In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ndimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds 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). Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.^{[1]} However, not every topological manifold can be endowed with a particular additional structure. For example, the E8 manifold is a topological manifold which cannot be endowed with a differentiable structure.
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Formal definition
A topological space X is called locally Euclidean if there is a nonnegative integer n such that every point in X has a neighborhood which is homeomorphic to real nspace R^{n}.^{[2]}
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^{[3]} or secondcountable.^{[2]}
In the remainder of this article a manifold will mean a topological manifold. An nmanifold will mean a topological manifold such that every point has a neighborhood homeomorphic to R^{n}.
Examples
nManifolds
 The real coordinate space R^{n} is an nmanifold.
 Any discrete space is a 0dimensional manifold.
 A circle is a compact 1manifold.
 A torus and a Klein bottle are compact 2manifolds (or surfaces).
 The ndimensional sphere S^{n} is a compact nmanifold.
 The ndimensional torus T^{n} (the product of n circles) is a compact nmanifold.
Projective manifolds
 Projective spaces over the reals, complexes, or quaternions are compact manifolds.
 Real projective space RP^{n} is a ndimensional manifold.
 Complex projective space CP^{n} is a 2ndimensional manifold.
 Quaternionic projective space HP^{n} is a 4ndimensional manifold.
 Manifolds related to projective space include Grassmannians, flag manifolds, and Stiefel manifolds.
Other manifolds
 Differentiable manifolds are a class of topological manifolds equipped with a differential structure.
 Lens spaces are a class of differentiable manifolds that are quotients of odddimensional spheres.
 Lie groups are a class of differentiable manifolds equipped with a compatible group structure.
 The E8 manifold is a topological manifold which cannot be given a differentiable structure.
Properties
The property of being locally Euclidean is preserved by local homeomorphisms. 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 spaces.
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 σcompactness and secondcountability are the same. Indeed, a Hausdorff manifold is a locally compact Hausdorff space, hence it is (completely) regular.^{[4]} Assume such a space X is σcompact. Then it is Lindelöf, and because Lindelöf + regular implies paracompact, X is metrizable. But in a metrizable space, secondcountability coincides with being Lindelöf, so X is secondcountable. Conversely, if X is a Hausdorff secondcountable manifold, it must be σcompact.^{[5]}
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 sets since manifolds are locallyconnected. Being locally path connected, a manifold is pathconnected if and only if it is connected. It follows that the pathcomponents 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 T_{1}.
An example of a nonHausdorff 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, nonparacompact manifolds are generally regarded as pathological. An example of a nonparacompact 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 secondcountable. This is precisely the condition required to ensure that the manifold embeds in some finitedimensional Euclidean space. For any manifold the properties of being secondcountable, Lindelöf, and σcompact are all equivalent.
Every secondcountable manifold is paracompact, but not vice versa. However, the converse is nearly true: a paracompact manifold is secondcountable 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 secondcountable. Every secondcountable manifold is separable and paracompact. Moreover, if a manifold is separable and paracompact then it is also secondcountable.
Every compact manifold is secondcountable and paracompact.
Dimensionality
By invariance of domain, a nonempty nmanifold cannot be an mmanifold for n ≠ m.^{[6]} The dimension of a nonempty nmanifold is n. Being an nmanifold is a topological property, meaning that any topological space homeomorphic to an nmanifold is also an nmanifold.^{[7]}
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 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 .
 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 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 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.
Classification of manifolds
Discrete Spaces (0Manifold)
A 0manifold is just a discrete space. A discrete space is secondcountable if and only if it is countable.^{[7]}
Curves (1Manifold)
Every nonempty, paracompact, connected 1manifold is homeomorphic either to R or the circle.^{[7]}
Surfaces (2Manifold)
Every nonempty, compact, connected 2manifold (or surface) is homeomorphic to the sphere, a connected sum of tori, or a connected sum of projective planes.^{[8]}
Volumes (3Manifold)
A classification of 3manifolds results from Thurston's geometrization conjecture, proven by Grigori Perelman in 2003. More specifically, Perelman's results provide an algorithm for deciding if two threemanifolds are homeomorphic to each other.^{[9]}
General nManifold
The full classification of nmanifolds 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.^{[10]}
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.^{[11]}^{[12]}
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 halfspace (for a fixed n):
Every topological manifold is a topological manifold with boundary, but not vice versa.^{[7]}
Constructions
There are several methods of creating manifolds from other manifolds.
Product Manifolds
If M is an mmanifold and N is an nmanifold, the Cartesian product M×N is a (m+n)manifold when given the product topology.^{[13]}
Disjoint Union
The disjoint union of a countable family of nmanifolds is a nmanifold (the pieces must all have the same dimension).^{[7]}
Connected Sum
The connected sum of two nmanifolds 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 nmanifold.^{[7]}
Submanifold
Any open subset of an nmanifold is an nmanifold with the subspace topology.^{[13]}
Footnotes
 ^ Rajendra Bhatia (6 June 2011). Proceedings of the International Congress of Mathematicians: Hyderabad, August 1927, 2010. World Scientific. pp. 477–. ISBN 9789814324359.
 ^ ^{a} ^{b} John M. Lee (6 April 2006). Introduction to Topological Manifolds. Springer Science & Business Media. ISBN 9780387227276.
 ^ Thierry Aubin (2001). A Course in Differential Geometry. American Mathematical Soc. pp. 25–. ISBN 9780821872147.
 ^ Topospaces subwiki, Locally compact Hausdorff implies completely regular
 ^ Stack Exchange, Hausdorff locally compact and second countable is sigmacompact
 ^ Tammo tom Dieck (2008). Algebraic Topology. European Mathematical Society. pp. 249–. ISBN 9783037190487.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} John Lee (25 December 2010). Introduction to Topological Manifolds. Springer Science & Business Media. pp. 64–. ISBN 9781441979407.
 ^ Jean Gallier; Dianna Xu (5 February 2013). A Guide to the Classification Theorem for Compact Surfaces. Springer Science & Business Media. ISBN 9783642343643.
 ^ Geometrisation of 3manifolds. European Mathematical Society. 2010. ISBN 9783037190821.
 ^ Lawrence Conlon (17 April 2013). Differentiable Manifolds: A First Course. Springer Science & Business Media. pp. 90–. ISBN 9781475722840.
 ^ Žubr A.V. (1988) Classification of simplyconnected topological 6manifolds. In: Viro O.Y., Vershik A.M. (eds) Topology and Geometry — Rohlin Seminar. Lecture Notes in Mathematics, vol 1346. Springer, Berlin, Heidelberg
 ^ Barden, D. "Simply Connected FiveManifolds." Annals of Mathematics, vol. 82, no. 3, 1965, pp. 365–385. JSTOR, www.jstor.org/stable/1970702.
 ^ ^{a} ^{b} Jeffrey Lee; Jeffrey Marc Lee (2009). Manifolds and Differential Geometry. American Mathematical Soc. pp. 7–. ISBN 9780821848159.
References
 Gauld, D. B. (1974). "Topological Properties of Manifolds". The American Mathematical Monthly. Mathematical Association of America. 81 (6): 633–636. doi:10.2307/2319220. JSTOR 2319220.
 Kirby, Robion C.; Siebenmann, Laurence C. (1977). Foundational Essays on Topological Manifolds. Smoothings, and Triangulations (PDF). Princeton: Princeton University Press. ISBN 0691081913.
 Lee, John M. (2000). Introduction to Topological Manifolds. Graduate Texts in Mathematics 202. New York: Springer. ISBN 0387987592.