Covering space

A covering map satisfies the local triviality condition. Intuitively, such maps locally project a "stack of pancakes" above an open region, U, onto U.

In mathematics, more specifically algebraic topology, a covering map is a continuous surjective function p ^a[›] from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p. This means that for each point x in X, there is associated an ordered pair, (K, U), where U is a neighborhood of x and where K is a collection of disjoint open sets in C, each of which gets mapped homeomorphically, via p, to U (as shown in the image). In particular, this means that every covering map is necessarily a local homeomorphism. Under this definition, C is called a covering space of X.

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 X is a "sufficiently good" topological space, there is a bijection from the collection of all isomorphism classes of connected coverings of X and subgroups of the fundamental group of X.

Formal definition

Let X be a topological space. A covering space of X is a space C together with a continuous surjective map

$p : C \to X\,$

such that for every x ∈ X, there exists an open neighborhood U of x, such that p⁻¹(U) (the inverse image of U under p) is a disjoint union of open sets in C, each of which is mapped homeomorphically onto U by p.

The map p is called the covering map: the space X is often called the base space of the covering and the space C is called the total space of the covering. For any point x in the base the inverse image of x in C is necessarily a discrete space called the fiber over x.

The special open neighborhoods U of x given in the definition are called evenly-covered neighborhoods. The evenly-covered neighborhoods form an open cover of the space X. The homeomorphic copies in C of an evenly-covered neighborhood U are called the sheets over U. One generally pictures C as "hovering above" X, with p mapping "downwards", the sheets over U being horizontally stacked above each other and above U, and the fiber over x consisting of those points of C that lie "vertically above" x. 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 U, to U X F, where F is the fiber, satisfying the local trivialization condition. That is, if we project this homeomorphism onto U (and thus the composition of the projection with this homeomorphism will be a map from the pre-image of U to U), the derived composition will equal p.

Alternative definitions

Many authors impose some connectivity conditions on the spaces X and C in the definition of a covering map. In particular, many authors require both spaces to be path-connected and locally path-connected.^{[citation needed]} 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 X is connected and C is nonempty then surjectivity of the covering map actually follows from the other axioms.

Examples

Consider the unit circle S¹ in R². Then the map p : R → S¹ with

p(t) = (cos(t),sin(t))

is a cover where each point of S¹ is covered infinitely often.

Consider the complex plane with the origin removed, denoted by C^×, and pick a non-zero integer n. Then q_n : C^× → C^× given by

q_n(z) = zⁿ

is a cover. Here every fiber has n elements. The map q_n leaves the unit circle S¹ invariant and the restriction of this map to S¹ is an n-fold cover of the circle by itself.

In fact, S¹ and R are the only connected covering spaces of the circle. To prove this, we first note that the fundamental group of the circle is isomorphic to the additive group of integers Z. As follows from the correspondence between equivalence classes of connected coverings and conjugacy classes of subgroups of the fundamental group of the base space discussed below, a connected covering f : C →S¹ is determined by a subgroup f_#(π₁(C)) of π₁(S¹) = Z, where f_# is the induced homomorphism. The group Z is abelian and it only has two kinds of subgroups: the trivial subgroup (which has infinite subgroup index in Z) and the subgroups H_n = { kn | k ∈ Z } for n =1, 2, 3...., where H_n has index n in Z. Each of the subgroups H_n of Z is realized by the covering q_n : S¹ → S¹ since one can check that (q_n)_# : Z → Z maps an integer k to kn and hence (q_n)_#(Z) = H_n. The trivial subgroup of Z is realized by the covering p : R → S¹ since R is simply connected and has trivial fundamental group and hence p_#(π₁(R)) = {0}, the trivial subgroup of Z. Since the total space of the coverings q_n is S¹ and since the total space of the covering p is R, this shows that every connected cover of S¹ is either S¹ or R.

A further example, originating from physics (see quantum mechanics), is the special orthogonal group SO(3) of rotations of R³, which has the "double" covering group SU(2) of unitary rotations of C² (in quantum mechanics acting as the group of spinor rotations). Both groups have identical Lie algebras, but only SU(2) is simply connected.

Properties

Common local properties: Every cover p : C → X is a local homeomorphism (i.e. to every $c\in C$ there exists an open set A in C containing c and an open set B in X such that the restriction of p to A yields a homeomorphism between A and B). This implies that C and X share all local properties. If X is simply connected and C is connected, then this holds globally as well, and the covering p is a homeomorphism.

Homeomorphism of the fibres: 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 U x 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).

The lifting property: 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 $c\in C$ 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 o Γ = γ) and with 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.

A more general lifting property is described as follows:

Let p : C → X be a cover and let f be a continuous map from Z to X where Z is path connected and locally path connected. Let z in Z be a base-point, let x = f(z) and let c in C be in the fiber over x, that is such that p(c)=x.

Then there exists a lift of f (that is, a continuous map g : Z → C such that p o g = f and g(z)=c) if and only if for the induced homomorphisms at the level of the fundamental groups we have

$f_\#(\pi_1(Z,z))\subset p_\#(\pi_1(C,c)).$    (♠)

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 π₁(Z,z) = 1), 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 the induced homomorphism p_# : π₁(C,c) → π₁(X,x) is injective and the induced homomorphisms p_# : π_n(C,c) → π_n(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 n ≥ 2 the sphere Sⁿ is simply connected and hence every continuous map from Sⁿ to X can be lifted to C.

Equivalence: 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 o 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 o p₂₁, then one says that p₂ dominates p₁.

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 (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 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.

Universal covers

A connected covering space is a universal cover 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 all connected covers 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.

The example R → S¹ given above is a universal cover. The map S³ → SO(3) from unit quaternions to rotations of 3D space described in quaternions and spatial rotation is also a universal cover.

If the space X carries some additional structure, then its universal cover normally inherits that structure:

• if the space X is a manifold, then so is its universal cover D
• if the space X is a Riemann surface, then so is its universal cover D, and p is a holomorphic map
• if the space X is a Lorentzian manifold, then so is its universal cover. Furthermore, suppose the subset p⁻¹(U) is a disjoint union of open sets each of which is diffeomorphic with U by the mapping p. If the space X contains a closed timelike curve, then the space X is timelike multiply connected (no CTC can be timelike homotopic to a point, as that point would not be causally well-behaved), its universal (diffeomorphic) cover is timelike simply connected (it does not contain a CTC).
• if X is a Lie group (as in the two examples above), then so is its universal cover D, and the mapping p is a homomorphism of Lie groups. In this case the universal cover is also called the universal covering group. This has particular application to representation theory and quantum mechanics, since ordinary representations of the universal covering group (D) are projective representations of the original (classical) group (X).

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. It is natural to ask under what conditions the projection, $X \to X/G$, 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 $Y \times Y$ by the twist action. 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.

Deck transformation group, regular covers

A deck transformation or automorphism of a cover p : C → X is a homeomorphism f : C → C such that p o f = p. The set of all deck transformations of p forms a group under composition, the deck transformation group Aut(p). 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 f is not the identity and C is path connected, then f has no fixed points.

Now suppose p : C → X is a covering map and C (and therefore also X) is connected and locally path connected. The action of Aut(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. Every such regular cover is a principal G-bundle, where G = Aut(p) is considered as a discrete topological group.

Every universal cover p : D → X is regular, with deck transformation group being isomorphic to the fundamental group π₁(X).

The example p : C^× → C^× with p(z) = zⁿ from above 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 C_n.

Another example: $\,p:\mathbf{C}^*\rightarrow\mathbf{C}^*$ with $p(z)\,=\,z^{n!}$ from above is regular. Here one has a hierarchy of deck transformation groups. In fact C_x! is a subgroup of C_y!, for $\,1 \le x \le y \le n$.

Monodromy action

Again suppose p : C → X 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. p(c) = x), and γ:[0,1]→X is a path with γ(0)=γ(1)=x, 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 π₁(X,x), and in this fashion we obtain a right group action of π₁(X,x) on the fiber over x. This is known as the monodromy action.

So there are two actions on the fiber over x: Aut(p) acts on the left and π₁(X,x) acts on the right. These two actions are compatible in the following sense:

f.(c.γ) = (f.c).γ

for all f in Aut(p), c in p ^-1(x) and γ in π₁(X,x).

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 $g\mapsto g^{-1}$ ).

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 $p_{\ast}(\pi_1(C, c))$ in π₁(X,x) over $p_{\ast}(\pi_1(C, c))$, where p(c) = x.

More on the group structure

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 o γ is null-homotopic in X, then consider a null-homotopy of p o 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 o γ. By the lifting property the map f lifts to a continuous map g : D² → C such that the restriction of f 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

$\Gamma_p(c) = \{ [\gamma] : \gamma_C \mbox{ is a closed curve in } C \mbox { passing through } c\in C \}$,

the set of homotopy classes of those closed curves γ based at x whose lifts γ_C 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 C 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(Γ_p) be the normalizer of Γ_p in π₁(X,x). The deck transformation group Aut(p) is isomorphic to the quotient group N(Γ_p)/Γ_p. If p is a universal covering, then Γ_p 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.

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 π₁X. 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. Proofs of these facts are given in the book `Topology and Groupoids' referenced below.

Relations with classifying spaces and group cohomology

If X is a connected cell complex with homotopy groups π_n(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 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 ZG-basis given by representatives of G-orbits of n-cells in T. In this case the standard topological chain complex

$\cdots \overset{\partial}{\to} C_n(T)\overset{\partial}{\to} C_{n-1}(T)\overset{\partial}{\to} \cdots \overset{\partial}{\to} C_0(T)\overset{\varepsilon}{\to} \mathbb Z,$

where ε is the augmentation map, is a free ZG-resolution of Z (where Z is equipped with the trivial ZG-module structure, g m = m for every g ∈ G and every m ∈ Z). This resolution can be used to compute group cohomology of G with arbitrary coefficients.

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.

Applications

Gimbal lock occurs because the map T³ → RP³ is not a covering space. 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, 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

• Bethe lattice is the universal cover of a Cayley graph
• Covering graph, a covering space for an undirected graph, and its special case the bipartite double cover.
• Covering group
• Galois connection

Notes

^ a: Some authors do not require covering maps to be surjective; see above for more details.

References

• Farkas, Hershel M.; Irwin Kra (1980). Riemann Surfaces (2nd ed. ed.). New York: Springer. ISBN 0-387-90465-4.  See chapter 1 for a simple review.
• Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0. http://www.math.cornell.edu/~hatcher/AT/ATpage.html. 
• Jost, Jurgen (2002). Compact Riemann Surfaces. New York: Springer. ISBN 3-540-43299-X.  See section 1.3
• Massey, William (1991). A Basic Course in Algebraic Topology. New York: Springer. ISBN 0-387-97430-X.  See chapter 5.
• Brown, Ronald (2006). Topology and Groupoids. Charleston, S. Carolina: Booksurge LLC. ISBN 1-4196-2722-8. http://www.bangor.ac.uk/r.brown/topgpds.html.  See chapter 10.
• Categories and groupoids, P.J. Higgins, downloadable reprint of van Nostrand Notes in Mathematics, 1971, which deal with applications of groupoids in group theory and topology.