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 U × F. 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 p : C → X. Thus, if every fiber has n elements, we speak of an n-fold covering (for the case n = 1, the covering is trivial; when n = 2, the covering is a double cover; when n = 3, the covering is a triple cover and so on).
If p : C → X is a cover and γ is a path in X (i.e. a continuous map from the unit interval [0, 1] into X) and c ∈ C is a point "lying over" γ(0) (i.e. p(c) = γ(0)), then there exists a unique path Γ in C lying over γ (i.e. p ∘ Γ = γ) such that Γ(0) = c. 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 f : Z → X be a continuous map to X from a path connected and locally path connected space Z. Fix a base-point z ∈ Z, and choose a point c ∈ C "lying over" f(z) (i.e. p(c) = f(z)). Then there exists a lift of f (that is, a continuous map g : Z → C for which p ∘ g = f and g(z) = c) if and only if the induced homomorphisms f# : π1(Z, z) → π1(X, f(z)) and p# : π1(C, c) → π1(X, f(z)) at the level of fundamental groups satisfy
In particular, if the space Z is assumed to be simply connected (so that π1(Z, z) is trivial), condition (♠) is automatically satisfied, and every continuous map from Z to X can be lifted. Since the unit interval [0, 1] is simply connected, the lifting property for paths is a special case of the lifting property for maps stated above.
If p : C → X is a covering and c ∈ C and x ∈ X are such that p(c) = x, then p# is injective at the level of fundamental groups, and the induced homomorphisms p# : πn(C, c) → πn(X, x) are isomorphisms for all n ≥ 2. Both of these statements can be deduced from the lifting property for continuous maps. Surjectivity of p# for n ≥ 2 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.
Let p1 : C1 → X and p2 : C2 → X be two coverings. One says that the two coverings p1 and p2 are equivalent if there exists a homeomorphism p21 : C2 → C1 and such that p2 = p1 ∘ p21. Equivalence classes of coverings correspond to conjugacy classes of subgroups of the fundamental group of X, as discussed below. If p21 : C2 → C1 is a covering (rather than a homeomorphism) and p2 = p1 ∘ p21, then one says that p2 dominates p1.
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 (x, y) with (2x, y/2). If p : C → X is the quotient map then it is a covering since the action of Z on C generated by f(x, y) = (2x, y/2) is free and wandering (but not properly discontinuous). The points p(1, 0) and p(0, 1) 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.
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 q: D → X is a universal cover of the space X and the mapping p : C → X is any cover of the space X where the covering space C is connected, then there exists a covering map f : D → C such that p ∘ f = q. 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 q(d) = x and a point c in the space C with p(c) = x, then there exists a unique covering map f : D → C such that p ∘ f = q and f(d) = c.
If the space X has a universal cover then that universal cover is essentially unique: if the mappings q1 : D1 → X and q2 : D2 → X are two universal covers of the space X then there exists a homeomorphism f : D1 → D2 such that q2 ∘ f = q1.
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 π1(X) as structure group.
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.
If p is a universal cover, then Aut(p) can be naturally identified with the opposite group of π1(X, x) so that the left action of the opposite group of π1(X, x) coincides with the action of Aut(p) on the fiber over x. Note that Aut(p) and π1(X, x) are naturally isomorphic in this case (as a group is always naturally isomorphic to its opposite through g ↦ g−1).
If p is a regular cover, then Aut(p) is naturally isomorphic to a quotient of π1(X, x).
In general (for good spaces), Aut(p) is naturally isomorphic to the quotient of the normalizer of p*(π1(C, c)) in π1(X, x) over p*(π1(C, c)), where p(c) = x.
Let p : C → X be a covering map where both X and C are path-connected. Let x ∈ X be a basepoint of X and let c ∈ C be one of its pre-images in C, that is p(c) = x. There is an induced homomorphism of fundamental groups p# : π1(C, c) → π1(X,x) which is injective by the lifting property of coverings. Specifically if γ is a closed loop at c such that p#([γ]) = 1, that is p ∘ γ is null-homotopic in X, then consider a null-homotopy of p ∘ γ as a map f : D2 → X from the 2-disc D2 to X such that the restriction of f to the boundary S1 of D2 is equal to p ∘ γ. By the lifting property the map f lifts to a continuous map g : D2 → C 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 p# : π1(C, c) → π1(X, x) is trivial and thus p# : π1(C, c) → π1(X, x) is an injective homomorphism.
Therefore, π1(C, c) is isomorphic to the subgroup p#(π1(C, c)) of π1(X, x). If c1 ∈ C is another pre-image of x in C then the subgroups p#(π1(C, c)) and p#(π1(C, c1)) are conjugate in π1(X, x) by p-image of a curve in C connecting c to c1. Thus a covering map p : C → X defines a conjugacy class of subgroups of π1(X, x) and one can show that equivalent covers of X define the same conjugacy class of subgroups of π1(X, x).
For a covering p : C → X the group p#(π1(C, c)) 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 [π1(X, x) : p#(π1(C, c))] of the subgroup p#(π1(C, c)) in π1(X, x).
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 π1(X, x). 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 π1(X, x). Once the existence of a universal cover C of X is established, if H ≤ π1(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 π1(X, x) are equivalent. Connected cell complexes and connected manifolds are examples of "sufficiently good" spaces.
Let N(Γp) be the normalizer of Γp in π1(X, x). The deck transformation group Aut(p) is isomorphic to the quotient group N(Γp)/Γp. If p is a universal covering, then Γp is the trivial group, and Aut(p) is isomorphic to π1(X).
Let us reverse this argument. Let N be a normal subgroup of π1(X, x). By the above arguments, this defines a (regular) covering p : C → X. 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.
between the category of covering spaces of a reasonably nice space X and the category of groupoid covering morphisms of π1(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.
If X is a connected cell complex with homotopy groups πn(X) = 0 for all n ≥ 2, 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 K(G, 1) for G = π1(X).
Moreover, for every n ≥ 0 the group of cellular n-chains Cn(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, Cn(T) 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 (where Z is equipped with the trivial ZG-module structure, gm = m for every g ∈ G and every m ∈ 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 K(G, 1) inductively at the same time as a contracting homotopy of this universal cover. It is the latter which gives the computational method.
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 h = fg 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 (MR) 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.
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.