Covering space

In mathematics, specifically algebraic topology, a covering map (also covering projection) is a continuous function formula_1 from a topological space formula_2 to a topological space 
formula_3 such that each point in formula_3 has an open neighborhood evenly covered by formula_1 (as shown in the image). In this case, formula_2 is called a covering space and formula_3 the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.
Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, 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 formula_3 is a "sufficiently good" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of formula_3 and the conjugacy classes of subgroups of the fundamental group of formula_3.
Formal definition.
Let formula_3 be a topological space. A covering space of formula_3 is a topological space formula_2 together with a continuous surjective map
such that for every formula_15, there exists an open neighborhood formula_16 of formula_17, such that formula_18 (the pre-image of formula_16 under is a union of disjoint open sets in each of which is mapped homeomorphically onto formula_16 by formula_1.
Equivalently, a covering space of formula_3 may be defined as a fiber bundle formula_14 with discrete fibers.
The map formula_1 is called the covering map, the space formula_3 is often called the base space of the covering, and the space formula_2 is called the total space of the covering. For any point formula_17 in the base the inverse image of formula_17 in formula_2 is necessarily a discrete space called the fiber over 
The special open neighborhoods formula_16 of formula_17 given in the definition are called evenly covered neighborhoods. The evenly covered neighborhoods form an open cover of the space formula_3. The homeomorphic copies in formula_2 of an evenly covered neighborhood formula_16 are called the sheets over formula_16. One generally pictures formula_2 as "hovering above" formula_3, with formula_1 mapping "downwards", the sheets over formula_16 being horizontally stacked above each other and above formula_16, and the fiber over formula_17 consisting of those points of formula_2 that lie "vertically above" 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, from the pre-image of an evenly covered neighborhood onto where formula_43 is the fiber, satisfying the local trivialization condition, which is that, if we project formula_44 onto so the composition of the projection formula_45 with the homeomorphism formula_46 will be a map formula_47 from the pre-image formula_18 onto then the derived composition formula_47 will equal formula_1 locally (within 
Alternative definitions.
Many authors impose some connectivity conditions on the spaces formula_3 and formula_2 in the definition of a covering map. In particular, many authors require both spaces to be path-connected and locally path-connected. 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 formula_3 is connected and formula_2 is nonempty then surjectivity of the covering map actually follows from the other axioms.
Properties.
Homeomorphism of the fibers.
For every "x" in "X", the fiber over "x" is a discrete subset of "C". On every connected component of "X", the fibers are homeomorphic.
If "X" is connected, there is a discrete space "F" such that for every "x" in "X" the fiber over "x" is homeomorphic to "F" and, moreover, for every "x" in "X" there is a neighborhood "U" of "x" such that its full pre-image "p"−1("U") is homeomorphic to . 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 . Thus, if every fiber has "n" elements, we speak of an "n"-fold covering (for the case , the covering is trivial; when , the covering is a double cover; when , the covering is a triple cover and so on).
Lifting properties.
If is a cover and γ is a path in "X" (i.e. a continuous map from the unit interval into "X") and is a point "lying over" γ(0) (i.e. , then there exists a unique path Γ in "C" lying over γ (i.e. ) such that . 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.
More generally, let be a continuous map to "X" from a path connected and locally path connected space "Z". Fix a base-point , and choose a point "lying over" "f"("z") (i.e. ). Then there exists a lift of "f" (that is, a continuous map for which and ) if and only if the induced homomorphisms and at the level of fundamental groups satisfy
Moreover, if such a lift "g" of "f" exists, it is unique.
In particular, if the space "Z" is assumed to be simply connected (so that is trivial), condition is automatically satisfied, and every continuous map from "Z" to "X" can be lifted. Since the unit interval is simply connected, the lifting property for paths is a special case of the lifting property for maps stated above.
If is a covering and and are such that , then "p"# is injective at the level of fundamental groups, and the induced homomorphisms are isomorphisms for all . Both of these statements can be deduced from the lifting property for continuous maps. Surjectivity of "p"# for follows from the fact that for all such "n", the "n"-sphere S"n" is simply connected and hence every continuous map from S"n" to "X" can be lifted to "C".
Equivalence.
Let and be two coverings. One says that the two coverings "p"1 and "p"2 are equivalent if there exists a homeomorphism and such that . Equivalence classes of coverings correspond to conjugacy classes of subgroups of the fundamental group of "X", as discussed below. If is a covering (rather than a homeomorphism) and , then one says that "p"2 dominates "p"1.
Covering of a manifold.
Since coverings are local homeomorphisms, 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 with . If is the quotient map then it is a covering since the action of "Z" on "C" generated by is properly discontinuous. The points and do not have disjoint neighborhoods in "X".
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 "n".
Universal covers.
A covering space is a universal covering space if it is simply connected. The name "universal cover" comes from the following important property: if the mapping is a universal cover of the space "X" and the mapping is any cover of the space "X" where the covering space "C" is connected, then there exists a covering map such that . This can be phrased as
The universal cover (of the space "X") covers any connected cover (of the space "X").
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 and a point "c" in the space "C" with , then there exists a unique covering map such that and .
If the space "X" has a universal cover then that universal cover is essentially unique: if the mappings and are two universal covers of the space "X" then there exists a homeomorphism such that .
The space "X" has a universal cover if it is connected, 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". More explicitly, it forms a principal bundle with the fundamental group as structure group.
The example given above is a universal cover. The map from unit quaternions to rotations of 3D space described in quaternions and spatial rotation is also a universal cover.
If the space formula_3 carries some additional structure, then its universal cover usually inherits that structure:
The universal cover first arose in the theory of analytic functions as the natural domain of an analytic continuation.
G-coverings.
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 H"g" of "X" onto itself, in such a way that H"g" "h" is always equal to H"g" ∘ 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 "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 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 (covering) transformation group, regular covers.
A covering transformation or deck transformation or automorphism of a cover formula_98 is a homeomorphism formula_99 such that formula_100. The set of all deck transformations of formula_1 forms a group under composition, the deck transformation group formula_102. Deck transformations are also called covering transformations. Every deck transformation 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 formula_103 is not the identity and formula_2 is path connected, then formula_103 has no fixed points.
Now suppose formula_98 is a covering map and formula_2 (and therefore also formula_3) is connected and locally path connected. The action of formula_102 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 formula_110 is considered as a discrete topological group.
Every universal cover formula_111 is regular, with deck transformation group being isomorphic to the fundamental group 
As another important example, consider formula_112 the complex plane and formula_113 the complex plane minus the origin. Then the map formula_114 with formula_115 is a regular cover. The deck transformations are multiplications with formula_65-th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group formula_117. Likewise, the map formula_118 with formula_119 is the universal cover.
Monodromy action.
Again suppose formula_120 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., formula_121), and formula_122 is a path with formula_123, 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 . In this fashion we obtain a right group action of on the fiber over "x". This is known as the monodromy action.
There are two actions on the fiber over acts on the left and acts on the right. These two actions are compatible in the following sense:
formula_124 for all "f" in Aut("p"), "c" in "p"−1("x") and γ in .
If "p" is a universal cover, then Aut("p") can be naturally identified with the opposite group of so that the left action of the opposite group of coincides with the action of Aut("p") on the fiber over "x". Note that Aut("p") and are naturally isomorphic in this case (as a group is always naturally isomorphic to its opposite through .
If "p" is a regular cover, then Aut("p") is naturally isomorphic to a quotient of .
In general (for good spaces), Aut("p") is naturally isomorphic to the quotient of the normalizer of in over , where .
More on the group structure.
Let be a covering map where both "X" and "C" are path-connected. Let be a basepoint of "X" and let be one of its pre-images in "C", that is . There is an induced homomorphism of fundamental groups which is injective by the lifting property of coverings. Specifically if "γ" is a closed loop at "c" such that , that is is null-homotopic in "X", then consider a null-homotopy of as a map from the 2-disc "D"2 to "X" such that the restriction of "f" to the boundary S1 of "D"2 is equal to . By the lifting property the map "f" lifts to a continuous map such that the restriction of "g" to the boundary S1 of "D"2 is equal to "γ". Therefore, "γ" is null-homotopic in "C", so that the kernel of is trivial and thus is an injective homomorphism.
Therefore, is isomorphic to the subgroup of . If is another pre-image of "x" in "C" then the subgroups and are conjugate in by "p"-image of a curve in "C" connecting "c" to "c"1. Thus a covering map defines a conjugacy class of subgroups of and one can show that equivalent covers of "X" define the same conjugacy class of subgroups of .
For a covering the group can also be seen to be equal to
the set of homotopy classes of those closed curves γ based at "x" whose lifts "γC" 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 the subgroup in .
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 . 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 . Once the existence of a universal cover "C" of "X" is established, if "H" ≤ 1("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 "X" corresponding to the same (conjugacy class of) subgroup of are equivalent. Connected cell complexes and connected manifolds are examples of "sufficiently good" spaces.
Let "N"("Γp") be the normalizer of Γ"p" in . The deck transformation group Aut("p") is isomorphic to the quotient group "N"(Γ"p")/Γ"p". If "p" is a universal covering, then "Γp" is the trivial group, and Aut("p") is isomorphic to 1("X").
Let us reverse this argument. Let "N" be a normal subgroup of . By the above arguments, this defines a (regular) covering . 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.
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
between the category of covering spaces of a reasonably nice space "X" and the category of groupoid covering morphisms of 1("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.
Relations with classifying spaces and group cohomology.
If "X" is a connected cell complex with homotopy groups for all , 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 for .
Moreover, for every the group of cellular "n"-chains "C""n"("T") (that is, a free abelian group with basis given by "n"-cells in "T") also has a natural ZG"-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""n"("T") is a free ZG"-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
where "ε" is the augmentation map, is a free ZG"-resolution of Z (where Z is equipped with the trivial ZG"-module structure, for every and every ). This resolution can be used to compute group cohomology of "G" with arbitrary coefficients.
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 inductively at the same time as a contracting homotopy of this universal cover. It is the latter which gives the computational method.
Generalizations.
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, difficulties arise. Some progress has been made for more complex spaces, such as the Hawaiian earring; see the references there for further information.
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 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.
Another generalisation is to actions of a group which are not free. Ross Geoghegan in his 1986 review () of two papers by M.A. Armstrong on the fundamental groups of orbit spaces 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.
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, 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.
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, 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.