Covering map

In mathematics, specifically topology, a covering map is a continuous surjective map ''p'' : ''C'' → ''X'', with ''C'' and ''X'' being topological spaces, which has the following property: :to every ''x'' in ''X'' there exists an open set neighborhood (topology) ''U'' such that ''p'' ^-1(''U'') is a union (set theory) of mutually disjoint open sets ''S''_''i'' (where ''i'' ranges over some index set ''I'') such that ''p'' restricted to ''S''_''i'' yields a homeomorphism from ''S''_''i'' onto ''U'' for every ''i'' in ''I''. A covering map is also simply called a cover; we say ''C'' is a covering space of ''X'' or ''C'' covers ''X''. For each ''x'' in ''X'', the set ''p'' ^-1(''x'') is called the fiber over x; the sets ''S''_''i'' 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''. A special case, called an open cover (or just cover (topology)) is when ''C'' is the disjoint union of a collection of open sets ''X''_''i'', with union ''X''. A cover of any set ''S'' is the special case of this idea, when ''S'' carries the discrete topology (so that any subset is open). == Examples == Consider the unit circle ''S''¹ in R². Then the map ''p'' : R → ''S''¹ with :''p''(''t'') = (cos(''t''),sin(''t'')) is a cover. Consider the complex plane with the origin removed, denoted by C^×, and pick a non-zero integer ''n''. Then ''p'' : C^× → C^× given by :''p''(''z'') = ''z''^''n'' is a cover. Here every fiber has ''n'' elements. If ''G'' is group (mathematics) (considered as a discrete space topological group), then every principal bundle is a covering map. Here every fiber can be identified with ''G''. == Elementary properties == Common local properties: Every cover ''p'' : ''C'' → ''X'' is a local homeomorphism (i.e. to every $c\backslash in\; C$ there exists an open set ''A'' in ''C'' containing ''c'' and an open set ''B'' in ''X'' such that the restriction of ''p'' to ''A'' yields a homeomorphism between ''A'' and ''B''). This implies that ''C'' and ''X'' share all local properties. If ''X'' is simply connected, then this holds globally as well, and the covering ''p'' is a homeomorphism. Cardinality: For every $x\backslash in\; X$, the fiber over ''x'' is a discrete space subset of ''C''. On every connected space of ''X'', the cardinality of the fibers is the same (possibly infinite). If every fiber has 2 elements, we speak of a double cover. The lifting property: If ''p'' : ''C'' → ''X'' is a cover and γ is a path in ''X'' (i.e. a continuous map from the unit interval [0,1] into ''X'') and $c\backslash in\; C$ is a point "lying over" γ(0) (i.e. ''p''(''c'') = γ(0)), then there exists a unique path ρ in ''C'' lying over γ (i.e. ''p'' o ρ = γ) and with ρ(0) = ''c''. The curve ρ 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. Equivalance: Let $p\_1:C\_1\backslash rightarrow\; X$ and $p\_2:C\_2\backslash rightarrow\; X$ be two coverings. One then says that the two coverings $(p\_1,C\_1)$ and $(p\_2,C\_2)$ are equivalent if there exists a homeomorphism $p\_\{21\}:C\_2\backslash rightarrow\; C\_1$ and $p\_2\; =\; p\_1\; \backslash circ\; p\_\{21\}$. Equivalence classes of coverings correspond to conjugacy classes, as discussed below. If $p\_\{21\}$ is a covering rather than a homeomorphism, then one says that $(p\_2,C\_2)$ dominates $(p\_1,C\_1)$ (given that $p\_2\; =\; p\_1\; \backslash circ\; p\_\{21\}$). == Universal covers == A cover ''q'' : ''D'' → ''X'' is a universal cover iff ''D'' is simply connected. The name comes from the following important universal property: if ''p'' : ''C'' → ''X'' is any cover of ''X'' with ''C'' connected, then there exists a covering map ''f'' : ''D'' → ''C'' such that ''p'' o ''f'' = ''q''. This can be phrased as "The universal cover of ''X'' covers all connected covers of ''X''." The map ''f'' is unique in the following sense: if we fix ''x''∈''X'' and ''d''∈''D'' with ''q''(''d'') = ''x'' and ''c''∈''C'' with ''p''(''c'') = ''x'', then there exists a unique covering map ''f'' : ''D'' → ''C'' such that ''p'' o ''f'' = ''q'' and ''f''(''d'') = ''c''. If ''X'' has a universal cover, then that universal cover is essentially unique: if ''q''₁ : ''D''₁ → ''X'' and ''q''₂ : ''D''₂ → ''X'' are two universal covers of ''X'', then there exists a homeomorphism ''f'' : ''D''₁ → ''D''₂ such that ''q''₂ o ''f'' = ''q''₁. The space ''X'' has a universal cover if and only if it is connectedness, connectedness and semi-locally simply connected. The universal cover of ''X'' can be constructed as a certain space of paths in ''X''. The example R → ''S''¹ given above is a universal cover. The map ''S''³ → SO(3) from quaternion to rotations of 3D space described in quaternions and spatial rotation is also a universal cover. If the space ''X'' carries some additional structure, then its universal cover normally inherits that structure: * if ''X'' is a manifold, then so is its universal cover ''C'' * if ''X'' is a Riemann surface, then so is its universal cover ''C'', and ''p'' is a holomorphic map * if ''X'' is a Lie group (as in the two examples above), then so is its universal cover ''C'', and ''p'' is a homomorphism of Lie groups. The universal cover first arose in the theory of analytic functions as the natural domain of an analytic continuation. == Deck transformation group, regular covers == A deck transformation or automorphism of a cover ''p'' : ''C'' → ''X'' is a homeomorphism ''f'' : ''C'' → ''C'' such that ''p'' o ''f'' = ''p''. The set of all deck transformations of ''p'' forms a group under function composition, the deck transformation group Aut(''p''). Every deck transformation permutation the elements of each fiber. This defines a group action of the deck transformation group on each fiber. Note that if ''f'' is not the identity, then ''f'' has no fixed point (mathematics) . Now suppose ''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. If this action is group action on some fiber, then it is transitive on all fibers, and we call the cover regular. Every such regular cover is a principal bundle, where ''G'' = Aut(''p'') is considered as a discrete topological group. Every universal cover ''p'' : ''D'' → ''X'' is regular, with deck transformation group being isomorphic to the Dual_(category_theory) of the fundamental group π(''X''). The example ''p'' : C^× → C^× with ''p''(''z'') = ''z''^''n'' from above is a regular cover. The deck transformations are multiplications with ''n''-th root of unity and the deck transformation group is therefore isomorphic to the cyclic group ''C''_''n''. == Monodromy action == Again suppose ''p'' : ''C'' → ''X'' is a covering map and ''C'' (and therefore also ''X'') is connected and locally path connected. If ''x''∈''X'' and ''c'' belongs to the fiber over ''x'' (i.e. ''p''(''c'') = ''x''), and γ:[0,1]→''X'' is a path with γ(0)=γ(1)=''x'', then this path Homotopy lifting property 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 π(''X'',''x''), and in this fashion we obtain a right group action of π(''X'',''x'') on the fiber over ''x''. This is known as the monodromy action. So there are two actions on the fiber over ''x'': Aut(''p'') acts on the left and π(''X'',''x'') acts on the right. These two actions are compatible in the following sense: :''f''.(''c''.γ) = (''f''.''c'').γ for all ''f''∈Aut(''p''), ''c''∈''p'' ^-1(''x'') and γ∈π(''X'',''x''). If ''p'' is a universal cover, then the monodromy action is regular; if we identify Aut(''p'') with the Dual_(category_theory) group of π(''X'',''x''), then the monodromy action coincides with the action of Aut(''p'') on the fiber over ''x''. ==Group structure redux== The deck transformation group and the monodromy action can be understood to relate the normal subgroups of the fundamental group $\backslash pi\_1(X)$ of ''X'' and the fundamental group $\backslash pi\_1(C)$ of the cover. Furthermore, these equate the conjugacy classes of subgroups of $\backslash pi\_1(X)$ and equivalence classes of coverings. As a result, one can conclude that ''X''=''C''/Aut(''p''), that is, the manifold ''X'' is given as the quotient of the covering manifold under the action of the deck transformation group. These inter-relationships are explored below. Let γ be a curve in ''X''. Denote by $\backslash gamma\_C$ the Homotopy lifting property of γ to ''C''. Consider the set :$\backslash Gamma\_p(c)\; =\; \backslash \{\; \backslash gamma\; :\; \backslash gamma\_C\; \backslash mbox\{\; is\; a\; closed\; curve\; in\; \}\; C\; \backslash mbox\; \{\; passing\; through\; \}\; c\backslash in\; C\; \backslash \}$ Note that $\backslash Gamma\_p(c)$ is a group (mathematics), and that is is a subgroup of $\backslash pi\_1(X,p(c))$. Note also that it depends on ''c'', and that different values of ''c'' in the same fiber yield different subgroups. Each such subgroups is conjugacy class to another by the monodromy action. To see this, pick two points $c\_1,\; c\_2$ in the same fiber: $p(c\_1)=p(c\_2)=x$ and let ''g'' be a curve in ''C'' connecting $c\_1$ to $c\_2$. Then p(g) is a closed curve in ''X''. If $\backslash gamma\_C$ is a closed curve in ''C'' passing through $c\_1$, then $g\backslash gamma\_C\; g^\{-1\}$ is a closed curve in ''C'' passing through $c\_2$. Thus, we have shown :$\backslash Gamma\_p(c\_2)\; =\; g\; \backslash Gamma\_p(c\_1)\; g^\{-1\}$ and so we have the result that $\backslash Gamma\_p(c\_1)$ and $\backslash Gamma\_p(c\_2)$ are conjugate subgroups of $\backslash pi\_1(X,x)$. All of the conjugate subgroups may be obtained in this way. It should be clear that two equivalent coverings lead to the same conjugacy class of subgroups of $\backslash pi\_1(X,x)$; there is a bijective correspondence between equivalence classes of coverings and conjugacy classes of subgroups of $\backslash pi\_1\; (X)$. Note that this implies that the fundamental group $\backslash pi\_1(C)$ is isomorphic to $\backslash Gamma\_p$. Let $N(\backslash Gamma\_p)$ be the normalizer of $\backslash Gamma\_p$ in $\backslash pi\_1(X)$. The deck transformation group Aut(''p'') is isomorphic to $N(\backslash Gamma\_p)/\backslash Gamma\_p$. If ''p'' is a universal covering, then $\backslash Gamma\_p$ is the trivial group, and Aut(''p'') is isomorphic to $\backslash pi\_1(X)$. As a corollary, let us reverse this argument. Let Γ be a normal subgroup of $\backslash pi\_1(X,x)$. By the above arguments, this defines a (regular) covering $p:X\backslash rightarrow\; C$. Let $c\_1$ in ''C'' be in the fiber of ''x''. Then for every other $c\_2$ in the fiber of ''x'', there is precisely one deck transformation that takes $c\_1$ to $c\_2$. This deck transformation corresponds to a curve ''g'' in ''C'' connecting $c\_1$ to $c\_2$. Note that Aut(''p'') operates free regular set on ''C'', and so we have that ''X''=''C''/Aut(''p''), that is, ''X'' is the manifold given by the quotient of the covering manifold by the deck transformation group. ==References== * Hershel M. Farkas and Irwin Kra, ''Riemann Surfaces'' (1980), Springer-Verlag, New York. ISBN 0-387-90465-4 ''(See Chapter 1 for a simple review)'' * Jurgen Jost, ''Compact Riemann Surfaces'' (2002), Springer-Verlag, New York. ISBN 3-540-43299-X ''(See Section 1.3)'' Topology Algebraic topology Homotopy theory Fiber bundles Topological graph theory

Covering map

User:Charles Matthews There is now some overlap between the content here and at local homeomorphism. The way covering map has been defined allows it not to be surjective (the condition holds vacuously for points with empty pre-image); the usual definition has a covering map being surjective. I think surjective should be added to the definition since that's what is needed for most purposes. :Just noticed that one property that a covering map is supposed to have, according to whoever made the page, is being surjective. So I'll add 'surjective' to the definition. ---- I removed this paragraph of mine: :The composition of two covering maps need not be a covering map: consider the unit circle ''S''¹ as a subset of the complex plane, and for any natural number ''n'' define ''p''_''n'' : ''S''¹ → ''S''¹ by ''p''_''n''(''z'') = ''z''^−''n''. Consider the map ''p'' : ''S''¹ × N → ''S''¹ × N by ''p''(''z'',''n'') = (''p''_''n''(''z''),''n''). If N is equipped with the discrete topology and ''S''¹ × N carries the product topology, then ''p'' is a covering map. The natural projection ''q'' : ''S''¹ × N → ''S''¹ defined by ''q''(''z'',''n'') = ''z'' is obviously a covering map. The composition ''qp'' : ''S''¹ × N → ''S''¹ is not: no matter how small an open set ''U'' you pick in ''S''¹, there will always be an ''n'' large enough so that ''p_n''⁻¹(''U'') = ''S''¹ which cannot be isomorphic to ''U''. The last statement, ''p_n''⁻¹(''U'') = ''S''¹, is false, and that kills the whole argument. I don't know if the composition of two covering maps is always again a covering map. User:AxelBoldt 14:52, 23 Nov 2003 (UTC) :Huh? I'm often muddle-headed and confused, but ... the last statement is perfectly true. What's false is the statement that ''p'' is a covering map. The problem being that ''p'' restricted to to the inverse image ''S''¹ does not produce a homeomorphism to ''U''. That is, one can always find an ''n'' large enough so that ''p_n''(''S''¹) is not equal to ''U''; thus ''p'' was never a covering to begin with. Changing -n to +n in the definition would make ''p'' into a covering. Interesting example, though. Non-trival fundamental group. User:Linas 15:47, 3 Apr 2005 (UTC) ---- Would someone who knows what is meant by the "opposite" of a group like to make a stub/redirection? I can't find anything on this. Special case of dual (category theory); anyway like defining g*h = hg. User:Charles Matthews 09:06, 26 Feb 2004 (UTC) Ah yes, of course. I've added links - that ok?

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