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 (as shown in the image); the precise definition is given below. In this case, C is called a covering space of X. 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 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

$p \colon C \to X\,$

such that for every xX, there exists an open neighborhood U of x, such that p−1(U) (the inverse image of U under p) is a union of disjoint open sets in C, each of which is mapped homeomorphically onto U by p.[1][2]

The map p is called the covering map,[2] 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[2] 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, h, from the pre-image p−1(U), of an evenly covered neighbourhood U, onto U×F, where F is the fiber, satisfying the local trivialization condition, which is that, if we project U×F onto U, π:U×FU, so the composition of the projection π with the homeomorphism h will be a map πh from the pre-image p−1(U) onto U, then the derived composition πh will equal p locally (within p−1(U)).

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 S1R2. The map p : RS1 with

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

is a cover where each point of S1 is covered infinitely often.[1]

As another example, take the complex plane with the origin removed (denoted by C*), and pick a non-zero integer n. Then qn : C* → C* given by

$q_n (z) = z^n$

is a cover. Here, every fiber has n elements. The map qn 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 : CS1 is determined by a subgroup f#1(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 Hn = {kn : kZ} for n = 1, 2, 3, ..., where Hn has index n in Z. Each of the subgroups Hn of Z is realized by the covering qn : S1S1 since one can check that (qn)# : ZZ maps an integer k to kn and hence, (qn)#(Z) = Hn. The trivial subgroup of Z is realized by the covering p : RS1, since R is simply connected and has trivial fundamental group and hence p#1(R)) = {0}, the trivial subgroup of Z. Since the total space of the coverings qn is S1 and 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: CX is a local homeomorphism[3]—that is, for every cC, there exists a neighborhood UC of c and a neighborhood VX of p(c) such that the restriction of p to U yields a homeomorphism from U to V. 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.
• If p: EB and p′: E′ → B′ are covering maps, then so is the map p × p′: E × E′ → B × B′ given by (p × p′)(e, e′) = (p(e), p′(e′)).[4]

Homeomorphism of the fibres

For every x in X, the fiber over x is a discrete subset of C.[2] 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 : CX. 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).

Lifting properties

If p: CX is a cover and γ is a path in X (i.e. a continuous map from the unit interval [0, 1] into X) and cC 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) such that 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.

More generally, let f: ZX be a continuous map to X from a path connected and locally path connected space Z. Fix a base-point zZ, and choose a point cC "lying over" f(z) (i.e. p(c) = f(z)). Then there exists a lift of f (that is, a continuous map g: ZC for which p o 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

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

()

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) 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: CX is a covering and cC and xX 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.

Equivalence

Let p1 : C1X and p2 : C2X be two coverings. One says that the two coverings p1 and p2 are equivalent if there exists a homeomorphism p21 : C2C1 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 : C2C1 is a covering (rather than a homeomorphism) and p2 = p1 o p21, 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. (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 : CX 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: DX is a universal cover of the space X and the mapping p: CX is any cover of the space X where the covering space C is connected, then there exists a covering map f : DC such that p o 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: DC such that p o 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 : D1X and q2: D2X are two universal covers of the space X then there exists a homeomorphism f: D1D2 such that q2 o 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 RS1 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:

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 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 group, regular covers

A deck transformation or automorphism of a cover p : CX is a homeomorphism f : CC 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 : CX 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 (or normal or Galois). Every such regular cover is a principal G-bundle, where G = Aut(p) is considered as a discrete topological group.

Every universal cover p : DX 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 Cn.

Another example: p : C* → C* with $p(z)\,=\,z^{n!}$ from above is regular. Here one has a hierarchy of deck transformation groups. In fact Cx! is a subgroup of Cy!, for 1 ≤ xyn.

Monodromy action

Again suppose p: CX 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). 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.

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: $f\cdot(c\cdot\gamma) = (f\cdot c)\cdot\gamma$ 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 gg−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*1(C, c)) in π1(X, x) over p*1(C, c)), where p(c) = x.

More on the group structure

Let p : CX be a covering map where both X and C are path-connected. Let xX be a basepoint of X and let cC 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 : D2X 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 : D2C 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#1(C, c)) of π1(X, x). If c1C is another pre-image of x in C then the subgroups p#1(C, c)) and p#1(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 : CX 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 : CX the group p#1(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 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 index1(X, x): p#1(C, c))] of the subgroup p#1(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(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 : CX . 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

$\pi_1: \mathrm{TopCov}(X) \to \mathrm{GpdCov}(\pi_1 X)$

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. 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 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

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

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 gG and every mZ). 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.

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 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 (MR0760769) 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

Gimbal lock occurs because the map T3RP3 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.

Notes

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

Citations

1. ^ a b Chernavskii 2001
2. ^ a b c d Munkres 2000, p. 336
3. ^ Munkres 2000, p. 338
4. ^ Munkres 2000, p. 339, Theorem 53.3

References

• Brazas, J., 'Semicoverings: a generalisation of covering space theory', Homology, Homotopy, and Applications, 14 (2012), No. 1, pp. 33–63.
• Ellis, G. "Homological Algebra Program
• Ellis, G. Computing group resolutions, J. Symbolic Comput. 38 (2004) 1077–1118.