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topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, a covering or covering projection is a map between
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms. If p : \tilde X \to X is a covering, (\tilde X, p) is said to be a covering space or cover of X, and X is said to be the base of the covering, or simply the base. By abuse of terminology, \tilde X and p may sometimes be called covering spaces as well. Since coverings are local homeomorphisms, a covering space is a special kind of étalé space. Covering spaces first arose in the context of
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
(specifically, the technique of
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ne ...
), where they were introduced by Riemann as domains on which naturally multivalued complex functions become single-valued. These spaces are now called Riemann surfaces. Covering spaces are an important tool in several areas of mathematics. In modern
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, covering spaces (or branched coverings, which have slightly weaker conditions) are used in the construction of
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
s, orbifolds, and the morphisms between them. In
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, covering spaces are closely related to the fundamental group: for one, since all coverings have the homotopy lifting property, covering spaces are an important tool in the calculation of homotopy groups. A standard example in this vein is the calculation of the fundamental group of the circle by means of the covering of S^1 by \mathbb (see below). Under certain conditions, covering spaces also exhibit a Galois correspondence with the subgroups of the fundamental group.


Definition

Let X be a topological space. A covering of X is a continuous map : \pi : \tilde X \rightarrow X such that for every x \in X there exists an open neighborhood U_x of x and a
discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
D_x such that \pi^(U_x)= \displaystyle \bigsqcup_ V_d and \pi, _:V_d \rightarrow U_x is a
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
for every d \in D_x . The open sets V_ are called sheets, which are uniquely determined up to homeomorphism if U_x is connected. For each x \in X the discrete set \pi^(x) is called the fiber of x. If X is connected (and \tilde X is non-empty), it can be shown that \pi is surjective, and the
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
of D_x is the same for all x \in X; this value is called the degree of the covering. If \tilde X is path-connected, then the covering \pi : \tilde X \rightarrow X is called a path-connected covering. This definition is equivalent to the statement that \pi is a locally trivial fiber bundle. Some authors also require that \pi be surjective in the case that X is not connected.


Examples

* For every topological space X, the
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
\operatorname:X \rightarrow X is a covering. Likewise for any discrete space D the projection \pi:X \times D \rightarrow X taking (x, i) \mapsto x is a covering. Coverings of this type are called trivial coverings; if D has finitely many (say k) elements, the covering is called the trivial ''k-sheeted'' covering of X. * The map r : \mathbb \to S^1 with r(t)=(\cos(2 \pi t), \sin(2 \pi t)) is a covering of the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
S^1. The base of the covering is S^1 and the covering space is \mathbb. For any point x = (x_1, x_2) \in S^1 such that x_1 > 0, the set U := \ is an open neighborhood of x. The preimage of U under r is *: r^(U)=\displaystyle\bigsqcup_ \left( n - \frac 1 4, n + \frac 1 4\right) : and the sheets of the covering are V_n = (n - 1/4, n+1/4) for n \in \mathbb. The fiber of x is :: r^(x) = \. * Another covering of the unit circle is the map q : S^1 \to S^1 with q(z)=z^ for some positive n \in \mathbb. For an open neighborhood U of an x \in S^1, one has: :: q^(U)=\displaystyle\bigsqcup_^ U. * A map which is a local homeomorphism but not a covering of the unit circle is p : \mathbb \to S^1 with p(t)=(\cos(2 \pi t), \sin(2 \pi t)). There is a sheet of an open neighborhood of (1,0), which is not mapped homeomorphically onto U.


Properties


Local homeomorphism

Since a covering \pi:E \rightarrow X maps each of the disjoint open sets of \pi^(U) homeomorphically onto U it is a local homeomorphism, i.e. \pi is a continuous map and for every e \in E there exists an open neighborhood V \subset E of e, such that \pi, _V : V \rightarrow \pi(V) is a homeomorphism. It follows that the covering space E and the base space X locally share the same properties. * If X is a connected and non-orientable manifold, then there is a covering \pi:\tilde X \rightarrow X of degree 2, whereby \tilde X is a connected and orientable manifold. * If X is a connected
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
, then there is a covering \pi:\tilde X \rightarrow X which is also a Lie group homomorphism and \tilde X := \ is a Lie group. * If X is a graph, then it follows for a covering \pi:E \rightarrow X that E is also a graph. * If X is a connected
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
, then there is a covering \pi:\tilde X \rightarrow X, whereby \tilde X is a connected and
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
manifold. * If X is a connected Riemann surface, then there is a covering \pi:\tilde X \rightarrow X which is also a holomorphic map and \tilde X is a connected and simply connected Riemann surface.


Factorisation

Let X, Y and E be path-connected, locally path-connected spaces, and p,q and r be continuous maps, such that the diagram commutes. * If p and q are coverings, so is r. * If p and r are coverings, so is q.


Product of coverings

Let X and X' be topological spaces and p:E \rightarrow X and p':E' \rightarrow X' be coverings, then p \times p':E \times E' \rightarrow X \times X' with (p \times p')(e, e') = (p(e), p'(e')) is a covering. However, coverings of X\times X' are not all of this form in general.


Equivalence of coverings

Let X be a topological space and p:E \rightarrow X and p':E' \rightarrow X be coverings. Both coverings are called equivalent, if there exists a homeomorphism h:E \rightarrow E', such that the diagram commutes. If such a homeomorphism exists, then one calls the covering spaces E and E'
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
.


Lifting property

All coverings satisfy the lifting property, i.e.: Let I be the unit interval and p:E \rightarrow X be a covering. Let F:Y \times I \rightarrow X be a continuous map and \tilde F_0:Y \times \ \rightarrow E be a lift of F, _, i.e. a continuous map such that p \circ \tilde F_0 = F, _. Then there is a uniquely determined, continuous map \tilde F:Y \times I \rightarrow E for which \tilde F(y,0) = \tilde F_0 and which is a lift of F, i.e. p \circ \tilde F = F. If X is a path-connected space, then for Y=\ it follows that the map \tilde F is a lift of a path in X and for Y=I it is a lift of a
homotopy In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...
of paths in X. As a consequence, one can show that the fundamental group \pi_(S^1) of the unit circle is an infinite cyclic group, which is generated by the homotopy classes of the loop \gamma: I \rightarrow S^1 with \gamma (t) = (\cos(2 \pi t), \sin(2 \pi t)). Let X be a path-connected space and p:E \rightarrow X be a connected covering. Let x,y \in X be any two points, which are connected by a path \gamma, i.e. \gamma(0)= x and \gamma(1)= y. Let \tilde \gamma be the unique lift of \gamma, then the map : L_:p^(x) \rightarrow p^(y) with L_(\tilde \gamma (0))=\tilde \gamma (1) is
bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
. If X is a path-connected space and p: E \rightarrow X a connected covering, then the induced
group homomorphism In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) whe ...
: p_: \pi_(E) \rightarrow \pi_(X) with p_(
gamma Gamma (; uppercase , lowercase ; ) is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter normally repr ...
= \circ \gamma/math>, is injective and the
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
p_(\pi_1(E)) of \pi_1(X) consists of the homotopy classes of loops in X, whose lifts are loops in E.


Branched covering


Definitions


Holomorphic maps between Riemann surfaces

Let X and Y be Riemann surfaces, i.e. one dimensional complex manifolds, and let f: X \rightarrow Y be a continuous map. f is holomorphic in a point x \in X, if for any charts \phi _x:U_1 \rightarrow V_1 of x and \phi_:U_2 \rightarrow V_2 of f(x), with \phi_x(U_1) \subset U_2, the map \phi _ \circ f \circ \phi^ _x: \mathbb \rightarrow \mathbb is holomorphic. If f is holomorphic at all x \in X, we say f is holomorphic. The map F =\phi _ \circ f \circ \phi^ _x is called the local expression of f in x \in X. If f: X \rightarrow Y is a non-constant, holomorphic map between compact Riemann surfaces, then f is surjective and an
open map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, ...
, i.e. for every open set U \subset X the
image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
f(U) \subset Y is also open.


Ramification point and branch point

Let f: X \rightarrow Y be a non-constant, holomorphic map between compact Riemann surfaces. For every x \in X there exist charts for x and f(x) and there exists a uniquely determined k_x \in \mathbb, such that the local expression F of f in x is of the form z \mapsto z^. The number k_x is called the ramification index of f in x and the point x \in X is called a ramification point if k_x \geq 2. If k_x =1 for an x \in X, then x is unramified. The image point y=f(x) \in Y of a ramification point is called a branch point.


Degree of a holomorphic map

Let f: X \rightarrow Y be a non-constant, holomorphic map between compact Riemann surfaces. The degree \operatorname(f) of f is the cardinality of the fiber of an unramified point y=f(x) \in Y, i.e. \operatorname(f):=, f^(y), . This number is well-defined, since for every y \in Y the fiber f^(y) is discrete and for any two unramified points y_1,y_2 \in Y, it is: , f^(y_1), =, f^(y_2), . It can be calculated by: : \sum_ k_x = \operatorname(f)


Branched covering


Definition

A continuous map f: X \rightarrow Y is called a branched covering, if there exists a
closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
with dense complement E \subset Y, such that f_:X \smallsetminus f^(E) \rightarrow Y \smallsetminus E is a covering.


Examples

* Let n \in \mathbb and n \geq 2, then f:\mathbb \rightarrow \mathbb with f(z)=z^n is a branched covering of degree n, where by z=0 is a branch point. * Every non-constant, holomorphic map between compact Riemann surfaces f: X \rightarrow Y of degree d is a branched covering of degree d.


Universal covering


Definition

Let p: \tilde X \rightarrow X be a
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
covering. If \beta : E \rightarrow X is another simply connected covering, then there exists a uniquely determined homeomorphism \alpha : \tilde X \rightarrow E, such that the diagram commutes. This means that p is, up to equivalence, uniquely determined and because of that
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
denoted as the universal covering of the space X.


Existence

A universal covering does not always exist. The following theorem guarantees its existence for a certain class of base spaces. Let X be a connected,
locally simply connected In mathematics, a locally simply connected space is a topological space that admits a Base (topology), basis of simply connected sets. Every locally simply connected space is also locally path-connected and locally connected. The circle is an exam ...
topological space. Then, there exists a universal covering p:\tilde X \rightarrow X. The set \tilde X is defined as \tilde X = \/\text, where x_0 \in X is any chosen base point. The map p:\tilde X \rightarrow X is defined by p(
gamma Gamma (; uppercase , lowercase ; ) is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter normally repr ...
=\gamma(1). The
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
on \tilde X is constructed as follows: Let \gamma:I \rightarrow X be a path with \gamma(0)=x_0. Let U be a simply connected neighborhood of the endpoint x=\gamma(1). Then, for every y \in U, there is a path \sigma_y inside U from x to y that is unique up to
homotopy In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...
. Now consider the set \tilde U=\/\text. The restriction p, _: \tilde U \rightarrow U with p( gamma\sigma_y=\gamma\sigma_y(1)=y is a bijection and \tilde U can be equipped with the final topology of p, _. The fundamental group \pi_(X,x_0) = \Gamma acts freely on \tilde X by (
gamma Gamma (; uppercase , lowercase ; ) is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter normally repr ...
tilde x \mapsto gamma\tilde x and the orbit space \Gamma \backslash \tilde X is homeomorphic to X through the map Gamma \tilde xmapsto\tilde x(1).


Examples

* r : \mathbb \to S^1 with r(t)=(\cos(2 \pi t), \sin(2 \pi t)) is the universal covering of the unit circle S^1. * p : S^n \to \mathbbP^n \cong \\backslash S^n with p(x)= /math> is the universal covering of the projective space \mathbbP^n for n>1. * q : \mathrm(n) \ltimes \mathbb \to U(n) with q(A,t)= \begin \exp(2 \pi i t) & 0\\ 0 & I_ \end_\vphantom A is the universal covering of the unitary group U(n). * Since \mathrm(2) \cong S^3, it follows that the quotient map f : \mathrm(2) \rightarrow \mathrm(2) / \mathbb \cong \mathrm(3) is the universal covering of \mathrm(3). * A topological space which has no universal covering is the Hawaiian earring: X = \bigcup_\left\ One can show that no neighborhood of the origin (0,0) is simply connected.


G-coverings

Let ''G'' be a
discrete group In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and ...
acting on the
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''X''. This means that each element ''g'' of ''G'' is associated to a homeomorphism H''g'' of ''X'' onto itself, in such a way that H''g'' ''h'' is always equal to H''g'' \circ H''h'' for any two elements ''g'' and ''h'' of ''G''. (Or in other words, a group action of the group ''G'' on the space ''X'' is just a group homomorphism of the group ''G'' into the group Homeo(''X'') of self-homeomorphisms of ''X''.) It is natural to ask under what conditions the projection from ''X'' to the
orbit space In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under fun ...
''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 by the twist action where the non-identity element acts by . Thus the study of the relation between the fundamental groups of ''X'' and ''X''/''G'' is not so straightforward. However the group ''G'' does act on the fundamental
groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fu ...
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.


Smooth coverings

Let and be
smooth manifolds In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas (topology ...
with or without boundary. A covering \pi : E \to M is called a smooth covering if it is a smooth map and the sheets are mapped ''diffeomorphically'' onto the corresponding open subset of . (This is in contrast to the definition of a covering, which merely requires that the sheets are mapped ''homeomorphically'' onto the corresponding open subset.)


Deck transformation


Definition

Let p:E \rightarrow X be a covering. A deck transformation is a homeomorphism d:E \rightarrow E, such that the diagram of continuous maps commutes. Together with the composition of maps, the set of deck transformation forms a group \operatorname(p), which is the same as \operatorname(p). Now suppose p:C \to X is a covering map and C (and therefore also X) is connected and locally path connected. The action of \operatorname(p) on each fiber is free. If this action is 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 , where G = \operatorname(p) is considered as a discrete topological group. Every universal cover p:D \to X is regular, with deck transformation group being isomorphic to the fundamental group


Examples

* Let q : S^1 \to S^1 be the covering q(z)=z^ for some n \in \mathbb , then the map d_k:S^1 \rightarrow S^1 : z \mapsto z \, e^ for k \in \mathbb is a deck transformation and \operatorname(q)\cong \mathbb/ n\mathbb. * Let r : \mathbb \to S^1 be the covering r(t)=(\cos(2 \pi t), \sin(2 \pi t)), then the map d_k:\mathbb \rightarrow \mathbb : t \mapsto t + k for k \in \mathbb is a deck transformation and \operatorname(r)\cong \mathbb. * As another important example, consider \Complex the complex plane and \Complex^ the complex plane minus the origin. Then the map p: \Complex^ \to \Complex^ with p(z) = z^ is a regular cover. The deck transformations are multiplications with n-th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group \Z/n\Z. Likewise, the map \exp : \Complex \to \Complex^ with \exp(z) = e^ is the universal cover.


Properties

Let X be a path-connected space and p:E \rightarrow X be a connected covering. Since a deck transformation d:E \rightarrow E is
bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
, it permutes the elements of a fiber p^(x) with x \in X and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber. Because of this property every deck transformation defines a
group action In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under ...
on E, i.e. let U \subset X be an open neighborhood of a x \in X and \tilde U \subset E an open neighborhood of an e \in p^(x), then \operatorname(p) \times E \rightarrow E: (d,\tilde U)\mapsto d(\tilde U) is a
group action In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under ...
.


Normal coverings


Definition

A covering p:E \rightarrow X is called normal, if \operatorname(p) \backslash E \cong X. This means, that for every x \in X and any two e_0,e_1 \in p^(x) there exists a deck transformation d:E \rightarrow E, such that d(e_0)=e_1.


Properties

Let X be a path-connected space and p:E \rightarrow X be a connected covering. Let H=p_(\pi_1(E)) be a
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
of \pi_1(X), then p is a normal covering iff H is a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
of \pi_1(X). If p:E \rightarrow X is a normal covering and H=p_(\pi_1(E)), then \operatorname(p) \cong \pi_1(X)/H. If p:E \rightarrow X is a path-connected covering and H=p_(\pi_1(E)), then \operatorname(p) \cong N(H)/H, whereby N(H) is the normaliser of H. Let E be a topological space. A group \Gamma acts ''discontinuously'' on E, if every e \in E has an open neighborhood V \subset E with V \neq \empty, such that for every d_1, d_2 \in \Gamma with d_1 V \cap d_2 V \neq \empty one has d_1 = d_2. If a group \Gamma acts discontinuously on a topological space E, then the quotient map q: E \rightarrow \Gamma \backslash E with q(e)=\Gamma e is a normal covering. Hereby \Gamma \backslash E = \ is the quotient space and \Gamma e = \ is the
orbit In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
of the group action.


Examples

* The covering q : S^1 \to S^1 with q(z)=z^ is a normal coverings for every n \in \mathbb. * Every simply connected covering is a normal covering.


Calculation

Let \Gamma be a group, which acts discontinuously on a topological space E and let q: E \rightarrow \Gamma \backslash E be the normal covering. * If E is path-connected, then \operatorname(q) \cong \Gamma. * If E is simply connected, then \operatorname(q)\cong \pi_1(\Gamma \backslash E).


Examples

* Let n \in \mathbb. The antipodal map g:S^n \rightarrow S^n with g(x)=-x generates, together with the composition of maps, a group D(g) \cong \mathbb and induces a group action D(g) \times S^n \rightarrow S^n, (g,x)\mapsto g(x), which acts discontinuously on S^n. Because of \mathbb \backslash S^n \cong \mathbbP^n it follows, that the quotient map q : S^n \rightarrow \mathbb\backslash S^n \cong \mathbbP^n is a normal covering and for n > 1 a universal covering, hence \operatorname(q)\cong \mathbb\cong \pi_1() for n > 1. * Let \mathrm(3) be the special orthogonal group, then the map f : \mathrm(2) \rightarrow \mathrm(3) \cong \mathbb \backslash \mathrm(2) is a normal covering and because of \mathrm(2) \cong S^3, it is the universal covering, hence \operatorname(f) \cong \mathbb \cong \pi_1(\mathrm(3)). * With the group action (z_1,z_2)*(x,y)=(z_1+(-1)^x,z_2+y) of \mathbb on \mathbb, whereby (\mathbb,*) is the semidirect product \mathbb \rtimes \mathbb , one gets the universal covering f: \mathbb \rightarrow (\mathbb \rtimes \mathbb) \backslash \mathbb \cong K of the
klein bottle In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...
K, hence \operatorname(f) \cong \mathbb \rtimes \mathbb \cong \pi_1(K). * Let T = S^1 \times S^1 be the
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
which is embedded in the \mathbb. Then one gets a homeomorphism \alpha: T \rightarrow T: (e^,e^) \mapsto (e^,e^), which induces a discontinuous group action G_ \times T \rightarrow T, whereby G_ \cong \mathbb. It follows, that the map f: T \rightarrow G_ \backslash T \cong K is a normal covering of the klein bottle, hence \operatorname(f) \cong \mathbb. * Let S^3 be embedded in the \mathbb. Since the group action S^3 \times \mathbb \rightarrow S^3: ((z_1,z_2), \mapsto (e^z_1,e^z_2) is discontinuously, whereby p,q \in \mathbb are
coprime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
, the map f:S^3 \rightarrow \mathbb \backslash S^3 =: L_ is the universal covering of the lens space L_, hence \operatorname(f) \cong \mathbb \cong \pi_1(L_).


Galois correspondence

Let X be a connected and
locally simply connected In mathematics, a locally simply connected space is a topological space that admits a Base (topology), basis of simply connected sets. Every locally simply connected space is also locally path-connected and locally connected. The circle is an exam ...
space, then for every
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
H\subseteq \pi_1(X) there exists a path-connected covering \alpha:X_H \rightarrow X with \alpha_(\pi_1(X_H))=H. Let p_1:E \rightarrow X and p_2: E' \rightarrow X be two path-connected coverings, then they are equivalent iff the subgroups H = p_(\pi_1(E)) and H'=p_(\pi_1(E')) are conjugate to each other. Let X be a connected and locally simply connected space, then, up to equivalence between coverings, there is a bijection: \begin \qquad \displaystyle \ & \longleftrightarrow & \displaystyle \ \\ H & \longrightarrow & \alpha:X_H \rightarrow X \\ p_\#(\pi_1(E))&\longleftarrow & p \\ \displaystyle \ & \longleftrightarrow & \displaystyle \ \end For a sequence of subgroups \displaystyle \ \subset H \subset G \subset \pi_1(X) one gets a sequence of coverings \tilde X \longrightarrow X_H \cong H \backslash \tilde X \longrightarrow X_G \cong G \backslash \tilde X \longrightarrow X\cong \pi_1(X) \backslash \tilde X . For a subgroup H \subset \pi_1(X) with
index Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ...
\displaystyle pi_1(X):H= d , the covering \alpha:X_H \rightarrow X has degree d.


Classification


Definitions


Category of coverings

Let X be a topological space. The objects of the category \boldsymbol are the coverings p:E \rightarrow X of X and the
morphisms In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
between two coverings p:E \rightarrow X and q:F\rightarrow X are continuous maps f:E \rightarrow F, such that the diagram commutes.


G-Set

Let G be a
topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
. The category \boldsymbol is the category of sets which are G-sets. The morphisms are G-maps \phi:X \rightarrow Y between G-sets. They satisfy the condition \phi(gx)=g \, \phi(x) for every g \in G.


Equivalence

Let X be a connected and locally simply connected space, x \in X and G = \pi_1(X,x) be the fundamental group of X. Since G defines, by lifting of paths and evaluating at the endpoint of the lift, a group action on the fiber of a covering, the
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
F:\boldsymbol \longrightarrow \boldsymbol: p \mapsto p^(x) is an
equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two Category (mathematics), categories that establishes that these categories are "essentially the same". There are numerous examples of cate ...
.


Applications

An important practical application of covering spaces occurs in charts on SO(3), the rotation group. This group occurs widely in engineering, due to 3-dimensional rotations being heavily used in
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the motion, movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navig ...
, nautical engineering, and aerospace engineering, among many other uses. Topologically, SO(3) is the real projective space RP3, with fundamental group Z/2, and only (non-trivial) covering space the hypersphere ''S''3, which is the group Spin(3), and represented by the unit quaternions. Thus quaternions are a preferred method for representing spatial rotations – see
quaternions and spatial rotation unit vector, Unit quaternions, known as versor, ''versors'', provide a convenient mathematics, mathematical notation for representing spatial Orientation (geometry), orientations and rotations of elements in three dimensional space. Specifically, th ...
. 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 for someone familiar with planar rotation, and because one can build a combination of three gimbals to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus ''T''3 of three angles to the real projective space RP3 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 Gimbal lock is the loss of one degree of freedom (mechanics), degree of freedom in a multi-dimensional mechanism at certain alignments of the axes. In a three-dimensional three-gimbal mechanism, gimbal lock occurs when the axes of two of the gi ...
, and is demonstrated in the animation at the right – at some points (when the axes are coplanar) the 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.


See also

* Bethe lattice is the universal cover of a
Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathematics), group. Its definition is sug ...
* Covering graph, a covering space for an undirected graph, and its special case the bipartite double cover *
Covering group In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous (topology), continuous group homomorphism. The map ''p'' is called the c ...
*
Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fun ...
* Quotient space (topology)


Literature

* * * * {{cite book , first = Wolfgang , last = Kühnel, title=Matrizen und Lie-Gruppen Eine geometrische Einführung , publisher = Vieweg+Teubner Verlag , publication-place=Wiesbaden , date=2011 , isbn=978-3-8348-9905-7 , oclc=706962685 , language=de , doi=10.1007/978-3-8348-9905-7


References

Algebraic topology Homotopy theory Fiber bundles Topological graph theory