Topological manifold
Encyclopedia
In mathematics
, a topological manifold is a topological space
(can even be a separated space
) which looks locally like Euclidean space
in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics.
A manifold
can mean a topological manifold, or more frequently, a topological manifold together with some additional structure. Differentiable manifold
s, for example, are topological manifolds equipped with a differential structure
. Every manifold has an underlying topological manifold, obtained simply by forgetting the additional structure. An overview of the manifold concept is given in that article. This article focuses purely on the topological aspects of manifolds.
X is called locally Euclidean if there is a non-negative integer n such that every point in X has a neighborhood which is homeomorphic to an open subset of Euclidean space
Rn.
A topological manifold is a locally Euclidean Hausdorff space
. It is common to place additional requirements on topological manifolds. In particular, many authors define them to be paracompact or second-countable. The reasons, and some equivalent conditions, are discussed below.
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 Rn. A non-trivial theorem states that for every connected
manifold X there is a unique integer n such that X is an n-manifold. This integer is called the dimension of X.
See also: List of manifolds
s. That is, if X is locally Euclidean of dimension n and f : X → Y is a local homeomorphism, then Y is locally Euclidean of dimension n. In particular, being locally Euclidean is a topological property
.
Manifolds inherit many of the local properties of Euclidean space. In particular, they are locally compact
, locally connected
, first countable
, locally contractible, and locally metrizable. Being locally compact Hausdorff spaces, manifolds are necessarily Tychonoff space
s.
A manifold need not be connected, but every manifold M is a disjoint union
of connected manifolds. These are just the connected components of M, which are open set
s since manifolds are locally-connected. Being locally path connected, a manifold is path-connected if and only if
it is connected. It follows that the path-components are the same as the components.
.
An example of a non-Hausdorff locally Euclidean space is the line with two origins. 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.
. An example of a non-paracompact manifold is given by the long line
. Paracompact manifolds have all the topological properties of metric spaces. In particular, they are perfectly normal Hausdorff spaces.
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 (see the Whitney embedding theorem). 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
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
manifold is second-countable and paracompact.
of a manifold is a topological property
, meaning that any manifold homeomorphic to an n-manifold also has dimension n. It follows from invariance of domain
that an n-manifold cannot be homeomorphic to an m-manifold for n ≠ m.
A 1-dimensional manifold is often called a curve
while a 2-dimensional manifold is called a surface
. Higher dimensional manifolds are usually just called n-manifolds. For n = 3, 4, or 5 see 3-manifold
, 4-manifold
, and 5-manifold
.
that Euclidean neighborhoods are always open sets. One can always find Euclidean neighborhoods that are homeomorphic to "nice" open sets in Rn. Indeed, a space M is locally Euclidean if and only if either of the following equivalent conditions holds:
A Euclidean neighborhood homeomorphic to an open ball in Rn is called a Euclidean ball. Euclidean balls form a basis for the topology of a locally Euclidean space.
For any Euclidean neighborhood U a homeomorphism φ : U → φ(U) ⊂ Rn 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
by Euclidean neighborhoods. A set of Euclidean neighborhoods that cover M, together with their coordinate charts, is called an atlas
on M. (The terminology comes from an analogy with cartography
whereby a spherical globe
can be described by an atlas
of flat maps or charts).
Given two charts φ and ψ with overlapping domains U and V there is a transition function
Such a map is a homeomorphism between open subsets of Rn. 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 diffeomorphism
s.
. Such spaces are classified by their cardinality. Every discrete space is paracompact. A discrete space is second-countable if and only if it is countable
.
Every paracompact, connected 1-manifold is homeomorphic either to R or the circle
. The unconnected ones are just disjoint union
s of these.
Every compact, connected 2-manifold (or surface
) is homeomorphic to the sphere
, a connected sum
of tori
, or a connected sum of projective plane
s. See the classification theorem for surfaces for more details.
The 3-dimensional case may be solved. Thurston's geometrization conjecture, if true, together with current knowledge, would imply a classification of 3-manifolds. Grigori Perelman
sketched a proof of this conjecture in 2003 which (as of 2011) appears to be essentially correct.
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
, which is known to be algorithmically undecidable
. In fact, there is no algorithm
for deciding whether a given manifold is simply connected. There is, however, a classification of simply connected manifolds of dimension ≥ 5.
in which every point has a neighborhood homeomorphic to an open subset of Euclidean half-space
(for a fixed n):
The terminology is somewhat confusing: every topological manifold is a topological manifold with boundary, but not vice-versa.
Let M be a manifold with boundary. The interior of M, denoted Int M, is the set of points in M which have neighborhoods homeomorphic to an open subset of Rn. The boundary of M, denoted ∂M, is the complement
of Int M in M. The boundary points can be characterized as those points which land on the boundary hyperplane (xn = 0) of Rn+ under some coordinate chart.
If M is a manifold with boundary of dimension n, then Int M is a manifold (without boundary) of dimension n and ∂M is a manifold (without boundary) of dimension n − 1.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a topological manifold is a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
(can even be a separated space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
) which looks locally like Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics.
A manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
can mean a topological manifold, or more frequently, a topological manifold together with some additional structure. Differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
s, for example, are topological manifolds equipped with a differential structure
Differential structure
In mathematics, an n-dimensional differential structure on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows us to do differential calculus on the manifold...
. Every manifold has an underlying topological manifold, obtained simply by forgetting the additional structure. An overview of the manifold concept is given in that article. This article focuses purely on the topological aspects of manifolds.
Formal definition
A topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X is called locally Euclidean if there is a non-negative integer n such that every point in X has a neighborhood which is homeomorphic to an open subset of Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn.
A topological manifold is a locally Euclidean Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
. It is common to place additional requirements on topological manifolds. In particular, many authors define them to be paracompact or second-countable. The reasons, and some equivalent conditions, are discussed below.
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 Rn. A non-trivial theorem states that for every connected
Connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected...
manifold X there is a unique integer n such that X is an n-manifold. This integer is called the dimension of X.
Examples
- Euclidean spaceEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn is the prototypical n-manifold. - Any discrete spaceDiscrete spaceIn topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
is a 0-dimensional manifold. - A circleCircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
is a 1-manifold. - A torusTorusIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
is a 2-manifold (or surfaceSurfaceIn mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
) as is the Klein bottleKlein bottleIn mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...
. - The n-dimensional sphere Sn is a compactCompact spaceIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
n-manifold. - The n-dimensional torus Tn (the product of n circleCircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
s) is a compact n-manifold. - Projective spaceProjective spaceIn mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
s over the realsReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
, complexesComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
, or quaternionQuaternionIn mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s are compact manifolds.- Real projective spaceReal projective spaceIn mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...
RPn is a n-dimensional manifold. - Complex projective spaceComplex projective spaceIn 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 complex projective space label the complex lines...
CPn is a 2n-dimensional manifold. - Quaternionic projective spaceQuaternionic projective spaceIn mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H. Quaternionic projective space of dimension n is usually denoted byand is a closed manifold of dimension 4n...
HPn is a 4n-dimensional manifold.
- Real projective space
- Manifolds related to projective space include GrassmannianGrassmannianIn mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...
s, flag manifoldFlag manifoldIn mathematics, a generalized 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 smooth or complex manifold, called a real or complex flag manifold...
s, and Stiefel manifoldStiefel manifoldIn mathematics, the Stiefel manifold Vk is the set of all orthonormal k-frames in Rn. That is, it is the set of ordered k-tuples of orthonormal vectors in Rn. It is named after Swiss mathematician Eduard Stiefel...
s. - Lens spaceLens spaceA 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....
s are a class of manifolds that are quotientQuotient spaceIn topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
s of odd-dimensional spheres. - Lie groupLie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s are manifolds endowed with a groupGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
structure. - Any open subset of an n-manifold is a n-manifold with the subspace topologySubspace topologyIn topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
. - If M is an m-manifold and N is an n-manifold, the product M × N is a (m+n)-manifold.
- The disjoint unionDisjoint union (topology)In general topology and related areas of mathematics, the disjoint union of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology...
of a family of n-manifolds is a n-manifold (the pieces must all have the same dimension). - The connected sumConnected sumIn 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...
of two n-manifolds results in another n-manifold.
See also: List of manifolds
Properties
The property of being locally Euclidean is preserved by local homeomorphismLocal homeomorphism
In mathematics, more specifically topology, a local homeomorphism is intuitively a function, f, between topological spaces that preserves local structure. Equivalently, one can cover the domain of this function by open sets, such that f restricted to each such open set is a homeomorphism onto its...
s. That is, if X is locally Euclidean of dimension n and f : X → Y is a local homeomorphism, then Y is locally Euclidean of dimension n. In particular, being locally Euclidean is a topological property
Topological property
In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic...
.
Manifolds inherit many of the local properties of Euclidean space. In particular, they are locally compact
Locally compact space
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.-Formal definition:...
, locally connected
Locally connected space
In topology and other branches of mathematics, a topological space X islocally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.-Background:...
, first countable
First-countable space
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...
, locally contractible, and locally metrizable. Being locally compact Hausdorff spaces, manifolds are necessarily Tychonoff space
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....
s.
A manifold need not be connected, but every manifold M is a disjoint union
Disjoint union (topology)
In general topology and related areas of mathematics, the disjoint union of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology...
of connected manifolds. These are just the connected components of M, which are open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
s since manifolds are locally-connected. Being locally path connected, a manifold is path-connected if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
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 T1T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
.
An example of a non-Hausdorff locally Euclidean space is the line with two origins. 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 if and only if it is paracompact. 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 pathologicalPathological (mathematics)
In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved....
. An example of a non-paracompact manifold is given by the long line
Long line (topology)
In topology, the long line is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore it serves as one of the basic counterexamples of topology...
. Paracompact manifolds have all the topological properties of metric spaces. In particular, they are perfectly normal Hausdorff spaces.
Manifolds are also commonly required to be second-countable
Second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
. This is precisely the condition required to ensure that the manifold embeds
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
in some finite-dimensional Euclidean space (see the Whitney embedding theorem). For any manifold the properties of being second-countable, Lindelöf
Lindelöf space
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....
, 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
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
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 space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
manifold is second-countable and paracompact.
Dimensionality
The dimensionDimension
In physics and mathematics, the dimension of a 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 because only one coordinate is needed to specify a point on it...
of a manifold is a topological property
Topological property
In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic...
, meaning that any manifold homeomorphic to an n-manifold also has dimension n. It follows from invariance of domain
Invariance of domain
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. It states:The theorem and its proof are due to L.E.J. Brouwer, published in 1912...
that an n-manifold cannot be homeomorphic to an m-manifold for n ≠ m.
A 1-dimensional manifold is often called a curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
while a 2-dimensional manifold is called a surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
. Higher dimensional manifolds are usually just called n-manifolds. For n = 3, 4, or 5 see 3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
, 4-manifold
4-manifold
In mathematics, 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...
, and 5-manifold
5-manifold
In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure.Non-simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups. Simply connected compact 5-manifolds were first...
.
Coordinate charts
By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open subset of Rn. Such neighborhoods are called Euclidean neighborhoods. It follows from invariance of domainInvariance of domain
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. It states:The theorem and its proof are due to L.E.J. Brouwer, published in 1912...
that Euclidean neighborhoods are always open sets. One can always find Euclidean neighborhoods that are homeomorphic to "nice" open sets in Rn. 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 Rn.
- every point of M has a neighborhood homeomorphic to Rn itself.
A Euclidean neighborhood homeomorphic to an open ball in Rn is called a Euclidean ball. Euclidean balls form a basis for the topology of a locally Euclidean space.
For any Euclidean neighborhood U a homeomorphism φ : U → φ(U) ⊂ Rn 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 (topology)
In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset. Formally, ifC = \lbrace U_\alpha: \alpha \in A\rbrace...
by Euclidean neighborhoods. A set of Euclidean neighborhoods that cover M, together with their coordinate charts, is called an atlas
Atlas (topology)
In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...
on M. (The terminology comes from an analogy with cartography
Cartography
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...
whereby a spherical globe
Globe
A globe is a three-dimensional scale model of Earth or other spheroid celestial body such as a planet, star, or moon...
can be described by an atlas
Atlas
An atlas is a collection of maps; it is typically a map of Earth or a region of Earth, but there are atlases of the other planets in the Solar System. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats...
of flat maps or charts).
Given two charts φ and ψ with overlapping domains U and V there is a transition function
- ψφ−1 : φ(U ∩ V) → ψ(U ∩ V).
Such a map is a homeomorphism between open subsets of Rn. 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 diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
s.
Classification of manifolds
A 0-manifold is just a discrete spaceDiscrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
. Such spaces are classified by their cardinality. Every discrete space is paracompact. A discrete space is second-countable if and only if it is countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
.
Every paracompact, connected 1-manifold is homeomorphic either to R or the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
. The unconnected ones are just disjoint union
Disjoint union
In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
s of these.
Every compact, connected 2-manifold (or surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
) is homeomorphic to the sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
, a connected sum
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...
of tori
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
, or a connected sum of projective plane
Real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
s. See the classification theorem for surfaces for more details.
The 3-dimensional case may be solved. Thurston's geometrization conjecture, if true, together with current knowledge, would imply a classification of 3-manifolds. Grigori Perelman
Grigori Perelman
Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...
sketched a proof of this conjecture in 2003 which (as of 2011) appears to be essentially correct.
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
Word problem for groups
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element...
in group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, which is known to be algorithmically undecidable
Undecidable problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is impossible to construct a single algorithm that always leads to a correct yes-or-no answer....
. In fact, there is no algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
for deciding whether a given manifold is simply connected. There is, however, a classification of simply connected manifolds of dimension ≥ 5.
Manifolds with boundary
A slightly more general concept is sometimes useful. A topological manifold with boundary is a Hausdorff spaceHausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
in which every point has a neighborhood homeomorphic to an open subset of Euclidean half-space
Half-space
In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional euclidean space. More generally, a half-space is either of the two parts into which a hyperplane divides an affine space...
(for a fixed n):
The terminology is somewhat confusing: every topological manifold is a topological manifold with boundary, but not vice-versa.
Let M be a manifold with boundary. The interior of M, denoted Int M, is the set of points in M which have neighborhoods homeomorphic to an open subset of Rn. The boundary of M, denoted ∂M, is the complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
of Int M in M. The boundary points can be characterized as those points which land on the boundary hyperplane (xn = 0) of Rn+ under some coordinate chart.
If M is a manifold with boundary of dimension n, then Int M is a manifold (without boundary) of dimension n and ∂M is a manifold (without boundary) of dimension n − 1.