Covering space

In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p {displaystyle p} from a topological space C {displaystyle C} to a topological space X {displaystyle X} such that each point in X {displaystyle X} has an open neighbourhood evenly covered by p {displaystyle p} (as shown in the image); the precise definition is given below. In this case, C {displaystyle C} is called a covering space and X {displaystyle X} the base space of the covering projection. The definition implies that every covering map is a local homeomorphism. f # ( π 1 ( Z , z ) ) ⊂ p # ( π 1 ( C , c ) ) . {displaystyle f_{#}(pi _{1}(Z,z))subset p_{#}(pi _{1}(C,c)).}     (♠)The universal cover (of the space X) covers any connected cover (of the space X). In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p {displaystyle p} from a topological space C {displaystyle C} to a topological space X {displaystyle X} such that each point in X {displaystyle X} has an open neighbourhood evenly covered by p {displaystyle p} (as shown in the image); the precise definition is given below. In this case, C {displaystyle C} is called a covering space and X {displaystyle 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 {displaystyle X} is a 'sufficiently good' topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X {displaystyle X} and the conjugacy classes of subgroups of the fundamental group of X {displaystyle X} . Let X {displaystyle X} be a topological space. A covering space of X {displaystyle X} is a topological space C {displaystyle C} together with a continuous surjective map such that for every x ∈ X {displaystyle xin X} , there exists an open neighborhood U {displaystyle U} of x {displaystyle x} , such that p − 1 ( U ) {displaystyle p^{-1}(U)} (the inverse image of U {displaystyle U} under p {displaystyle p} ) is a union of disjoint open sets in C {displaystyle C} , each of which is mapped homeomorphically onto U {displaystyle U} by p {displaystyle p} . Equivalently, a covering space of X {displaystyle X} may be defined as a fiber bundle p : C → X {displaystyle pcolon C o X} with discrete fibers. The map p {displaystyle p} is called the covering map, the space X {displaystyle X} is often called the base space of the covering, and the space C {displaystyle C} is called the total space of the covering. For any point x {displaystyle x} in the base the inverse image of x {displaystyle x} in C {displaystyle C} is necessarily a discrete space called the fiber over x {displaystyle x} . The special open neighborhoods U {displaystyle U} of x {displaystyle x} given in the definition are called evenly covered neighborhoods. The evenly covered neighborhoods form an open cover of the space X {displaystyle X} . The homeomorphic copies in C {displaystyle C} of an evenly covered neighborhood U {displaystyle U} are called the sheets over U {displaystyle U} . One generally pictures C {displaystyle C} as 'hovering above' X {displaystyle X} , with p {displaystyle p} mapping 'downwards', the sheets over U {displaystyle U} being horizontally stacked above each other and above U {displaystyle U} , and the fiber over x {displaystyle x} consisting of those points of C {displaystyle C} that lie 'vertically above' x {displaystyle 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 {displaystyle h} , from the pre-image p − 1 ( U ) {displaystyle p^{-1}(U)} , of an evenly covered neighbourhood U {displaystyle U} , onto U × F {displaystyle U imes F} , where F {displaystyle F} is the fiber, satisfying the local trivialization condition, which is that, if we project U × F {displaystyle U imes F} onto U {displaystyle U} , π : U × F → U {displaystyle pi colon U imes F o U} , so the composition of the projection π {displaystyle pi } with the homeomorphism h {displaystyle h} will be a map π ∘ h {displaystyle pi circ h} from the pre-image p − 1 ( U ) {displaystyle p^{-1}(U)} onto U {displaystyle U} , then the derived composition π ∘ h {displaystyle pi circ h} will equal p {displaystyle p} locally (within p − 1 ( U ) {displaystyle p^{-1}(U)} ). Many authors impose some connectivity conditions on the spaces X {displaystyle X} and C {displaystyle C} in the definition of a covering map. In particular, many authors require both spaces to be path-connected and locally path-connected. 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 {displaystyle X} is connected and C {displaystyle C} is nonempty then surjectivity of the covering map actually follows from the other axioms. For instance the diamond crystal as an abstract graph is the maximal abelian covering graph of the dipole graph D4