ...
, a non-empty ''n''-manifold cannot be an ''m''-manifold for ''n'' ≠ ''m''.
The dimension of a non-empty ''n''-manifold is ''n''. Being an ''n''-manifold is a
topological property
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spa ...
, meaning that any topological space homeomorphic to an ''n''-manifold is also an ''n''-manifold.
Coordinate charts
By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open subset of
. Such neighborhoods are called Euclidean neighborhoods. It follows from
invariance of domain
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space \R^n.
It states:
:If U is an open subset of \R^n and f : U \rarr \R^n is an injective continuous map, then V := f(U) is open in \R^n and f is a homeomorph ...
that Euclidean neighborhoods are always open sets. One can always find Euclidean neighborhoods that are homeomorphic to "nice" open sets in
. Indeed, a space ''M'' is locally Euclidean if and only if either of the following equivalent conditions holds:
*every point of ''M'' has a neighborhood homeomorphic to an
open ball
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are defi ...
in
.
*every point of ''M'' has a neighborhood homeomorphic to
itself.
A Euclidean neighborhood homeomorphic to an open ball in
is called a Euclidean ball. Euclidean balls form a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
for the topology of a locally Euclidean space.
For any Euclidean neighborhood ''U'', a homeomorphism
is called a coordinate chart on ''U'' (although the word ''chart'' is frequently used to refer to the domain or range of such a map). A space ''M'' is locally Euclidean if and only if it can be
covered
Cover or covers may refer to:
Packaging
* Another name for a lid
* Cover (philately), generic term for envelope or package
* Album cover, the front of the packaging
* Book cover or magazine cover
** Book design
** Back cover copy, part of co ...
by Euclidean neighborhoods. A set of Euclidean neighborhoods that cover ''M'', together with their coordinate charts, is called an
atlas
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth.
Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geograp ...
on ''M''. (The terminology comes from an analogy with
cartography
Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an im ...
whereby a spherical
globe
A globe is a spherical model of Earth, of some other celestial body, or of the celestial sphere. Globes serve purposes similar to maps, but unlike maps, they do not distort the surface that they portray except to scale it down. A model globe ...
can be described by an
atlas
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth.
Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geograp ...
of flat maps or charts).
Given two charts
and
with overlapping domains ''U'' and ''V'', there is a transition function
:
Such a map is a homeomorphism between open subsets of
. That is, coordinate charts agree on overlaps up to homeomorphism. Different types of manifolds can be defined by placing restrictions on types of transition maps allowed. For example, for
differentiable manifolds the transition maps are required to be
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
.
Classification of manifolds
Discrete Spaces (0-Manifold)
A 0-manifold is just a
discrete space. A discrete space is second-countable if and only if it is
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
.
Curves (1-Manifold)
Every nonempty, paracompact, connected 1-manifold is homeomorphic either to R or the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
.
Surfaces (2-Manifold)
Every nonempty, compact, connected 2-manifold (or
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 ...
) is homeomorphic to the
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, a
connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...
of
tori, or a connected sum of
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
s.
Volumes (3-Manifold)
A classification of 3-manifolds results from
Thurston's geometrization conjecture
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensi ...
, proven by
Grigori Perelman
Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
in 2003. More specifically, Perelman's results provide an algorithm for deciding if two three-manifolds are homeomorphic to each other.
General ''n''-Manifold
The full classification of ''n''-manifolds for ''n'' greater than three is known to be impossible; it is at least as hard as the
word problem in
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, which is known to be
algorithmically undecidable.
In fact, there is no
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
for deciding whether a given manifold is
simply connected. There is, however, a classification of simply connected manifolds of dimension ≥ 5.
[Barden, D. "Simply Connected Five-Manifolds." Annals of Mathematics, vol. 82, no. 3, 1965, pp. 365–385. JSTOR, www.jstor.org/stable/1970702.]
Manifolds with boundary
A slightly more general concept is sometimes useful. A topological manifold with boundary is a
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
in which every point has a neighborhood homeomorphic to an open subset of Euclidean
half-space (for a fixed ''n''):
:
Every topological manifold is a topological manifold with boundary, but not vice versa.
Constructions
There are several methods of creating manifolds from other manifolds.
Product Manifolds
If ''M'' is an ''m''-manifold and ''N'' is an ''n''-manifold, the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
''M''×''N'' is a (''m''+''n'')-manifold when given the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-s ...
.
Disjoint Union
The
disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
of a countable family of ''n''-manifolds is a ''n''-manifold (the pieces must all have the same dimension).
Connected Sum
The
connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...
of two ''n''-manifolds is defined by removing an open ball from each manifold and taking 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 disjoint union of the resulting manifolds with boundary, with the quotient taken with regards to a homeomorphism between the boundary spheres of the removed balls. This results in another ''n''-manifold.
Submanifold
Any open subset of an ''n''-manifold is an ''n''-manifold with the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
.
Footnotes
References
*
*
*
External links
*
{{Authority control
Manifolds
Properties of topological spaces