|
|
| Line 1: |
Line 1: |
| In [[topology]], a branch of [[mathematics]], a '''topological manifold''' is a [[topological space]] (which may also be a [[Hausdorff space|separated space]]) which locally resembles [[real numbers|real]] ''n''-[[dimension (mathematics)|dimensional]] space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics.
| | Luke Bryan is really a superstar during the generating as well as the career advancement to start with 2nd to his 3rd place album, And , is definitely the proof. He burst open to the picture in 2011 with his funny mixture of downward-house convenience, movie superstar fantastic seems and lines, is set t inside a major way. The latest record Top around the country graph and #2 on the take graphs, creating it the next top very first during that time of 2000 for a region designer. <br><br><br><br>The kid of your , knows patience and dedication are important elements in terms of a successful job- . His initially record, Remain Me, made the Top strikes “All My Friends cheap tickets luke bryan ([http://www.museodecarruajes.org museodecarruajes.org]) Say” and “Country Man,” when his energy, Doin’ Point, identified the artist-about three straight No. 8 men and women: More Calling Is often a Excellent Point.”<br><br>In the drop of 2000, Concerts: Luke & that have an impressive list of , which includes Urban. “It’s much like you are getting a authorization buy luke bryan tickets ([http://minioasis.com look at here now]) to visit to another level, affirms all those musicians that have been an element of the Concert tourabove in a larger sized amount of designers.” It packaged as the best excursions within its ten-calendar year background.<br><br>my page ... concert dates for luke bryan ([http://lukebryantickets.lazintechnologies.com look at here now]) |
| | |
| 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.
| |
| | |
| == Formal definition ==
| |
| | |
| A [[topological space]] ''X'' is called '''locally Euclidean''' if there is a non-negative integer ''n'' such that every point in ''X'' has a [[neighborhood (topology)|neighborhood]] which is [[homeomorphic]] to the [[Euclidean space]] '''E'''<sup>''n''</sup> (or, equivalently, to the [[real coordinate space|real ''n''-space]] '''R'''<sup>''n''</sup>, or to some [[domain (mathematical analysis)|connected open subset]] of either of two).<ref>The [[topology (structure)|topology]] of '''E'''<sup>''n''</sup> is identical to the [[standard topology]] of '''R'''<sup>''n''</sup>, so these two spaces are not distinguished in topology. Also, any non-empty [[open subset]] of '''E'''<sup>''n''</sup> contains an Euclidean [[open ball]], which is homeomorphic to the entire space.</ref>
| |
| | |
| 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 '''R'''<sup>''n''</sup>.
| |
| | |
| == Examples ==
| |
| | |
| * The [[real coordinate space]] '''R'''<sup>''n''</sup> is the prototypical ''n''-manifold.
| |
| * Any [[discrete space]] is a 0-dimensional manifold.
| |
| * A [[circle]] is a [[compact space|compact]] 1-manifold.
| |
| * A [[torus]] and a [[Klein bottle]] are compact 2-manifolds (or [[surface]]s).
| |
| * The [[n-sphere|''n''-dimensional sphere]] ''S''<sup>''n''</sup> is a compact ''n''-manifold.
| |
| * The [[n-torus|''n''-dimensional torus]] '''T'''<sup>''n''</sup> (the product of ''n'' circles) is a compact ''n''-manifold.
| |
| * [[Projective space]]s over the [[real number|reals]], [[complex number|complexes]], or [[quaternion]]s are compact manifolds.
| |
| ** [[Real projective space]] '''RP'''<sup>''n''</sup> is a ''n''-dimensional manifold.
| |
| ** [[Complex projective space]] '''CP'''<sup>''n''</sup> is a 2''n''-dimensional manifold.
| |
| ** [[Quaternionic projective space]] '''HP'''<sup>''n''</sup> is a 4''n''-dimensional manifold.
| |
| * Manifolds related to projective space include [[Grassmannian]]s, [[flag manifold]]s, and [[Stiefel manifold]]s.
| |
| * [[Lens space]]s are a class of manifolds that are [[quotient space|quotient]]s of odd-dimensional spheres.
| |
| * [[Lie group]]s are manifolds endowed with a [[group (mathematics)|group]] structure.
| |
| * Any open subset of an ''n''-manifold is a ''n''-manifold with the [[subspace topology]].
| |
| * If ''M'' is an ''m''-manifold and ''N'' is an ''n''-manifold, the [[product (topology)|product]] ''M'' × ''N'' is a (''m''+''n'')-manifold.
| |
| * The [[disjoint union (topology)|disjoint union]] of a family of ''n''-manifolds is a ''n''-manifold (the pieces must all have the same dimension).
| |
| * The [[connected sum]] 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 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 space|locally compact]], [[locally connected space|locally connected]], [[first-countable space|first countable]], [[locally contractible space|locally contractible]], and [[locally metrizable space|locally metrizable]]. Being locally compact Hausdorff spaces, manifolds are necessarily [[Tychonoff space]]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-compact]]ness and second-countability are the same. Indeed, a Hausdorff manifold is a locally compact Hausdorff space, hence it is (completely) regular [http://topospaces.subwiki.org/wiki/Locally_compact_Hausdorff_implies_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 [http://math.stackexchange.com/questions/57348/hausdorff-locally-compact-and-second-countable-is-sigma-compact].
| |
| | |
| A manifold need not be connected, but every manifold ''M'' is a [[disjoint union (topology)|disjoint union]] of connected manifolds. These are just the [[connected component (topology)|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 space|T<sub>1</sub>]].
| |
| | |
| 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 space|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 (mathematics)|pathological]]. An example of a non-paracompact manifold is given by the [[long line (topology)|long line]]. Paracompact manifolds have all the topological properties of metric spaces. In particular, they are [[perfectly normal Hausdorff space]]s.
| |
| | |
| Manifolds are also commonly required to be [[second-countable space|second-countable]]. This is precisely the condition required to ensure that the manifold [[embedding|embeds]] in some finite-dimensional Euclidean space. For any manifold the properties of being second-countable, [[Lindelöf space|Lindelöf]], and [[σ-compact space|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 set|countable]] number of [[connected component (topology)|connected component]]s. In particular, a connected manifold is paracompact if and only if it is second-countable.
| |
| Every second-countable manifold is [[separable space|separable]] and paracompact. Moreover, if a manifold is separable and paracompact then it is also second-countable.
| |
| | |
| Every [[compact space|compact]] manifold is second-countable and paracompact.
| |
| | |
| ===Dimensionality===
| |
| | |
| 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.
| |
| | |
| 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]].
| |
| | |
| ==Coordinate charts==
| |
| | |
| By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open subset of '''R'''<sup>''n''</sup>. 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 '''R'''<sup>''n''</sup>. 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 '''R'''<sup>''n''</sup>.
| |
| *every point of ''M'' has a neighborhood homeomorphic to '''R'''<sup>''n''</sup> itself.
| |
| A Euclidean neighborhood homeomorphic to an open ball in '''R'''<sup>''n''</sup> is called a '''Euclidean ball'''. Euclidean balls form a [[basis (topology)|basis]] for the topology of a locally Euclidean space.
| |
| | |
| For any Euclidean neighborhood ''U'' a homeomorphism φ : ''U'' → φ(''U'') ⊂ '''R'''<sup>''n''</sup> 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 [[cover (topology)|covered]] by Euclidean neighborhoods. A set of Euclidean neighborhoods that cover ''M'', together with their coordinate charts, is called an '''[[atlas (topology)|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'''
| |
| :ψφ<sup>−1</sup> : φ(''U'' ∩ ''V'') → ψ(''U'' ∩ ''V'').
| |
| Such a map is a homeomorphism between open subsets of '''R'''<sup>''n''</sup>. 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.
| |
| | |
| ==Classification of manifolds==
| |
| | |
| A 0-manifold is just a [[discrete space]]. 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 set|countable]].
| |
| | |
| Every nonempty, paracompact, connected 1-manifold is homeomorphic either to '''R''' or the [[circle]]. The unconnected ones are just [[disjoint union]]s of these.
| |
| | |
| Every nonempty, compact, connected 2-manifold (or [[surface]]) is homeomorphic to the [[sphere]], a [[connected sum]] of [[torus|tori]], or a connected sum of [[real projective plane|projective plane]]s. See the [[classification theorem for surfaces]] for more details.
| |
| | |
| A classification of 3-manifolds results from
| |
| [[Thurston's geometrization conjecture]]{{clarify|date=April 2013|reason=Existence of at least one global differential structure on any topological 3-manifold is not an obvious fact}} whose proof was sketched by [[Grigori Perelman]]. The details have been provided by other members of the mathematical community.
| |
| | |
| 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 for groups|word problem]] in [[group theory]], which is known to be [[undecidable problem|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.
| |
| | |
| ==Manifolds with boundary==
| |
| {{main|Manifold 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 (geometry)|half-space]] (for a fixed ''n''):
| |
| :<math>\mathbb R^n_{+} = \{(x_1,\ldots,x_n) \in \mathbb R^n : x_n \ge 0\}.</math>
| |
| The terminology is somewhat confusing: every topological manifold is a topological manifold with boundary, but not vice-versa.
| |
| | |
| ==See also==
| |
| | |
| *[[3-manifold]]
| |
| *[[4-manifold]]
| |
| *[[5-manifold]]
| |
| | |
| == Footnotes ==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| | |
| *{{cite journal | last = Gauld | first = D. B. | year = 1974 | title = Topological Properties of Manifolds | journal = The American Mathematical Monthly | volume = 81 | issue = 6 | pages = 633–636 | doi = 10.2307/2319220 | publisher = Mathematical Association of America | jstor = 2319220}}
| |
| *{{cite book | last = Kirby | first = Robion C. | authorlink = Robion Kirby | coauthors = Siebenmann, Laurence C. | title = Foundational Essays on Topological Manifolds. Smoothings, and Triangulations | location = Princeton | publisher = Princeton University Press | year = 1977 | isbn = 0-691-08191-3 | url = http://www.maths.ed.ac.uk/~aar/papers/ks.pdf}}
| |
| *{{cite book | first = John M. | last = Lee | year = 2000 | title = Introduction to Topological Manifolds | series = Graduate Texts in Mathematics '''202''' | publisher = Springer | location = New York | isbn = 0-387-98759-2}}
| |
| | |
| [[Category:Manifolds]]
| |
| [[Category:Properties of topological spaces]]
| |
Luke Bryan is really a superstar during the generating as well as the career advancement to start with 2nd to his 3rd place album, And , is definitely the proof. He burst open to the picture in 2011 with his funny mixture of downward-house convenience, movie superstar fantastic seems and lines, is set t inside a major way. The latest record Top around the country graph and #2 on the take graphs, creating it the next top very first during that time of 2000 for a region designer.
The kid of your , knows patience and dedication are important elements in terms of a successful job- . His initially record, Remain Me, made the Top strikes “All My Friends cheap tickets luke bryan (museodecarruajes.org) Say” and “Country Man,” when his energy, Doin’ Point, identified the artist-about three straight No. 8 men and women: More Calling Is often a Excellent Point.”
In the drop of 2000, Concerts: Luke & that have an impressive list of , which includes Urban. “It’s much like you are getting a authorization buy luke bryan tickets (look at here now) to visit to another level, affirms all those musicians that have been an element of the Concert tourabove in a larger sized amount of designers.” It packaged as the best excursions within its ten-calendar year background.
my page ... concert dates for luke bryan (look at here now)