En matemáticas , específicamente en la topología algebraica , un mapa de cobertura (que también cubre la proyección ) es una función continua. desde un espacio topológico a un espacio topológico tal que cada punto en tiene un vecindario abierto cubierto uniformemente por(como se muestra en la imagen). [1] En este caso,se llama espacio de cobertura yel espacio de la base de la proyección de la cubierta. La definición implica que cada mapa de cobertura es un homeomorfismo local .
Los espacios de cobertura juegan un papel importante en la teoría de la homotopía , el análisis armónico , la geometría de Riemann y la topología diferencial . En la geometría de Riemann, por ejemplo, la ramificación es una generalización de la noción de mapas de cobertura. Los espacios de cobertura también están profundamente entrelazados con el estudio de los grupos de homotopía y, en particular, el grupo fundamental . Una aplicación importante proviene del resultado de que, sies un espacio topológico "suficientemente bueno" , hay una biyección entre la colección de todas las clases de isomorfismos de revestimientos conectados dey las clases de conjugación de subgrupos del grupo fundamental de.
Definicion formal
Dejar ser un espacio topológico . Un espacio de cobertura de es un espacio topológico junto con un mapa de sobreyección continuo
tal que por cada , existe un barrio abierto de , tal que (la imagen previa de debajo ) es una unión de conjuntos abiertos disjuntos en, cada uno de los cuales se asigna homeomórficamente en por . [2] [3]
De manera equivalente, un espacio de cobertura de puede definirse como un haz de fibras con fibras discretas.
El mapa se llama mapa de cobertura , [3] el espacioa menudo se llama el espacio base de la cubierta, y el espaciose llama el espacio total de la cubierta. Por cualquier punto en la base la imagen inversa de en es necesariamente un espacio discreto [3] llamado fibra sobre.
Los barrios abiertos especiales de que se dan en la definición se denominan barrios con cobertura uniforme . Los vecindarios cubiertos de manera uniforme forman una cubierta abierta del espacio. Las copias homeomorfas en de un vecindario cubierto uniformemente se llaman las sábanas sobre. Uno generalmente imágenes como "flotando por encima" , con mapeando "hacia abajo", las hojas sobre estar apilados horizontalmente uno encima del otro y por encima y la fibra sobre que consta de esos puntos de que se encuentran "verticalmente arriba" . En particular, los mapas de cobertura son localmente triviales. Esto significa que localmente, cada mapa de cobertura es 'isomorfo' a una proyección en el sentido de que hay un homeomorfismo,, de la imagen previa, de un barrio cubierto uniformemente, sobre, dondees la fibra, satisfaciendo la condición de trivialización local , que es que, si proyectamos sobre , , entonces la composición de la proyección con el homeomorfismo será un mapa de la imagen previa sobre , luego la composición derivada será igual localmente (dentro de ).
Definiciones alternativas
Muchos autores imponen algunas condiciones de conectividad a los espacios y en la definición de un mapa de cobertura. En particular, muchos autores requieren que ambos espacios estén conectados por caminos y localmente conectados por caminos . [4] [5] Esto puede resultar útil porque muchos teoremas se cumplen solo si los espacios en cuestión tienen estas propiedades. Algunos autores omiten el supuesto de sobrejetividad, porque si está conectado y no está vacío, entonces la sobrejetividad del mapa de cobertura se sigue en realidad de los otros axiomas.
Ejemplos de
- Cada espacio se cubre trivialmente.
- Un espacio topológico conectado y conectado de forma local tiene una cubierta universal si y solo si se conecta de forma semilocal de forma sencilla .
- es la cubierta universal del círculo
- El grupo de giro es una tapa doble del grupo ortogonal especial y una tapa universal cuando. Los isomorfismos accidentales o excepcionales para los grupos de Lie dan entonces isomorfismos entre los grupos de espín en la dimensión baja y los grupos de Lie clásicos.
- El grupo unitario tiene cubierta universal .
- La n-esfera es una doble cubierta de espacio proyectivo real y es una funda universal para .
- Cada colector tiene una doble tapa orientable que se conecta si y solo si el colector no es orientable.
- El teorema de la uniformización afirma que cada superficie de Riemann tiene una cubierta universal equivalente conforme a la esfera de Riemann , el plano complejo o el disco unitario.
- La funda universal de una cuña de círculos es el gráfico de Cayley del grupo libre engeneradores, es decir, una celosía Bethe .
- El toro es una doble tapa de la botella de Klein . Esto se puede ver usando los polígonos del toro y la botella de Klein, y observando que la doble tapa del círculo (incrustado en enviando ).
- Cada gráfico tiene una cubierta doble bipartita . Dado que cada gráfico es homotópico a una cuña de círculos, su cobertura universal es un gráfico de Cayley.
- Cada inmersión de un colector compacto a un colector de la misma dimensión es una cobertura de su imagen.
- Otra herramienta eficaz para la construcción de espacios de cobertura es el uso de cocientes mediante acciones libres de grupos finitos.
- Por ejemplo, el espacio se define por el cociente de (incrustado en ) mediante el -acción . Este espacio, llamado espacio de lentes , tiene un grupo fundamental y tiene funda universal .
- El mapa de esquemas afines forma un espacio de cobertura con como su grupo de transformaciones de cubierta. Este es un ejemplo de una cubierta cíclica de Galois .
Propiedades
Propiedades locales comunes
- Cada portada es un homeomorfismo local ; [6] es decir, para cada, existe un barrio de cy un barrio de tal que la restricción de p a T produce un homeomorfismo de U a V . Esto implica que C y X comparten todas las propiedades locales. Si X está simplemente conectado y C está conectado, entonces esto también es válido globalmente, y la cobertura p es un homeomorfismo.
- Si y están cubriendo mapas, entonces también lo es el mapa dada por . [7]
Homeomorfismo de las fibras
Para cada x en X , la fibra sobre x es un discreto subconjunto de C . [3] En cada componente conectado de X , las fibras son homeomórficas.
Si X está conectado, hay un espacio discreto F tal que para cada x en X la fibra sobre x es homeomorfa a F y, además, para cada x en X hay un vecindario U de x tal que su preimagen completa p -1 ( U ) es homeomorfo a U × F . En particular, la cardinalidad de la fibra sobre x es igual a la cardinalidad de F y se llama el grado de la cubierta p : C → X . Así, si cada fibra tiene n elementos, hablamos de una cubierta de n pliegues (para el caso n = 1 , la cubierta es trivial; cuando n = 2 , la cubierta es una cubierta doble ; cuando n = 3 , la cubierta es una cubierta triple y así sucesivamente).
Propiedades de elevación
Si p : C → X es una cobertura y γ es una ruta en X (es decir, un mapa continuo desde el intervalo unitario [0, 1] en X ) y c ∈ C es un punto "que se encuentra sobre" γ (0) (es decir p ( c ) = γ (0)) , entonces existe una ruta única Γ en C que se encuentra sobre γ (es decir, p ∘ Γ = γ ) tal que Γ (0) = c . La curva Γ se llama elevación de γ. Si x y y son dos puntos en X conectadas por un camino, entonces ese camino proporciona una biyección entre la fibra sobre x y la fibra sobre y a través de la propiedad de elevación.
De manera más general, sea f : Z → X un mapa continuo a X desde un espacio Z conectado con una ruta y con una ruta conectada localmente . Fije un punto base z ∈ Z y elija un punto c ∈ C "que se encuentre sobre" f ( z ) (es decir, p ( c ) = f ( z ) ). Entonces existe una elevación de f (es decir, un mapa continuo g : Z → C para el cual p ∘ g = f y g ( z ) = c ) si y solo si los homomorfismos inducidos f # : π 1 ( Z , z ) → π 1 ( X , f ( z )) y p # : π 1 ( C , c ) → π 1 ( X , f ( z )) a nivel de grupos fundamentales satisfacer
(♠)
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# : 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 : 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.
Cubiertas universales
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.
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 carries some additional structure, then its universal cover usually inherits that structure:
- If the space is a manifold, then so is its universal cover D.
- If the space is a Riemann surface, then so is its universal cover D, and is a holomorphic map.
- If the space is a Riemannian manifold, then so is its universal cover, and is a local isometry.
- If the space 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 . If the space contains a closed timelike curve (CTC), then the space 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.
Recubrimientos G
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.
Grupo de transformación de cubierta (cubierta), cubiertas regulares
A covering transformation or deck transformation or automorphism of a cover is a homeomorphism such that . The set of all deck transformations of forms a group under composition, the deck transformation group . Deck transformations are also called covering transformations. Every deck transformation permutes the elements of each fiber. This defines a group action of the deck transformation group on each fiber. Note that by the unique lifting property, if is not the identity and is path connected, then has no fixed points.
Now suppose is a covering map and (and therefore also ) is connected and locally path connected. The action of on each fiber is free. If this action is transitive on some fiber, then it is transitive on all fibers, and we call the cover regular (or normal or Galois). Every such regular cover is a principal G {\displaystyle G} -bundle, where is considered as a discrete topological group.
Every universal cover is regular, with deck transformation group being isomorphic to the fundamental group .
As another important example, consider the complex plane and the complex plane minus the origin. Then the map with is a regular cover. The deck transformations are multiplications with -th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group . Likewise, the map with is the universal cover.
Acción monodromía
Again suppose 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., ), and is a path with , then this path lifts to a unique path in C with starting point c. The end point of this lifted path need not be c, but it must lie in the fiber over x. It turns out that this end point only depends on the class of γ in the fundamental group 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: 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.
Más sobre la estructura del grupo
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.
Relaciones con los grupoides
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 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.
Relaciones con espacios de clasificación y cohomología grupal
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.
Generalizaciones
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.
Aplicaciones
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.
Ver también
- 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
- Quotient space (topology)
Notas
- ^ Spanier 1994, p. 62
- ^ a b c d Munkres 2000, p. 336
- ^ Lickorish (1997). An Introduction to Knot Theory. pp. 66–67.
- ^ Bredon, Glen (1997). Topology and Geometry. ISBN 978-0387979267.
- ^ Munkres 2000, p. 338
- ^ Munkres 2000, p. 339, Theorem 53.3
Referencias
- Brown, Ronald (2006). Topology and Groupoids. Charleston, S. Carolina: Booksurge LLC. ISBN 1-4196-2722-8. See chapter 10.
- Chernavskii, A.V. (2001) [1994], "Covering", Encyclopedia of Mathematics, EMS Press
- Farkas, Hershel M.; Kra, Irwin (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.
- Higgins, Philip J. (1971). Notes on categories and groupoids. Mathematical Studies. 32. London-New York-Melbourne: Van Nostrand Reinhold. MR 0327946.
- Jost, Jürgen (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, Jeremy (2012). "Semicoverings: a generalization of covering space theory". Homology, Homotopy and Applications. 14 (1): 33–63. arXiv:1108.3021. doi:10.4310/HHA.2012.v14.n1.a3. MR 2954666. S2CID 55921193.
- Ellis, Graham. "Homological Algebra Programming".
- Ellis, Graham (2004). "Computing group resolutions". Journal of Symbolic Computation. 38 (3): 1077–1118. doi:10.1016/j.jsc.2004.03.003.
- Spanier, Edwin (1994) [1966]. Algebraic Topology. Springer. ISBN 0-387-94426-5.