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 (also covering projection) is a continuous function p ^[1] from a topological space, C, to a topological space, X, such that each point in X has an open 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 and X 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 X is a "sufficiently good" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.

Formal definition

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

$p\colon C\to X\,$

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 union of disjoint open sets in C, each of which is mapped homeomorphically onto U by p.^[2][3]

The map p is called the covering map,^[3] 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^[3] 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 × F → U, so the composition of the projection π with the homeomorphism h will be a map π ∘ h from the pre-image p⁻¹(U) onto U, then the derived composition π ∘ h 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.^[4][5] 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

• Every space trivially covers itself.
• A connected and locally path-connected topological space X has a universal cover if and only if it is semi-locally simply connected.
• ℝ is the universal cover of the unit circle S¹.
• The spin group Spin(n) is a double cover of the special orthogonal group and a universal cover when n > 2. The accidental, or exceptional isomorphisms for Lie groups then give isomorphisms between spin groups in low dimension and classical Lie groups.
• The unitary group U(n) has universal cover SU(n) × ℝ.
• The n-sphere Sⁿ is a double cover of real projective space RPⁿ and is a universal cover for n > 1.
• Every manifold has an orientable double cover that is connected if and only if the manifold is non-orientable.
• The uniformization theorem asserts that every Riemann surface has a universal cover conformally equivalent to the Riemann sphere, the complex plane, or the unit disc.
• The universal cover of a wedge of n circles is the Cayley graph of the free group on n generators, i.e. a Bethe lattice.
• The torus is a double cover of the Klein bottle.
• Every graph has a bipartite double cover. Since every graph is homotopic to a wedge of circles, its universal cover is a Cayley graph.
• Every immersion from a compact manifold to a manifold of the same dimension is a covering of its image.
• Infinite-fold abelian covering graphs of finite graphs are regarded as abstractions of crystal structures.^[6] For instance the diamond crystal as an abstract graph is the maximal abelian covering graph of the dipole graph D₄

Properties

Common local properties

• Every cover p: C → X is a local homeomorphism^[7]—that is, for every c ∈ C, there exists a neighborhood U ⊆ C of c and a neighborhood V ⊆ X 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 : E → B 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′)).^[8]

Homeomorphism of the fibres

For every x in X, the fiber over x is a discrete subset of C.^[3] 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 × 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).

Lifting properties

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

$f_{\#}(\pi _{1}(Z,z))\subset p_{\#}(\pi _{1}(C,c)).$

(♠)

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 : 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_# : 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 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 : 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.

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 : 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 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.

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 usually 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 (CTC), 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 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 : C → X is a homeomorphism f : C → C such that p ∘ 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 (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 : D → X is regular, with deck transformation group being isomorphic to the fundamental group π₁(X).

As another important example, consider C the complex plane and C^× the complex plane minus the origin. Then the map p : C^× → C^× with p(z) = zⁿ 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. Likewise, the map exp : C → C^× with exp(z) = e^z is the universal cover.

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). 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 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.

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 ∘ 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

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

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 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(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

$\pi _{1}:\mathrm {TopCov} (X)\to \mathrm {GpdCov} (\pi _{1}X)$

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 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

$\cdots {\overset {\partial }{\to }}C_{n}(T){\overset {\partial }{\to }}C_{n-1}(T){\overset {\partial }{\to }}\cdots {\overset {\partial }{\to }}C_{0}(T){\overset {\varepsilon }{\to }}\mathbf {Z} ,$

where ε is the augmentation map, is a free ZG-resolution of Z (where Z is equipped with the trivial ZG-module structure, 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.

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 any map T³ → RP³ is not a covering map. In particular, the relevant map carries any element of T³, that is, an ordered triple (a,b,c) of angles (real numbers mod 2π), to the composition of the three coordinate axis rotations R_x(a)∘R_y(b)∘R_z(c) by those angles, respectively. Each of these rotations, and their composition, is an element of the rotation group SO(3), which is topologically RP³. 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

1. Spanier, Edwin (1966). Algebraic Topology. McGraw-Hill. p. 62.
2. Chernavskii 2001
3. Munkres 2000, p. 336
4. Lickorish (1997). An Introduction to Knot Theory. pp. 66–67.
5. Bredon (1997). Topology and Geometry. ISBN 978-0387979267.
6. Sunada T. (2012), Topological Crystallography ---With a View Towards Discrete Geometric Analysis---", Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 6, Springer
7. Munkres 2000, p. 338
8. Munkres 2000, p. 339, Theorem 53.3

References

• Brown, Ronald (2006). Topology and Groupoids. Charleston, S. Carolina: Booksurge LLC. ISBN 1-4196-2722-8. See chapter 10.
• Chernavskii, A.V. (2001), "Covering", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Farkas, Hershel M.; Irwin Kra (1980). Riemann Surfaces (2nd 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.
• 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.
• 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.
• Munkres, James R. (2000). Topology (2. ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0131816292.
• 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.
• Spanier, Edwin (December 1994). Algebraic Topology. Springer. ISBN 0-387-94426-5.