, a branch of mathematics
, a topological manifold is a topological space
which locally resembles real
Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifold
s are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable manifold
s are topological manifolds equipped with a differential structure
). Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.
A topological space
''X'' is called locally Euclidean if there is a non-negative integer
''n'' such that every point in ''X'' has a neighbourhood
which is homeomorphic
to real ''n''-space
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
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''
* The real coordinate space
is an ''n''-manifold.
* Any discrete space
is a 0-dimensional manifold.
* A circle
is a compact
* A torus
and a Klein bottle
are compact 2-manifolds (or surface
* The ''n''-dimensional sphere
is a compact ''n''-manifold.
* The ''n''-dimensional torus
(the product of ''n'' circles) is a compact ''n''-manifold.
* Projective space
s over the reals
, or quaternion
s are compact manifolds.
** Real projective space
is a ''n''-dimensional manifold.
** Complex projective space
is a 2''n''-dimensional manifold.
** Quaternionic projective space
is a 4''n''-dimensional manifold.
* Manifolds related to projective space include Grassmannian
s, flag manifold
s, and Stiefel manifold
* Lens space
s are a class of manifolds that are quotient
s of odd-dimensional spheres.
* Lie group
s are manifolds endowed with a group
The property of being locally Euclidean is preserved by local homeomorphism
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
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
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-compact
ness 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
of connected manifolds. These are just the connected component
s 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.
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
. 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 pathological
. 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 space
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
number of connected component
s. 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.
manifold is second-countable and paracompact.
By invariance of domain
, 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
, meaning that any topological space homeomorphic to an ''n''-manifold is also an ''n''-manifold.
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
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
*every point of ''M'' has a neighborhood homeomorphic to
A Euclidean neighborhood homeomorphic to an open ball in
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
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
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 diffeomorphism
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
Every nonempty, paracompact, connected 1-manifold is homeomorphic either to R or the circle
is a 2-manifold.
Every nonempty, compact, connected 2-manifold (or surface
) is homeomorphic to the sphere
, a connected sum
, or a connected sum of projective plane
A classification of 3-manifolds results from
Thurston's geometrization conjecture
, proven by Grigori Perelman
in 2003. More specifically, Perelman's results provide an algorithm for deciding if two three-manifolds are homeomorphic to each other.
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.
[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 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.
There are several methods of creating manifolds from other manifolds.
If ''M'' is an ''m''-manifold and ''N'' is an ''n''-manifold, the Cartesian product
''M''×''N'' is a (''m''+''n'')-manifold when given the product topology
The disjoint union
of a countable family of ''n''-manifolds is a ''n''-manifold (the pieces must all have the same dimension).
The connected sum
of two ''n''-manifolds is defined by removing an open ball from each manifold and taking the quotient
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.
Any open subset of an ''n''-manifold is an ''n''-manifold with the subspace topology
Category:Properties of topological spaces