Covering space


In mathematics, specifically algebraic topology, a covering map is a continuous function from a topological space to a topological space
such that each point in has an open neighbourhood evenly covered by . In this case, is called a covering space and 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 is a "sufficiently good" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of and the conjugacy classes of subgroups of the fundamental group of.

Formal definition

Let be a topological space. A covering space of is a topological space together with a continuous surjective map
such that for every, there exists an open neighborhood of, such that is a union of disjoint open sets in, each of which is mapped homeomorphically onto by.
Equivalently, a covering space of may be defined as a fiber bundle with discrete fibers.
The map is called the covering map, the space is often called the base space of the covering, and the space is called the total space of the covering. For any point in the base the inverse image of in is necessarily a discrete space called the fiber over.
The special open neighborhoods of given in the definition are called evenly covered neighborhoods. The evenly covered neighborhoods form an open cover of the space. The homeomorphic copies in of an evenly covered neighborhood are called the sheets over. One generally pictures as "hovering above", with mapping "downwards", the sheets over being horizontally stacked above each other and above, and the fiber over consisting of those points of 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 neighbourhood, onto, where is the fiber, satisfying the local trivialization condition, which is that, if we project onto,, so the composition of the projection with the homeomorphism will be a map from the pre-image onto, then the derived composition will equal locally.

Alternative definitions

Many authors impose some connectivity conditions on the spaces and 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 is connected and is nonempty then surjectivity of the covering map actually follows from the other axioms.

Examples

Common local properties

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 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.

Lifting properties

If is a cover and γ is a path in X and is a point "lying over" γ 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. Then there exists a lift of f 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, 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 Sn is simply connected and hence every continuous map from Sn to X can be lifted to C.

Equivalence

Let and be two coverings. One says that the two coverings p1 and p2 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 and, then one says that p2 dominates p1.

Covering of a manifold

Since coverings are local homeomorphisms, a covering of a topological n-manifold is an n-manifold. 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 differentiable structure that turns p 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 covers any connected cover.

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 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 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. 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 is a homeomorphism such that. The set of all deck transformations of forms a group under composition, the deck transformation group. 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 is not the identity and is path connected, then has no fixed points.
Now suppose is a covering map and is connected and locally path connected. The action of 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. 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 .
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.

Monodromy action

Again suppose is a covering map and C is connected and locally path connected. If x is in X and c belongs to the fiber over x, and is a path with, 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:
for all f in Aut, c in p−1 and γ in.
If p is a universal cover, then Aut 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 on the fiber over x. Note that Aut and are naturally isomorphic in this case is naturally isomorphic to a quotient of.
In general, Aut 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 D2 to X such that the restriction of f to the boundary S1 of D2 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 D2 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 c1. 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 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 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 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 subgroup of are equivalent. Connected cell complexes and connected manifolds are examples of "sufficiently good" spaces.
Let N be the normalizer of Γp in. The deck transformation group Aut is isomorphic to the quotient group N/Γp. If p is a universal covering, then Γp is the trivial group, and Aut is isomorphic to 1.
Let us reverse this argument. Let N be a normal subgroup of. By the above arguments, this defines a covering. Let c1 in C be in the fiber of x. Then for every other c2 in the fiber of x, there is precisely one deck transformation that takes c1 to c2. This deck transformation corresponds to a curve g in C connecting c1 to c2.

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. 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 Cn 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, Cn is a free ZG-module with free ZG-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. 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, 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 is the real projective space RP3, with fundamental group Z/2, and only covering space the hypersphere S3, which is the group Spin, 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, 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 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.