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| [[File:Covering space diagram.svg|thumb|250px|right|A covering map satisfies the local triviality condition. Intuitively, such maps locally project a "stack of pancakes" above an [[open set|open region]], ''U'', onto ''U''.]]
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| In [[mathematics]], more specifically [[algebraic topology]], a '''covering map''' (also '''covering projection''') is a [[continuous function|continuous]] [[Function (mathematics)|function]] ''p'' <ref>{{cite book |last=Spanier |first=Edwin |title=Algebraic Topology |year=1966 |publisher=McGraw-Hill |page=62|isbn=}}</ref> from a [[topological space]], ''C'', to a topological space, ''X'', such that each point in ''X'' has an [[open set|open]] neighbourhood '''evenly covered''' by ''p'' (as shown in the image); the precise definition is given below. In this case, ''C'' is called a '''covering space''' and ''X'' the '''base space''' of the covering projection. The definition implies that every covering map is a [[local homeomorphism]].
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| Covering spaces play an important role in [[homotopy theory]], [[harmonic analysis]], [[Riemannian geometry]] and [[differential topology]]. In Riemannian geometry for example, [[Ramification#In_algebraic_topology|ramification]] is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the [[fundamental group]]. An important application comes from the result that, if ''X'' is a "sufficiently good" [[topological space]], there is a [[bijection]] from the collection of all isomorphism classes of [[connected space|connected coverings]] of ''X'' and subgroups of the [[fundamental group]] of ''X''.
| | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
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| == Formal definition ==
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| Let ''X'' be a [[topological space]]. A '''covering space''' of ''X'' is a space ''C'' together with a [[continuous function (topology)|continuous]] [[surjective]] map
| | :<math forcemathmode="png">E=mc^2</math> |
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| :<math>p \colon C \to X\,</math> | | '''source''' |
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| such that for every {{nowrap|''x'' ∈ ''X''}}, there exists an [[open set|open]] [[neighborhood (topology)|neighborhood]] ''U'' of ''x'', such that ''p''<sup>−1</sup>(''U'') (the [[inverse image]] of ''U'' under ''p'') is a union of disjoint open sets in ''C'', each of which is mapped [[homeomorphism|homeomorphically]] onto ''U'' by ''p''.<ref name="Chernavskii">{{harvnb|Chernavskii|2001}}</ref><ref name="Munkres p336">{{harvnb|Munkres|2000|p=336}}</ref>
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
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| The map ''p'' is called the '''covering map''',<ref name="Munkres p336"/> the space ''X'' is often called the '''base space''' of the covering, and the space ''C'' is called the '''total space''' of the covering. For any point ''x'' in the base the inverse image of ''x'' in ''C'' is necessarily a [[discrete space]]<ref name="Munkres p336"/> called the [[Fiber (mathematics)|fiber]] over ''x''.
| | ==Demos== |
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| The special open neighborhoods ''U'' of ''x'' given in the definition are called '''evenly-covered neighborhoods'''. The evenly-covered neighborhoods form an [[open cover]] of the space ''X''. The homeomorphic copies in ''C'' of an evenly-covered neighborhood ''U'' are called the '''sheets''' over ''U''. One generally pictures ''C'' as "hovering above" ''X'', with ''p'' mapping "downwards", the sheets over ''U'' being horizontally stacked above each other and above ''U'', and the fiber over ''x'' consisting of those points of ''C'' that lie "vertically above" ''x''. In particular, covering maps are locally trivial. This means that locally, each covering map is 'isomorphic' to a projection in the sense that there is a homeomorphism, ''h'', from the pre-image ''p''<sup>−1</sup>(''U''), of an evenly covered neighbourhood ''U'', onto {{nowrap|''U'' × ''F''}}, where ''F'' is the fiber, satisfying the '''local trivialization condition''', which is that, if we project {{nowrap|''U'' × ''F''}} onto ''U'', {{nowrap|''π'' : ''U'' × ''F'' → ''U''}}, so the composition of the projection ''π'' with the homeomorphism ''h'' will be a map {{nowrap|''π'' ∘ ''h''}} from the pre-image ''p''<sup>−1</sup>(''U'') onto ''U'', then the derived composition {{nowrap|''π'' ∘ ''h''}} will equal ''p'' locally (within ''p''<sup>−1</sup>(''U'')).
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| ===Alternative definitions===
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| Many authors impose some [[connectedness|connectivity]] conditions on the spaces ''X'' and ''C'' in the definition of a covering map. In particular, many authors require both spaces to be [[path-connected]] and [[locally path-connected]].<ref>{{Cite book|title = An Introduction to Knot Theory|date = 1997|last = Lickorish|pages = 66-67}}</ref><ref>{{Cite book|title = Topology and Geometry|last = Bredon|year = 1997|isbn = 978-0387979267}}</ref> This can prove helpful because many theorems hold only if the spaces in question have these properties. Some authors omit the assumption of surjectivity, for if ''X'' is connected and ''C'' is nonempty then surjectivity of the covering map actually follows from the other axioms.
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| == Examples ==
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| * Every space trivially covers itself. | | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)] |
| | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| * A topological space ''X'' has a [[universal cover]] if and only if it is connected, locally path connected, and semi-locally simply connected.
| | ==Test pages == |
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| * <math>\mathbb{R}</math> is the universal cover of the unit circle ''S<sup>1</sup>''
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| | *[[Displaystyle]] |
| | *[[MathAxisAlignment]] |
| | *[[Styling]] |
| | *[[Linebreaking]] |
| | *[[Unique Ids]] |
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| * The [[spin group]] ''Spin(n)'' is a double cover of the [[special orthogonal group]] and a universal cover when ''n > 2''. The accidental, or [[exceptional isomorphism|exceptional isomorphisms]] for Lie groups then give isomorphisms between spin groups in low dimension and classical Lie groups. | | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| * The [[unitary group]] ''U(n)'' has universal cover ''SU(n) x R''
| | ==Bug reporting== |
| | | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
| * The [[n-sphere]] ''S<sup>n</sup>'' is a double cover of real projective space ''RP<sup>n</sup>'' and is a universal cover for ''n >1''.
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| * Every manifold has an [[orientable double cover]] that is connected if and only if the manifold is non-orientable.
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| * The [[uniformization theorem]] asserts that every compact Riemann surface has a universal cover conformally equivalent to the [[Riemann sphere]], the complex plane, or the unit disc.
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| * The universal cover of a wedge of ''n'' circles is the [[Cayley graph]] of the free group on ''n'' generators, i.e. a [[Bethe lattice]].
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| * The [[torus]] is a double cover of the [[Klein bottle]]
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| * Every graph has a [[bipartite double cover]]. Since every graph is homotopic to a wedge of circles, its universal cover is a Cayley graph.
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| * Every immersion from a compact manifold to a manifold of the same dimension is a covering of its image.
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| == Properties ==
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| ===Common local properties===
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| * Every cover {{nowrap|''p'': ''C'' → ''X''}} is a [[local homeomorphism]]<ref>{{harvnb|Munkres|2000|p=338}}</ref>—that is, for every {{nowrap|''c'' ∈ ''C''}}, there exists a neighborhood {{nowrap|''U'' ⊆ ''C''}} of ''c'' and a neighborhood {{nowrap|''V'' ⊆ ''X''}} of ''p''(''c'') such that the restriction of ''p'' to ''U'' yields a [[homeomorphism]] from ''U'' to ''V''. This implies that ''C'' and ''X'' share all local properties. If ''X'' is [[simply connected]] and ''C'' is connected, then this holds globally as well, and the covering ''p'' is a homeomorphism.
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| * If {{nowrap|''p'' : ''E'' → ''B''}} and {{nowrap|''p''′ : ''E''′ → ''B''′}} are covering maps, then so is the map {{nowrap|''p'' × ''p''′ : ''E'' × ''E''′ → ''B'' × ''B′''}} given by {{nowrap|1=(''p'' × ''p''′)(''e'', ''e''′) = (''p''(''e''), ''p''′(''e''′))}}.<ref>{{harvnb|Munkres|2000|p=339|loc=Theorem 53.3}}</ref> | |
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| ===Homeomorphism of the fibres===
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| For every ''x'' in ''X'', the fiber over ''x'' is a [[discrete space|discrete]] subset of ''C''.<ref name="Munkres p336"/> On every [[connected space|connected component]] of ''X'', the fibers are homeomorphic.
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| If ''X'' is connected, there is a discrete space ''F'' such that for every ''x'' in ''X'' the fiber over ''x'' is [[homeomorphism|homeomorphic]] to ''F'' and, moreover, for every ''x'' in ''X'' there is a neighborhood ''U'' of ''x'' such that its full pre-image ''p''<sup>−1</sup>(''U'') is homeomorphic to {{nowrap|''U'' × ''F''}}. In particular, the [[cardinality]] of the fiber over ''x'' is equal to the cardinality of ''F'' and it is called the '''degree of the cover''' {{nowrap|''p'' : ''C'' → ''X''}}. Thus, if every fiber has ''n'' elements, we speak of an '''''n''-fold covering''' (for the case {{nowrap|1=''n'' = 1}}, the covering is trivial; when {{nowrap|1=''n'' = 2}}, the covering is a '''double cover'''; when {{nowrap|1=''n'' = 3}}, the covering is a '''triple cover''' and so on).
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| ===Lifting properties===
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| {{see also|Homotopy lifting property}}
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| If {{nowrap|1=''p'' : ''C'' → ''X''}} is a cover and γ is a path in ''X'' (i.e. a continuous map from the [[unit interval]] {{nowrap|[0, 1]}} into ''X'') and {{nowrap|''c'' ∈ ''C''}} is a point "lying over" γ(0) (i.e. {{nowrap|1=''p''(''c'') = γ(0))}}, then there exists a unique path Γ in ''C'' lying over γ (i.e. {{nowrap|1=''p'' ∘ Γ = γ}}) such that with {{nowrap|1=Γ(0) = ''c''}}. The curve Γ is called the '''lift''' of γ. If ''x'' and ''y'' are two points in ''X'' connected by a path, then that path furnishes a [[bijection]] between the fiber over ''x'' and the fiber over ''y'' via the lifting property.
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| More generally, let {{nowrap|''f'' : ''Z'' → ''X''}} be a continuous map to ''X'' from a [[path connected]] and [[locally path connected]] space ''Z''. Fix a base-point {{nowrap|''z'' ∈ ''Z''}}, and choose a point {{nowrap|''c'' ∈ ''C''}} "lying over" ''f''(''z'') (i.e. {{nowrap|1=''p''(''c'') = ''f''(''z'')}}). Then there exists a '''lift''' of ''f'' (that is, a continuous map {{nowrap|''g'' : ''Z'' → ''C''}} for which {{nowrap|1=''p'' ∘ ''g'' = ''f''}} and {{nowrap|1=''g''(''z'') = ''c''}}) [[if and only if]] the [[induced homomorphism]]s {{nowrap|''f''<sub>#</sub> : π<sub>1</sub>(''Z'', ''z'') → π<sub>1</sub>(''X'', ''f''(''z''))}} and {{nowrap|''p''<sub>#</sub> : π<sub>1</sub>(''C'', ''c'') → π<sub>1</sub>(''X'', ''f''(''z''))}} at the level of [[fundamental groups]] satisfy
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| {{NumBlk|::|<math>f_\#(\pi_1(Z,z))\subset p_\#(\pi_1(C,c)). </math>|{{EquationRef|♠}}}}
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| Moreover, if such a lift ''g'' of ''f'' exists, it is unique.
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| In particular, if the space ''Z'' is assumed to be [[simply connected]] (so that {{nowrap|π<sub>1</sub>(''Z'', ''z'')}} is trivial), condition {{EquationNote|♠|(♠)}} is automatically satisfied, and every continuous map from ''Z'' to ''X'' can be lifted. Since the unit interval {{nowrap|[0, 1]}} is simply connected, the lifting property for paths is a special case of the lifting property for maps stated above.
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| If {{nowrap|''p'' : ''C'' → ''X''}} is a covering and {{nowrap|''c'' ∈ ''C''}} and {{nowrap|''x'' ∈ ''X''}} are such that {{nowrap|1=''p''(''c'') = ''x''}}, then ''p''<sub>#</sub> is injective at the level of [[fundamental groups]], and the induced homomorphisms {{nowrap|''p''<sub>#</sub> : π<sub>''n''</sub>(''C'', ''c'') → π<sub>''n''</sub>(''X'', ''x'')}} are [[group isomorphism|isomorphisms]] for all {{nowrap|''n'' ≥ 2}}. Both of these statements can be deduced from the lifting property for continuous maps. Surjectivity of ''p''<sub>#</sub> for {{nowrap|''n'' ≥ 2}} follows from the fact that for all such ''n'', the ''n''-sphere '''S'''<sup>''n''</sup> is simply connected and hence every continuous map from '''S'''<sup>''n''</sup> to ''X'' can be lifted to ''C''.
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| ===Equivalence===
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| Let {{nowrap|''p''<sub>1</sub> : ''C''<sub>1</sub> → ''X''}} and {{nowrap|''p''<sub>2</sub> : ''C''<sub>2</sub> → ''X''}} be two coverings. One says that the two coverings ''p''<sub>1</sub> and ''p''<sub>2</sub> are '''equivalent''' if there exists a homeomorphism {{nowrap|''p''<sub>21</sub> : ''C''<sub>2</sub> → ''C''<sub>1</sub>}} and such that {{nowrap|1=''p''<sub>2</sub> = ''p''<sub>1</sub> ∘ ''p''<sub>21</sub>}}. Equivalence classes of coverings correspond to conjugacy classes of subgroups of the [[fundamental group]] of ''X'', as discussed below. If {{nowrap|''p''<sub>21</sub> : ''C''<sub>2</sub> → ''C''<sub>1</sub>}} is a covering (rather than a homeomorphism) and {{nowrap|1=''p''<sub>2</sub> = ''p''<sub>1</sub> ∘ ''p''<sub>21</sub>}}, then one says that ''p''<sub>2</sub> '''dominates''' ''p''<sub>1</sub>.
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| ===Covering of a manifold=== | |
| Since coverings are local [[homeomorphism]]s, a covering of a topological ''n''-[[manifold]] is an ''n''-manifold. (One can prove that the covering space is [[second-countable]] from the fact that the [[fundamental group]] of a manifold is always [[countable]].) However a space covered by an ''n''-manifold may be a [[non-Hausdorff manifold]]. An example is given by letting ''C'' be the plane with the origin deleted and ''X'' the quotient space obtained by identifying every point {{nowrap|(''x'', ''y'')}} with {{nowrap|(2''x'', ''y''/2)}}. If {{nowrap|''p'' : ''C'' → ''X''}} is the quotient map then it is a covering since the action of ''Z'' on ''C'' generated by {{nowrap|1=''f''(''x'', ''y'') = (2''x'', ''y''/2)}} is [[properly discontinuous]]. The points {{nowrap|''p''(1, 0)}} and {{nowrap|''p''(0, 1)}} do not have disjoint neighborhoods in ''X''.
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| Any covering space of a differentiable manifold may be equipped with a (natural) differentiable structure that turns ''p'' (the covering map in question) into a [[local diffeomorphism]] – a map with constant [[Rank (differential topology)|rank]] ''n''.
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| == Universal covers ==
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| A covering space is a '''universal covering space''' if it is [[simply connected]]. The name ''[[universal property|universal]] cover'' comes from the following important property: if the mapping {{nowrap|''q'': ''D'' → ''X''}} is a universal cover of the space ''X'' and the mapping {{nowrap|''p'' : ''C'' → ''X''}} is any cover of the space ''X'' where the covering space ''C'' is connected, then there exists a covering map {{nowrap|''f'' : ''D'' → ''C''}} such that {{nowrap|1=''p'' ∘ ''f'' = ''q''}}. This can be phrased as
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| <BLOCKQUOTE >The universal cover of the space ''X'' covers all connected covers of the space ''X''.</BLOCKQUOTE >
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| The map ''f'' is unique in the following sense: if we fix a point ''x'' in the space ''X'' and a point ''d'' in the space ''D'' with {{nowrap|1=''q''(''d'') = ''x''}} and a point ''c'' in the space ''C'' with {{nowrap|1=''p''(''c'') = ''x''}}, then there exists a unique covering map {{nowrap|''f'' : ''D'' → ''C''}} such that {{nowrap|1=''p'' ∘ ''f''= ''q''}} and {{nowrap|1=''f''(''d'') = ''c''}}.
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| If the space ''X'' has a universal cover then that universal cover is essentially unique: if the mappings {{nowrap|''q''<sub>1</sub> : ''D''<sub>1</sub> → ''X''}} and {{nowrap|''q''<sub>2</sub> : ''D''<sub>2</sub> → ''X''}} are two universal covers of the space ''X'' then there exists a homeomorphism {{nowrap|''f'' : ''D''<sub>1</sub> → ''D''<sub>2</sub>}} such that {{nowrap|1=''q''<sub>2</sub> ∘ ''f'' = ''q''<sub>1</sub>}}.
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| The space ''X'' has a universal cover if it is [[Connected space|connected]], [[Connected space#Local connectedness|locally path-connected]] and [[semi-locally simply connected]]. The universal cover of the space ''X'' can be constructed as a certain space of paths in the space ''X''.
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| The example {{nowrap|'''R''' → '''S'''<sup>1</sup>}} given above is a universal cover. The map {{nowrap|'''S'''<sup>3</sup> → SO(3)}} from [[quaternion|unit quaternions]] to [[rotation]]s of 3D space described in [[quaternions and spatial rotation]] is also a universal cover.
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| If the space ''X'' carries some additional structure, then its universal cover usually inherits that structure:
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| * if the space ''X'' is a [[manifold]], then so is its universal cover ''D''
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| * if the space ''X'' is a [[Riemann surface]], then so is its universal cover ''D'', and ''p'' is a [[holomorphic]] map
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| * if the space ''X'' is a [[Lorentzian manifold]], then so is its universal cover. Furthermore, suppose the subset ''p''<sup>−1</sup>(''U'') is a [[disjoint union]] of open sets each of which is diffeomorphic with ''U'' by the mapping ''p''. If the space ''X'' contains a [[closed timelike curve]] (CTC), then the space ''X'' is [[timelike multiply connected]] (no CTC can be [[timelike homotopic]] to a point, as that point would not be causally well-behaved), its universal (diffeomorphic) cover is [[timelike simply connected]] (it does not contain a CTC).
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| * if ''X'' is a [[Lie group]] (as in the two examples above), then so is its universal cover ''D'', and the mapping ''p'' is a homomorphism of Lie groups. In this case the universal cover is also called the ''[[universal covering group]]''. This has particular application to [[representation theory]] and [[quantum mechanics]], since ordinary [[Group representation|representations]] of the universal covering group (''D'') are [[projective representation]]s of the original (classical) group (''X'').
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| The universal cover first arose in the theory of [[analytic functions]] as the natural domain of an [[analytic continuation]].
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| == G-coverings ==
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| Let ''G'' be a [[discrete group]] [[group action|acting]] on the [[topological space]] ''X''. It is natural to ask under what conditions the projection from ''X'' to the [[orbit space]] ''X''/''G'' is a covering map. This is not always true since the action may have fixed points. An example for this is the cyclic group of order 2 acting on a product {{nowrap|''X'' × ''X''}} by the twist action where the non-identity element acts by {{nowrap|(''x'', ''y'') ↦ (''y'', ''x'')}}. Thus the study of the relation between the fundamental groups of ''X'' and ''X''/''G'' is not so straightforward.
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| However the group ''G'' does act on the fundamental groupoid of ''X'', and so the study is best handled by considering groups acting on groupoids, and the corresponding ''orbit groupoids''. The theory for this is set down in Chapter 11 of the book ''Topology and groupoids'' referred to below. The main result is that for discontinuous actions of a group ''G'' on a Hausdorff space ''X'' which admits a universal cover, then the fundamental groupoid of the orbit space ''X''/''G'' is isomorphic to the orbit groupoid of the fundamental groupoid of ''X'', i.e. the quotient of that groupoid by the action of the group ''G''. This leads to explicit computations, for example of the fundamental group of the symmetric square of a space.
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| == {{anchor|Deck transformation}} Deck transformation group, regular covers ==
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| A '''deck transformation''' or '''automorphism''' of a cover {{nowrap|''p'' : ''C'' → ''X''}} is a [[homeomorphism]] {{nowrap|''f'' : ''C'' → ''C''}} such that {{nowrap|1=''p'' ∘ ''f'' = ''p''}}. The set of all deck transformations of ''p'' forms a group under [[function composition|composition]], the '''deck transformation group''' Aut(''p''). Deck transformations are also called '''covering transformations'''. Every deck transformation [[permutation|permutes]] the elements of each fiber. This defines a [[group action]] of the deck transformation group on each fiber. Note that by the unique lifting property, if ''f'' is not the identity and ''C'' is path connected, then ''f'' has no [[fixed point (mathematics)|fixed points]].
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| Now suppose {{nowrap|''p'' : ''C'' → ''X''}} is a covering map and ''C'' (and therefore also ''X'') is connected and locally path connected. The action of Aut(''p'') on each fiber is [[group action|free]]. If this action is [[group action#Types of actions|transitive]] on some fiber, then it is transitive on all fibers, and we call the cover '''regular''' (or '''normal''' or '''Galois'''). Every such regular cover is a [[principal bundle|principal ''G''-bundle]], where {{nowrap|1=''G'' = Aut(''p'')}} is considered as a discrete topological group.
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| Every universal cover {{nowrap|''p'' : ''D'' → ''X''}} is regular, with deck transformation group being isomorphic to the [[fundamental group]] π<sub>1</sub>(''X'').
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| The example {{nowrap|''p'' : '''C'''<sup>×</sup> → '''C'''<sup>×</sup>}} with {{nowrap|1=''p''(''z'') = ''z''<sup>''n''</sup>}} from above is a regular cover. The deck transformations are multiplications with ''n''-th [[root of unity|roots of unity]] and the deck transformation group is therefore isomorphic to the [[cyclic group]] ''C''<sub>''n''</sub>.
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| Another example: {{nowrap|''p'' : '''C'''* → '''C'''*}} with {{nowrap|1=''p''(''z'') = ''z''<sup>''n''!</sup>}} from above is regular. Here one has a hierarchy of deck transformation groups. In fact ''C''<sub>''x''!</sub> is a subgroup of ''C''<sub>''y''!</sub>, for {{nowrap|1 ≤ ''x'' ≤ ''y'' ≤ ''n''}}.
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| == Monodromy action ==
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| {{main|Monodromy}}
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| Again suppose {{nowrap|''p'' : ''C'' → ''X''}} is a covering map and ''C'' (and therefore also ''X'') is connected and locally path connected. If ''x'' is in ''X'' and ''c'' belongs to the fiber over ''x'' (i.e. {{nowrap|1=''p''(''c'') = ''x''}}), and {{nowrap|γ : [0, 1] → ''X''}} is a path with {{nowrap|1=γ(0) = γ(1) = ''x''}}, then this path lifts to a unique path in ''C'' with starting point ''c''. The end point of this lifted path need not be ''c'', but it must lie in the fiber over ''x''. It turns out that this end point only depends on the class of γ in the fundamental group {{nowrap|π<sub>1</sub>(''X'', ''x'')}}. In this fashion we obtain a right [[group action]] of {{nowrap|π<sub>1</sub>(''X'', ''x'')}} on the fiber over ''x''. This is known as the '''monodromy action'''.
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| There are two actions on the fiber over {{nowrap|''x'' : Aut(''p'')}} acts on the left and {{nowrap|π<sub>1</sub>(''X'', ''x'')}} acts on the right. These two actions are compatible in the following sense:
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| <math>f\cdot(c\cdot\gamma) = (f\cdot c)\cdot\gamma</math> for all ''f'' in Aut(''p''), ''c'' in ''p''<sup>−1</sup>(''x'') and γ in {{nowrap|π<sub>1</sub>(''X'', ''x'')}}.
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| If ''p'' is a universal cover, then Aut(''p'') can be naturally identified with the [[Dual (category theory)|opposite]] group of {{nowrap|π<sub>1</sub>(''X'', ''x'')}} so that the left action of the opposite group of {{nowrap|π<sub>1</sub>(''X'', ''x'')}} coincides with the action of Aut(''p'') on the fiber over ''x''. Note that Aut(''p'') and {{nowrap|π<sub>1</sub>(''X'', ''x'')}} are naturally isomorphic in this case (as a group is always naturally isomorphic to its opposite through {{nowrap|''g'' ↦ ''g''<sup>−1</sup>)}}.
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| If ''p'' is a [[Covering_space#Deck_transformation_group.2C_regular_covers|regular]] cover, then Aut(''p'') is naturally isomorphic to a quotient of {{nowrap|π<sub>1</sub>(''X'', ''x'')}}.
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| In general (for good spaces), Aut(''p'') is naturally isomorphic to the quotient of the [[Centralizer and normalizer|normalizer]] of {{nowrap|''p''<sub>*</sub>(π<sub>1</sub>(''C'', ''c''))}} in {{nowrap|π<sub>1</sub>(''X'', ''x'')}} over {{nowrap|''p''<sub>*</sub>(π<sub>1</sub>(''C'', ''c''))}}, where {{nowrap|1=''p''(''c'') = ''x''}}.
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| ==More on the group structure==
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| Let {{nowrap|''p'' : ''C'' → ''X''}} be a covering map where both ''X'' and ''C'' are path-connected. Let {{nowrap|''x'' ∈ ''X''}} be a basepoint of ''X'' and let {{nowrap|''c'' ∈ ''C''}} be one of its pre-images in ''C'', that is {{nowrap|1=''p''(''c'') = ''x''}}. There is an [[induced homomorphism]] of [[fundamental group]]s {{nowrap|''p''<sub>#</sub> : π<sub>1</sub>(''C'', ''c'') → π<sub>1</sub>(''X'',''x'')}} which is injective by the lifting property of coverings. Specifically if ''γ'' is a closed loop at ''c'' such that {{nowrap|1=''p''<sub>#</sub>([''γ'']) = 1}}, that is {{nowrap|''p'' ∘ ''γ''}} is [[null-homotopic]] in ''X'', then consider a null-homotopy of {{nowrap|''p'' ∘ ''γ''}} as a map {{nowrap|''f'' : ''D''<sup>2</sup> → ''X''}} from the 2-disc ''D''<sup>2</sup> to ''X'' such that the restriction of ''f'' to the boundary '''S'''<sup>1</sup> of ''D''<sup>2</sup> is equal to {{nowrap|''p'' ∘ ''γ''}}. By the lifting property the map ''f'' lifts to a continuous map {{nowrap|''g'' : ''D''<sup>2</sup> → ''C''}} such that the restriction of ''f'' to the boundary '''S'''<sup>1</sup> of ''D''<sup>2</sup> is equal to ''γ''. Therefore ''γ'' is [[null-homotopic]] in ''C'', so that the [[Kernel (algebra)|kernel]] of {{nowrap|''p''<sub>#</sub> : π<sub>1</sub>(''C'', ''c'') → π<sub>1</sub>(''X'', ''x'')}} is trivial and thus {{nowrap|''p''<sub>#</sub> : π<sub>1</sub>(''C'', ''c'') → π<sub>1</sub>(''X'', ''x'')}} is an injective homomorphism.
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| Therefore {{nowrap|π<sub>1</sub>(''C'', ''c'')}} is isomorphic to the subgroup {{nowrap|''p''<sub>#</sub>(π<sub>1</sub>(''C'', ''c''))}} of {{nowrap|π<sub>1</sub>(''X'', ''x'')}}. If {{nowrap|''c''<sub>1</sub> ∈ ''C''}} is another pre-image of ''x'' in ''C'' then the [[subgroup]]s {{nowrap|''p''<sub>#</sub>(π<sub>1</sub>(''C'', ''c''))}} and {{nowrap|''p''<sub>#</sub>(π<sub>1</sub>(''C'', ''c''<sub>1</sub>))}} are [[conjugacy class|conjugate]] in {{nowrap|π<sub>1</sub>(''X'', ''x'')}} by ''p''-image of a curve in ''C'' connecting ''c'' to ''c''<sub>1</sub>. Thus a covering map {{nowrap|''p'' : ''C'' → ''X''}} defines a conjugacy class of subgroups of {{nowrap|π<sub>1</sub>(''X'', ''x'')}} and one can show that equivalent covers of ''X'' define the same conjugacy class of subgroups of {{nowrap|π<sub>1</sub>(''X'', ''x'')}}.
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| For a covering {{nowrap|''p'' : ''C'' → ''X''}} the group {{nowrap|''p''<sub>#</sub>(π<sub>1</sub>(''C'', ''c''))}} can also be seen to be equal to
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| :<math>\Gamma_p(c) = \{ [\gamma] : \gamma_C \mbox{ is a closed curve in } C \mbox { passing through } c\in C \}</math>,
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| the set of [[homotopy class]]es of those closed curves γ based at ''x'' whose lifts ''γ<sub>C</sub>'' in ''C'', starting at ''c'', are closed curves at ''c''. If ''X'' and ''C'' are path-connected, the degree of the cover ''p'' (that is, the cardinality of any fiber of ''p'') is equal to the [[Index of a subgroup|index]] [{{nowrap|π<sub>1</sub>(''X'', ''x'') : ''p''<sub>#</sub>(π<sub>1</sub>(''C'', ''c''))}}] of the [[subgroup]] {{nowrap|''p''<sub>#</sub>(π<sub>1</sub>(''C'', ''c''))}} in {{nowrap|π<sub>1</sub>(''X'', ''x'')}}.
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| A key result of the covering space theory says that for a "sufficiently good" space ''X'' (namely, if ''X'' is path-connected, locally path-connected and [[semi-locally simply connected]]) there is in fact a bijection between equivalence classes of path-connected covers of ''X'' and the conjugacy classes of subgroups of the fundamental group {{nowrap|π<sub>1</sub>(''X'', ''x'')}}. The main step in proving this result is establishing the existence of a universal cover, that is a cover corresponding to the trivial subgroup of {{nowrap|π<sub>1</sub>(''X'', ''x'')}}. Once the existence of a universal cover ''C'' of ''X'' is established, if ''H'' ≤ π<sub>1</sub>(''X'', ''x'') is an arbitrary subgroup, the space ''C''/''H'' is the covering of ''X'' corresponding to ''H''. One also needs to check that two covers of ''C'' corresponding to the same (conjugacy class of) subgroup of {{nowrap|π<sub>1</sub>(''X'', ''x'')}} are equivalent. Connected [[cell complex]]es and connected [[manifold]]s are examples of "sufficiently good" spaces.
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| Let ''N''(''Γ<sub>p</sub>'') be the [[normalizer]] of ''Γ<sub>p</sub>'' in {{nowrap|π<sub>1</sub>(''X'', ''x'')}}. The deck transformation group Aut(''p'') is isomorphic to the [[quotient group]] ''N''(''Γ<sub>p</sub>'')/''Γ<sub>p</sub>''. If ''p'' is a universal covering, then ''Γ<sub>p</sub>'' is the [[trivial group]], and Aut(''p'') is isomorphic to π<sub>1</sub>(''X'').
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| Let us reverse this argument. Let ''N'' be a [[normal subgroup]] of {{nowrap|π<sub>1</sub>(''X'', ''x'')}}. By the above arguments, this defines a (regular) covering {{nowrap|''p'' : ''C'' → ''X''}}. Let ''c''<sub>1</sub> in ''C'' be in the fiber of ''x''. Then for every other ''c''<sub>2</sub> in the fiber of ''x'', there is precisely one deck transformation that takes ''c''<sub>1</sub> to ''c''<sub>2</sub>. This deck transformation corresponds to a curve ''g'' in ''C'' connecting ''c''<sub>1</sub> to ''c''<sub>2</sub>.
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| ==Relations with groupoids== | |
| One of the ways of expressing the algebraic content of the theory of covering spaces is using [[groupoids]] and the [[fundamental groupoid]]. The latter functor gives an equivalence of categories
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| <math>\pi_1: \mathrm{TopCov}(X) \to \mathrm{GpdCov}(\pi_1 X)</math>
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| between the category of covering spaces of a reasonably nice space ''X'' and the category of groupoid covering morphisms of π<sub>1</sub>(''X''). Thus a particular kind of ''map'' of spaces is well modelled by a particular kind of ''morphism'' of groupoids. The category of covering morphisms of a groupoid ''G'' is also equivalent to the category of actions of ''G'' on sets, and this allows the recovery of more traditional classifications of coverings. Proofs of these facts are given in the book 'Topology and Groupoids' referenced below.
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| ==Relations with classifying spaces and group cohomology==
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| If ''X'' is a connected [[cell complex]] with [[homotopy group]]s {{nowrap|1=π<sub>''n''</sub>(''X'') = 0}} for all {{nowrap|''n'' ≥ 2}}, then the universal covering space ''T'' of ''X'' is contractible, as follows from applying the [[Whitehead theorem]] to ''T''. In this case ''X'' is a [[classifying space]] or {{nowrap|''K''(''G'', 1)}} for {{nowrap|1=''G'' = π<sub>''1''</sub>(''X'')}}.
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| Moreover, for every {{nowrap|''n'' ≥ 0}} the group of cellular ''n''-chains ''C''<sub>''n''</sub>(''T'') (that is, a [[free abelian group]] with basis given by ''n''-cells in ''T'') also has a natural '''Z'''''G''-[[Module (mathematics)|module]] structure. Here for an ''n''-cell ''σ'' in ''T'' and for ''g'' in ''G'' the cell ''g'' σ is exactly the translate of σ by a covering transformation of ''T'' corresponding to ''g''. Moreover, ''C''<sub>''n''</sub>(''T'') is a [[Free module|free]] '''Z'''''G''-module with free '''Z'''''G''-basis given by representatives of ''G''-orbits of ''n''-cells in ''T''. In this case the standard topological chain complex
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| :<math> \cdots \overset{\partial}{\to} C_n(T)\overset{\partial}{\to} C_{n-1}(T)\overset{\partial}{\to} \cdots \overset{\partial}{\to} C_0(T)\overset{\varepsilon}{\to} \mathbf Z, </math>
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| where ε is the [[augmentation map]], is a [[free resolution|free '''Z'''''G''-resolution]] of '''Z''' (where '''Z''' is equipped with the trivial '''Z'''''G''-module structure, {{nowrap|1=''gm'' = ''m''}} for every {{nowrap|''g'' ∈ ''G''}} and every {{nowrap|''m'' ∈ '''Z'''}}). This resolution can be used to compute [[group cohomology]] of ''G'' with arbitrary coefficients.
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| The method of Graham Ellis for computing group resolutions and other aspects of homological algebra, as shown in his paper in J. Symbolic Comp. and his web page listed below, is to build a universal cover of a prospective {{nowrap|''K''(''G'', 1)}} inductively at the same time as a contracting homotopy of this universal cover. It is the latter which gives the computational method.
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| ==Generalizations==
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| As a homotopy theory, the notion of covering spaces works well when the deck transformation group is discrete, or, equivalently, when the space is [[locally path-connected]]. However, when the deck transformation group is a [[topological group]] whose topology is not [[discrete topology|discrete]], difficulties arise. Some progress has been made for more complex spaces, such as the [[Hawaiian earring]]; see the references there for further information.
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| A number of these difficulties are resolved with the notion of ''semicovering'' due to Jeremy Brazas, see the paper cited below. Every covering map is a semicovering, but semicoverings satisfy the "2 out of 3" rule: given a composition {{nowrap|1=''h'' = ''fg''}} of maps of spaces, if two of the maps are semicoverings, then so also is the third. This rule does not hold for coverings, since the composition of covering maps need not be a covering map.
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| Another generalisation is to actions of a group which are not free. Ross Geoghegan in his 1986 review (MR0760769) of two papers by M.A. Armstrong on the fundamental groups of [[orbit space]]s wrote: "These two papers show which parts of elementary covering space theory carry over from the free to the nonfree case. This is the kind of basic material that ought to have been in standard textbooks on fundamental groups for the last fifty years." At present, "Topology and Groupoids" listed below seems to be the only basic topology text to cover such results.
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| ==Applications==
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| [[Image:Rotating gimbal-xyz.gif|thumb|300px|[[Gimbal lock]] occurs because any map {{nowrap|''T''<sup>3</sup> → '''RP'''<sup>3</sup>}} is not a covering map. In particular, the relevant map carries any element of ''T''<sup>3</sup>, that is, an ordered triple (a,b,c) of angles (real numbers mod 2π), to the composition of the three coordinate axis rotations R<sub>x</sub>(a)∘R<sub>y</sub>(b)∘R<sub>z</sub>(c) by those angles, respectively. Each of these rotations, and their composition, is an element of the rotation group '''SO'''(3), which is topologically '''RP'''<sup>3</sup>.
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| This animation shows a set of three gimbals mounted together to allow ''three'' degrees of freedom. When all three gimbals are lined up (in the same plane), the system can only move in two dimensions from this configuration, not three, and is in ''gimbal lock''. In this case it can pitch or yaw, but not roll (rotate in the plane that the axes all lie in).]]
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| An important practical application of covering spaces occurs in [[charts on SO(3)]], the [[rotation group SO(3)|rotation group]]. This group occurs widely in engineering, due to 3-dimensional rotations being heavily used in [[navigation]], [[nautical engineering]], and [[aerospace engineering]], among many other uses. Topologically, SO(3) is the [[real projective space]] '''RP'''<sup>3</sup>, with fundamental group '''Z'''/2, and only (non-trivial) covering space the hypersphere ''S''<sup>3</sup>, which is the group [[spin group|Spin(3)]], and represented by the unit [[quaternions]]. Thus quaternions are a preferred method for representing spatial rotations – see [[quaternions and spatial rotation]].
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| However, it is often desirable to represent rotations by a set of three numbers, known as [[Euler angles]] (in numerous variants), both because this is conceptually simpler, and because one can build a combination of three [[gimbal]]s to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus ''T''<sup>3</sup> of three angles to the real projective space '''RP'''<sup>3</sup> of rotations, and the resulting map has imperfections due to this map being unable to be a covering map. Specifically, the failure of the map to be a local homeomorphism at certain points is referred to as [[gimbal lock]], and is demonstrated in the animation at the right – at some points (when the axes are coplanar) the [[Rank (differential topology)|rank]] of the map is 2, rather than 3, meaning that only 2 dimensions of rotations can be realized from that point by changing the angles. This causes problems in applications, and is formalized by the notion of a covering space.
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| ==See also==
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| *[[Bethe lattice]] is the universal cover of a [[Cayley graph]]
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| *[[Covering graph]], a covering space for an [[undirected graph]], and its special case the [[bipartite double cover]].
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| *[[Covering group]]
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| *[[Galois connection]]
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| ==Notes==
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| <references/>
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| ==References==
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| *{{cite book |ref=harv | first = Ronald | last = Brown| authorlink=Ronald Brown (mathematician) | title = Topology and Groupoids| year = 2006 | publisher = Booksurge LLC | location = Charleston, S. Carolina | isbn = 1-4196-2722-8 | url = http://www.bangor.ac.uk/r.brown/topgpds.html}} See chapter 10.
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| *{{SpringerEOM
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| | title=Covering
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| | id=Covering&oldid=13627
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| | last=Chernavskii
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| | first=A.V.
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| }}
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| *{{cite book | ref=harv | first = Hershel M. | last = Farkas | coauthors = [[Irwin Kra]] | title = Riemann Surfaces | year = 1980 | edition = 2nd ed. | publisher = Springer | location = New York | isbn = 0-387-90465-4}} See chapter 1 for a simple review.
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| *{{cite book | ref=harv | first = Allen | last = Hatcher | authorlink = Allen Hatcher | year = 2002 | title = Algebraic Topology | publisher = Cambridge University Press | isbn = 0-521-79540-0 | url = http://www.math.cornell.edu/~hatcher/AT/ATpage.html}}
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| *[http://138.73.27.39/tac/reprints/articles/7/tr7abs.html Categories and groupoids, P.J. Higgins], downloadable reprint of van Nostrand Notes in Mathematics, 1971, which deal with applications of groupoids in group theory and topology.
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| *{{cite book | ref=harv | first = Jurgen | last = Jost | title = Compact Riemann Surfaces | year = 2002 | publisher = Springer | location = New York | isbn = 3-540-43299-X}} See section 1.3
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| *{{cite book | ref=harv | first = William | last = Massey | authorlink = William S. Massey | title = A Basic Course in Algebraic Topology | year = 1991 | publisher = Springer | location = New York | isbn = 0-387-97430-X}} See chapter 5.
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| *{{cite book|ref=harv|last=Munkres|first=James R.|authorlink=James Munkres|title=Topology|year=2000|publisher=Prentice Hall|location=Upper Saddle River, NJ|isbn=0131816292|edition=2. ed.}}
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| * Brazas, J., 'Semicoverings: a generalisation of covering space theory', Homology, Homotopy, and Applications, 14 (2012), No. 1, pp. 33–63.
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| * Ellis, G. [http://hamilton.nuigalway.ie/Hap/www/ "Homological Algebra Program]
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| * Ellis, G. Computing group resolutions, J. Symbolic Comput. 38 (2004) 1077–1118.
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| *{{cite book |last=Spanier |first=Edwin |title=Algebraic Topology |date=December 1994 |publisher=Springer |isbn=0-387-94426-5}}
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| [[Category:Algebraic topology]]
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| [[Category:Homotopy theory]]
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| [[Category:Fiber bundles]]
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| [[Category:Topological graph theory]]
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