Intopology, atopological manifold is atopological space that locally resemblesrealn-dimensionalEuclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. Allmanifolds 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 adifferential 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, theE8 manifold is a topological manifold which cannot be endowed with a differentiable structure.

Atopological spaceX is calledlocally Euclidean if there is a non-negativeintegern such that every point inX has aneighborhood which ishomeomorphic to anopen subset ofrealn-spaceRⁿ.^[2]

Atopological manifold is a locally EuclideanHausdorff space. It is common to place additional requirements on topological manifolds. In particular, many authors define them to beparacompact^[3] orsecond-countable.^[2]

In the remainder of this article amanifold will mean a topological manifold. Ann-manifold will mean a topological manifold such that every point has a neighborhood homeomorphic toRⁿ.

• Thereal coordinate spaceRⁿ is ann-manifold.
• Anydiscrete space is a 0-dimensional manifold.
• Acircle is acompact 1-manifold.
• Atorus and aKlein bottle are compact 2-manifolds (orsurfaces).
• Then-dimensional sphereSⁿ is a compactn-manifold.
• Then-dimensional torusTⁿ (the product ofn circles) is a compactn-manifold.

• Projective spaces over thereals,complexes, orquaternions are compact manifolds.
  • Real projective spaceRPⁿ is an-dimensional manifold.
  • Complex projective spaceCPⁿ is a 2n-dimensional manifold.
  • Quaternionic projective spaceHPⁿ is a 4n-dimensional manifold.
• Manifolds related to projective space includeGrassmannians,flag manifolds, andStiefel manifolds.

• Differentiable manifolds are a class of topological manifolds equipped with adifferential structure.
• Lens spaces are a class of differentiable manifolds that arequotients of odd-dimensional spheres.
• Lie groups are a class of differentiable manifolds equipped with a compatiblegroup structure.
• TheE8 manifold is a topological manifold which cannot be given a differentiable structure.

The property of being locally Euclidean is preserved bylocal homeomorphisms. That is, ifX is locally Euclidean of dimensionn andf :Y →X is a local homeomorphism, thenY is locally Euclidean of dimensionn. In particular, being locally Euclidean is atopological property.

Manifolds inherit many of the local properties of Euclidean space. In particular, they arelocally compact,locally connected,first countable,locally contractible, andlocally metrizable. Being locally compact Hausdorff spaces, manifolds are necessarilyTychonoff 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 ofσ-compactness and second-countability are the same. Indeed, aHausdorff manifold is a locally compact Hausdorff space, hence it is (completely) regular.^[4] 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.^[5]

A manifold need not be connected, but every manifoldM is adisjoint union of connected manifolds. These are just theconnected components ofM, which areopen sets since manifolds are locally-connected. Being locally path connected, a manifold is path-connectedif and only if it is connected. It follows that the path-components are the same as the components.

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 isT₁.

An example of a non-Hausdorff locally Euclidean space is theline with two origins. This space is created by replacing the origin of the real line withtwo 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.

A manifold ismetrizable if and only if it isparacompact. Thelong line is an example anormalHausdorff 1-dimensional topological manifold that is not metrizable nor 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 aspathological. Paracompact manifolds have all the topological properties of metric spaces. In particular, they areperfectly normal Hausdorff spaces.

Manifolds are also commonly required to besecond-countable. This is precisely the condition required to ensure that the manifoldembeds in some finite-dimensional Euclidean space. For any manifold the properties of being second-countable,Lindelöf, andσ-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 acountable number ofconnected components. In particular, a connected manifold is paracompact if and only if it is second-countable. Every second-countable manifold isseparable and paracompact. Moreover, if a manifold is separable and paracompact then it is also second-countable.

Everycompact manifold is second-countable and paracompact.

Byinvariance of domain, a non-emptyn-manifold cannot be anm-manifold forn ≠m.^[6] The dimension of a non-emptyn-manifold isn. Being ann-manifold is atopological property, meaning that any topological space homeomorphic to ann-manifold is also ann-manifold.^[7]

By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open subset of${\displaystyle {\mathbb{R}}^n}$. Such neighborhoods are calledEuclidean neighborhoods. It follows frominvariance of domain that Euclidean neighborhoods are always open sets. One can always find Euclidean neighborhoods that are homeomorphic to "nice" open sets in${\displaystyle {\mathbb{R}}^n}$. Indeed, a spaceM is locally Euclidean if and only if either of the following equivalent conditions holds:

• every point ofM has a neighborhood homeomorphic to anopen ball in${\displaystyle {\mathbb{R}}^n}$.
• every point ofM has a neighborhood homeomorphic to${\displaystyle {\mathbb{R}}^n}$ itself.

A Euclidean neighborhood homeomorphic to an open ball in${\displaystyle {\mathbb{R}}^n}$ is called aEuclidean ball. Euclidean balls form abasis for the topology of a locally Euclidean space.

For any Euclidean neighborhoodU, a homeomorphism${\displaystyle \varphi :U\to \varphi \left(U\right)\subset {\mathbb{R}}^n}$ is called acoordinate chart onU (although the wordchart is frequently used to refer to the domain or range of such a map). A spaceM is locally Euclidean if and only if it can becovered by Euclidean neighborhoods. A set of Euclidean neighborhoods that coverM, together with their coordinate charts, is called anatlas onM. (The terminology comes from an analogy withcartography whereby a sphericalglobe can be described by anatlas of flat maps or charts).

Given two charts${\displaystyle \varphi }$ and${\displaystyle \psi }$ with overlapping domainsU andV, there is atransition function

${\displaystyle \psi {\varphi }^{-1}:\varphi \left(U\cap V\right)\to \psi \left(U\cap V\right)}$

Such a map is a homeomorphism between open subsets of${\displaystyle {\mathbb{R}}^n}$. 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, fordifferentiable manifolds the transition maps are required to besmooth.

A 0-manifold is just adiscrete space. A discrete space is second-countable if and only if it iscountable.^[7]

Every nonempty, paracompact, connected 1-manifold is homeomorphic either toR or thecircle.^[7]

Every nonempty, compact, connected 2-manifold (orsurface) is homeomorphic to thesphere, aconnected sum oftori, or a connected sum ofprojective planes.^[8]

A classification of 3-manifolds results fromThurston's geometrization conjecture, proven byGrigori Perelman in 2003. More specifically, Perelman's results provide an algorithm for deciding if two three-manifolds are homeomorphic to each other.^[9]

The full classification ofn-manifolds forn greater than three is known to be impossible; it is at least as hard as theword problem ingroup theory, which is known to bealgorithmically undecidable.^[10]

In fact, there is noalgorithm for deciding whether a given manifold issimply connected. There is, however, a classification of simply connected manifolds of dimension ≥ 5.^[11][12]

A slightly more general concept is sometimes useful. Atopological manifold with boundary is aHausdorff space in which every point has a neighborhood homeomorphic to an open subset of Euclideanhalf-space (for a fixedn):

${\displaystyle {\mathbb{R}}_{+}^n=\{(x_1,\dots ,x_n)\in {\mathbb{R}}^n:x_n\ge 0\}.}$

Every topological manifold is a topological manifold with boundary, but not vice versa.^[7]

There are several methods of creating manifolds from other manifolds.

IfM is anm-manifold andN is ann-manifold, theCartesian productM×N is a (m+n)-manifold when given theproduct topology.^[13]

Thedisjoint union of a countable family ofn-manifolds is an-manifold (the pieces must all have the same dimension).^[7]

Theconnected sum of twon-manifolds is defined by removing an open ball from each manifold and taking thequotient 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 anothern-manifold.^[7]

Any open subset of ann-manifold is ann-manifold with thesubspace topology.^[13]

• Gauld, D. B. (1974). "Topological Properties of Manifolds".The American Mathematical Monthly.81 (6). Mathematical Association of America: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 0-691-08191-3.
• Lee, John M. (2000).Introduction to Topological Manifolds. Graduate Texts in Mathematics202. New York: Springer.ISBN 0-387-98759-2.

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