Covering space

Summary

In topology, a covering or covering projection is a map between topological spaces 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 is a covering, is said to be a covering space or cover of , and is said to be the base of the covering, or simply the base. By abuse of terminology, and may sometimes be called covering spaces as well. Since coverings are local homeomorphisms, a covering space is a special kind of étale space.

Intuitively, a covering locally projects a "stack of pancakes" above an open neighborhood onto

Covering spaces first arose in the context of complex analysis (specifically, the technique of analytic continuation), where they were introduced by Riemann as domains on which naturally multivalued complex functions become single-valued. These spaces are now called Riemann surfaces.[1]: 10 

Covering spaces are an important tool in several areas of mathematics. In modern geometry, covering spaces (or branched coverings, which have slightly weaker conditions) are used in the construction of manifolds, orbifolds, and the morphisms between them. In algebraic topology, 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 by (see below).[2]: 29  Under certain conditions, covering spaces also exhibit a Galois correspondence with the subgroups of the fundamental group.

Definition

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Let   be a topological space. A covering of   is a continuous map

 

such that for every   there exists an open neighborhood   of   and a discrete space   such that   and   is a homeomorphism for every  . The open sets   are called sheets, which are uniquely determined up to homeomorphism if   is connected.[2]: 56  For each   the discrete set   is called the fiber of  . If   is connected, it can be shown that   is surjective, and the cardinality of   is the same for all  ; this value is called the degree of the covering. If   is path-connected, then the covering   is called a path-connected covering. This definition is equivalent to the statement that   is a locally trivial Fiber bundle.

Some authors also require that   be surjective in the case that   is not connected.[3]

Examples

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  • For every topological space  , the identity map   is a covering. Likewise for any discrete space   the projection   taking   is a covering. Coverings of this type are called trivial coverings; if   has finitely many (say  ) elements, the covering is called the trivial  -sheeted covering of  .
 
The space   is a covering space of  . The disjoint open sets   are mapped homeomorphically onto  . The fiber of   consists of the points  .
  • The map   with   is a covering of the unit circle  . The base of the covering is   and the covering space is  . For any point   such that  , the set   is an open neighborhood of  . The preimage of   under   is
     
and the sheets of the covering are   for   The fiber of   is
 
  • Another covering of the unit circle is the map   with   for some   For an open neighborhood   of an  , one has:
 .
  • A map which is a local homeomorphism but not a covering of the unit circle is   with  . There is a sheet of an open neighborhood of  , which is not mapped homeomorphically onto  .

Properties

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Local homeomorphism

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Since a covering   maps each of the disjoint open sets of   homeomorphically onto   it is a local homeomorphism, i.e.   is a continuous map and for every   there exists an open neighborhood   of  , such that   is a homeomorphism.

It follows that the covering space   and the base space   locally share the same properties.

  • If   is a connected and non-orientable manifold, then there is a covering   of degree  , whereby   is a connected and orientable manifold.[2]: 234 
  • If   is a connected Lie group, then there is a covering   which is also a Lie group homomorphism and   is a Lie group.[4]: 174 
  • If   is a graph, then it follows for a covering   that   is also a graph.[2]: 85 
  • If   is a connected manifold, then there is a covering  , whereby   is a connected and simply connected manifold.[5]: 32 
  • If   is a connected Riemann surface, then there is a covering   which is also a holomorphic map[5]: 22  and   is a connected and simply connected Riemann surface.[5]: 32 

Factorisation

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Let   and   be path-connected, locally path-connected spaces, and   and   be continuous maps, such that the diagram

 

commutes.

  • If   and   are coverings, so is  .
  • If   and   are coverings, so is  .[6]: 485 

Product of coverings

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Let   and   be topological spaces and   and   be coverings, then   with   is a covering.[6]: 339  However covering of   are not all of this form in general.

Equivalence of coverings

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Let   be a topological space and   and   be coverings. Both coverings are called equivalent, if there exists a homeomorphism  , such that the diagram

 

commutes. If such a homeomorphism exists, then one calls the covering spaces   and   isomorphic.

Lifting property

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All coverings satisfy the lifting property, i.e.:

Let   be the unit interval and   be a covering. Let   be a continuous map and   be a lift of  , i.e. a continuous map such that  . Then there is a uniquely determined, continuous map   for which   and which is a lift of  , i.e.  .[2]: 60 

If   is a path-connected space, then for   it follows that the map   is a lift of a path in   and for   it is a lift of a homotopy of paths in  .

As a consequence, one can show that the fundamental group   of the unit circle is an infinite cyclic group, which is generated by the homotopy classes of the loop   with  .[2]: 29 

Let   be a path-connected space and   be a connected covering. Let   be any two points, which are connected by a path  , i.e.   and  . Let   be the unique lift of  , then the map

  with  

is bijective.[2]: 69 

If   is a path-connected space and   a connected covering, then the induced group homomorphism

  with  ,

is injective and the subgroup   of   consists of the homotopy classes of loops in  , whose lifts are loops in  .[2]: 61 

Branched covering

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Definitions

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Holomorphic maps between Riemann surfaces

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Let   and   be Riemann surfaces, i.e. one dimensional complex manifolds, and let   be a continuous map.   is holomorphic in a point  , if for any charts   of   and   of  , with  , the map   is holomorphic.

If   is holomorphic at all  , we say   is holomorphic.

The map   is called the local expression of   in  .

If   is a non-constant, holomorphic map between compact Riemann surfaces, then   is surjective and an open map,[5]: 11  i.e. for every open set   the image   is also open.

Ramification point and branch point

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Let   be a non-constant, holomorphic map between compact Riemann surfaces. For every   there exist charts for   and   and there exists a uniquely determined  , such that the local expression   of   in   is of the form  .[5]: 10  The number   is called the ramification index of   in   and the point   is called a ramification point if  . If   for an  , then   is unramified. The image point   of a ramification point is called a branch point.

Degree of a holomorphic map

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Let   be a non-constant, holomorphic map between compact Riemann surfaces. The degree   of   is the cardinality of the fiber of an unramified point  , i.e.  .

This number is well-defined, since for every   the fiber   is discrete[5]: 20  and for any two unramified points  , it is:  

It can be calculated by:

  [5]: 29 

Branched covering

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Definition

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A continuous map   is called a branched covering, if there exists a closed set with dense complement  , such that   is a covering.

Examples

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  • Let   and  , then   with   is branched covering of degree  , where by   is a branch point.
  • Every non-constant, holomorphic map between compact Riemann surfaces   of degree   is a branched covering of degree  .

Universal covering

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Definition

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Let   be a simply connected covering. If   is another simply connected covering, then there exists a uniquely determined homeomorphism  , such that the diagram

 

commutes.[6]: 482 

This means that   is, up to equivalence, uniquely determined and because of that universal property denoted as the universal covering of the space  .

Existence

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A universal covering does not always exist, but the following properties guarantee its existence:

Let   be a connected, locally simply connected topological space; then, there exists a universal covering  .

  is defined as   and   by  .[2]: 64 

The topology on   is constructed as follows: Let   be a path with  . Let   be a simply connected neighborhood of the endpoint  , then for every   the paths   inside   from   to   are uniquely determined up to homotopy. Now consider  , then   with   is a bijection and   can be equipped with the final topology of  .

The fundamental group   acts freely through   on   and   with   is a homeomorphism, i.e.  .

Examples

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The Hawaiian earring. Only the ten largest circles are shown.
  •   with   is the universal covering of the unit circle  .
  •   with   is the universal covering of the projective space   for  .
  •   with
     
    is the universal covering of the unitary group  .[7]: 5, Theorem 1 
  • Since  , it follows that the quotient map
     
    is the universal covering of the  .
  • A topological space which has no universal covering is the Hawaiian earring:
     
    One can show that no neighborhood of the origin   is simply connected.[6]: 487, Example 1 

G-coverings

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Let G be a discrete group acting on the topological space X. This means that each element g of G is associated to a homeomorphism Hg of X onto itself, in such a way that Hg h is always equal to Hg ∘ Hh 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 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 X × X by the twist action where the non-identity element acts by (x, y) ↦ (y, x). 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 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.

Deck transformation

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Definition

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Let   be a covering. A deck transformation is a homeomorphism  , such that the diagram of continuous maps

 

commutes. Together with the composition of maps, the set of deck transformation forms a group  , which is the same as  .

Now suppose   is a covering map and   (and therefore also  ) is connected and locally path connected. The action of   on each fiber is transitive. If this action is free on some fiber, then it is free on all fibers, and we call the cover regular (or normal or Galois). Every such regular cover is a principal  -bundle, where   is considered as a discrete topological group.

Every universal cover   is regular, with deck transformation group being isomorphic to the fundamental group  .

Examples

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  • Let   be the covering   for some  , then the map   is a deck transformation and  .
  • Let   be the covering  , then the map   with   is a deck transformation and  .
  • As another important example, consider   the complex plane and   the complex plane minus the origin. Then the map   with   is a regular cover. The deck transformations are multiplications with  -th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group  . Likewise, the map   with   is the universal cover.

Properties

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Let   be a path-connected space and   be a connected covering. Since a deck transformation   is bijective, it permutes the elements of a fiber   with   and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber.[2]: 70  Because of this property every deck transformation defines a group action on  , i.e. let   be an open neighborhood of a   and   an open neighborhood of an  , then   is a group action.

Normal coverings

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Definition

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A covering   is called normal, if  . This means, that for every   and any two   there exists a deck transformation  , such that  .

Properties

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Let   be a path-connected space and   be a connected covering. Let   be a subgroup of  , then   is a normal covering iff   is a normal subgroup of  .

If   is a normal covering and  , then  .

If   is a path-connected covering and  , then  , whereby   is the normaliser of  .[2]: 71 

Let   be a topological space. A group   acts discontinuously on  , if every   has an open neighborhood   with  , such that for every   with   one has  .

If a group   acts discontinuously on a topological space  , then the quotient map   with   is a normal covering.[2]: 72  Hereby   is the quotient space and   is the orbit of the group action.

Examples

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  • The covering   with   is a normal coverings for every  .
  • Every simply connected covering is a normal covering.

Calculation

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Let   be a group, which acts discontinuously on a topological space   and let   be the normal covering.

  • If   is path-connected, then  .[2]: 72 
  • If   is simply connected, then  .[2]: 71 

Examples

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  • Let  . The antipodal map   with   generates, together with the composition of maps, a group   and induces a group action  , which acts discontinuously on  . Because of   it follows, that the quotient map   is a normal covering and for   a universal covering, hence   for  .
  • Let   be the special orthogonal group, then the map   is a normal covering and because of  , it is the universal covering, hence  .
  • With the group action   of   on  , whereby   is the semidirect product  , one gets the universal covering   of the klein bottle  , hence  .
  • Let   be the torus which is embedded in the  . Then one gets a homeomorphism  , which induces a discontinuous group action  , whereby  . It follows, that the map   is a normal covering of the klein bottle, hence  .
  • Let   be embedded in the  . Since the group action   is discontinuously, whereby   are coprime, the map   is the universal covering of the lens space  , hence  .

Galois correspondence

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Let   be a connected and locally simply connected space, then for every subgroup   there exists a path-connected covering   with  .[2]: 66 

Let   and   be two path-connected coverings, then they are equivalent iff the subgroups   and   are conjugate to each other.[6]: 482 

Let   be a connected and locally simply connected space, then, up to equivalence between coverings, there is a bijection:

 

For a sequence of subgroups   one gets a sequence of coverings  . For a subgroup   with index  , the covering   has degree  .

Classification

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Definitions

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Category of coverings

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Let   be a topological space. The objects of the category   are the coverings   of   and the morphisms between two coverings   and   are continuous maps  , such that the diagram

 

commutes.

G-Set

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Let   be a topological group. The category   is the category of sets which are G-sets. The morphisms are G-maps   between G-sets. They satisfy the condition   for every  .

Equivalence

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Let   be a connected and locally simply connected space,   and   be the fundamental group of  . Since   defines, by lifting of paths and evaluating at the endpoint of the lift, a group action on the fiber of a covering, the functor   is an equivalence of categories.[2]: 68–70 

Applications

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Gimbal lock occurs because any map T3RP3 is not a covering map. In particular, the relevant map carries any element of T3, that is, an ordered triple (a,b,c) of angles (real numbers mod 2π), to the composition of the three coordinate axis rotations Rx(a)∘Ry(b)∘Rz(c) by those angles, respectively. Each of these rotations, and their composition, is an element of the rotation group SO(3), which is topologically RP3. 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).

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, 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 S3, 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.

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 T3 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, 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

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Literature

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  • Hatcher, Allen (2002). Algebraic topology. Cambridge: Cambridge University Press. ISBN 0-521-79160-X. OCLC 45420394.
  • Forster, Otto (1981). Lectures on Riemann surfaces. New York. ISBN 0-387-90617-7. OCLC 7596520.{{cite book}}: CS1 maint: location missing publisher (link)
  • Munkres, James R. (2018). Topology. New York, NY. ISBN 978-0-13-468951-7. OCLC 964502066.{{cite book}}: CS1 maint: location missing publisher (link)
  • Kühnel, Wolfgang (2011). Matrizen und Lie-Gruppen Eine geometrische Einführung (in German). Wiesbaden: Vieweg+Teubner Verlag. doi:10.1007/978-3-8348-9905-7. ISBN 978-3-8348-9905-7. OCLC 706962685.

References

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  1. ^ Forster, Otto (1981). "Chapter 1: Covering Spaces". Lectures on Riemann Surfaces. GTM. Translated by Bruce Gillian. New York: Springer. ISBN 9781461259633.
  2. ^ a b c d e f g h i j k l m n o p Hatcher, Allen (2001). Algebraic Topology. Cambridge: Cambridge Univ. Press. ISBN 0-521-79160-X.
  3. ^ Rowland, Todd. "Covering Map." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CoveringMap.html
  4. ^ Kühnel, Wolfgang (6 December 2010). Matrizen und Lie-Gruppen. Stuttgart: Springer Fachmedien Wiesbaden GmbH. ISBN 978-3-8348-9905-7.
  5. ^ a b c d e f g Forster, Otto (1991). Lectures on Riemann surfaces. München: Springer Berlin. ISBN 978-3-540-90617-9.
  6. ^ a b c d e Munkres, James (2000). Topology. Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-468951-7.
  7. ^ Aguilar, Marcelo Alberto; Socolovsky, Miguel (23 November 1999). "The Universal Covering Group of U(n) and Projective Representations". International Journal of Theoretical Physics. 39 (4). Springer US (published April 2000): 997–1013. arXiv:math-ph/9911028. Bibcode:1999math.ph..11028A. doi:10.1023/A:1003694206391. S2CID 18686364.