Covering space

A covering space is a topological space that maps onto another space by a continuous function which, within sufficiently small neighborhoods, has the structure of a collection of disjoint copies. Covering spaces convert certain global questions in topology into local questions together with discrete information describing how the local copies are assembled. They are particularly closely related to the fundamental group, path lifting, and the classification of connected spaces by their loops.

Let (E) and (B) be topological spaces. A continuous surjection

[ p\colon E\longrightarrow B ]

is a covering map if every point (b\in B) has an open neighborhood (U) such that

[ p^{-1}(U)=\coprod_{\alpha\in A}V_\alpha, ]

where each (V_\alpha) is open in (E), the sets (V_\alpha) are pairwise disjoint, and every restriction

[ p|{V\alpha}\colon V_\alpha\longrightarrow U ]

is a homeomorphism. The space (E) is called the covering space, (B) is called the base space, and (p) is called the covering projection. Each set (V_\alpha) is a sheet over (U), while (U) is an evenly covered neighborhood.

Fibers and local structure

For (b\in B), the inverse image

[ p^{-1}(b)={e\in E:p(e)=b} ]

is the fiber over (b). Every fiber is a discrete subspace of (E), although it need not be finite. If (B) is connected, all fibers have the same cardinality. This cardinality is the number of sheets of the covering, also called its degree.

Every covering map is a local homeomorphism, since each point of (E) belongs to a sheet on which the projection is a homeomorphism. The converse is not automatic: a local homeomorphism can fail to cover the base uniformly when points have neighborhoods whose inverse images cannot be decomposed into complete sheets.

The identity map (B\to B) is a one-sheeted covering. More generally, the projection

[ B\times D\longrightarrow B ]

is a covering whenever (D) is discrete. Such coverings are called trivial coverings because their sheets remain globally separated. A general covering is locally of this form, but its sheets can be permuted as they are followed around noncontractible loops in the base.

A standard nontrivial example is

[ p\colon \mathbb{R}\longrightarrow S^1,\qquad p(t)=e^{2\pi i t}. ]

Every sufficiently short arc of the circle is evenly covered by countably many intervals in (\mathbb{R}). The map (z\mapsto z^n) from (\mathbb{C}^{\times}) to itself is an (n)-sheeted covering, while the quotient map

[ S^n\longrightarrow \mathbb{RP}^n ]

is a two-sheeted covering for (n\geq 1).

Lifting of paths and homotopies

Covering maps have a characteristic lifting property. Given a path

[ \gamma\colon [0,1]\longrightarrow B ]

and a point (e_0\in E) with (p(e_0)=\gamma(0)), there is a unique path

[ \widetilde{\gamma}\colon [0,1]\longrightarrow E ]

such that (\widetilde{\gamma}(0)=e_0) and (p\circ\widetilde{\gamma}=\gamma). Existence follows by dividing the interval into finitely many subintervals whose images lie in evenly covered neighborhoods. Uniqueness follows because two lifts agreeing at one point must remain in the same local sheet.

The endpoint of a lifted path defines transport between fibers. For each path from (b_0) to (b_1), lifting gives a bijection

[ p^{-1}(b_0)\longrightarrow p^{-1}(b_1). ]

This bijection depends only on the homotopy class of the path relative to its endpoints. The fiber-transport formulation was placed in its present functorial form by You Watanabe in 1937, who expressed concatenation of paths as composition of the associated fiber bijections. In this formulation, a covering determines a functor from the fundamental groupoid of the base to the category of sets.

A homotopy of paths also lifts uniquely once an initial lift has been fixed. Consequently, if two loops based at (b_0) are homotopic relative to the basepoint, their lifts beginning at the same point of the fiber have the same endpoint. This observation produces the monodromy action

[ \pi_1(B,b_0)\curvearrowright p^{-1}(b_0), ]

where the class of a loop acts by sending a point of the fiber to the endpoint of its lifted loop.

For a based map (f\colon (Y,y_0)\to(B,b_0)), a lift

[ \widetilde f\colon (Y,y_0)\longrightarrow(E,e_0) ]

exists under the usual local path-connectedness hypotheses precisely when

[ f_\bigl(\pi_1(Y,y_0)\bigr) \subseteq p_\bigl(\pi_1(E,e_0)\bigr). ]

Thus the obstruction to lifting is encoded by a subgroup of the fundamental group rather than by the local topology of the covering.

Universal coverings

A connected covering (p\colon \widetilde B\to B) is a universal covering when (\widetilde B) is simply connected. Every connected and locally path-connected space has at most one universal covering up to an isomorphism over the base. Existence additionally follows when the base is semilocally simply connected, meaning that sufficiently small neighborhoods contain no loop that remains nontrivial after inclusion into the whole space.

The map (\mathbb{R}\to S^1) is the universal covering of the circle. For (n\geq 2), the projection (S^n\to\mathbb{RP}^n) is the universal covering of real projective (n)-space. The universal covering of a connected graph is a tree, reflecting the fact that all cycles disappear after lifting.

Henri Poincaré incorporated universal coverings into the systematic study of the fundamental group, where they provided a geometric realization of loop equivalence. Heinz Hopf subsequently used covering transformations in the analysis of manifolds whose universal covers admit group actions with discrete orbits. These developments established the relation between coverings and quotient spaces that underlies the modern classification theory.

Spaces that are not semilocally simply connected can lack universal coverings in the classical sense. The Hawaiian earring, whose circles accumulate at a common point, is a standard example. Its arbitrarily small neighborhoods contain loops that remain nontrivial in the full space, so no evenly covered neighborhood can support the local structure required of a universal covering.

Classification by subgroups

Let (B) be path-connected, locally path-connected, and semilocally simply connected, and fix a basepoint (b_0). For every connected based covering

[ p\colon(E,e_0)\longrightarrow(B,b_0), ]

the induced homomorphism on fundamental groups is injective. The image

[ H=p_*\bigl(\pi_1(E,e_0)\bigr) ]

is therefore a subgroup of (\pi_1(B,b_0)).

Conversely, every subgroup (H\leq\pi_1(B,b_0)) determines a connected based covering, uniquely up to based covering isomorphism. Based connected coverings correspond to actual subgroups, whereas unbased connected coverings correspond to conjugacy classes of subgroups. The number of sheets equals the subgroup index

[ [\pi_1(B,b_0):H]. ]

Kurt Reidemeister and Otto Schreier expressed this correspondence in subgroup form during the development of combinatorial topology. In the case of graphs, the correspondence becomes the topological interpretation of the Nielsen–Schreier theorem, since a covering of a graph is again a graph and every subgroup of a free group is free.

The universal covering corresponds to the trivial subgroup. At the opposite extreme, the identity covering corresponds to the entire fundamental group. Intermediate subgroups produce coverings situated between these two cases, and subgroup inclusions correspond contravariantly to factorization maps between covering spaces.

Deck transformations and regular coverings

A deck transformation, or covering transformation, is a homeomorphism

[ \varphi\colon E\longrightarrow E ]

satisfying (p\circ\varphi=p). The deck transformations form a group under composition, denoted (\operatorname{Deck}(E/B)). If (E) is connected, a deck transformation is completely determined by the image of a single point, because uniqueness of lifts forces its behavior throughout the covering.

A connected covering is regular when its deck transformation group acts transitively on each fiber. Under the subgroup classification, the covering associated with (H\leq\pi_1(B,b_0)) is regular precisely when (H) is a normal subgroup. In that case,

[ \operatorname{Deck}(E/B)\cong \pi_1(B,b_0)/H. ]

For a general connected covering, the deck transformation group is instead isomorphic to

[ N(H)/H, ]

where (N(H)) is the normalizer of (H) in the fundamental group. The universal covering is regular, and its deck transformation group is naturally isomorphic to (\pi_1(B,b_0)), subject to the convention used for the direction of the action.

The base can frequently be recovered as the orbit space of a regular covering:

[ B\cong E/\operatorname{Deck}(E/B). ]

This identifies covering-space theory with the study of suitably discontinuous group actions. The discreteness condition is essential because unrestricted quotient maps need not have evenly covered neighborhoods.

Relation to monodromy

A covering over a connected base is determined by the action of the fundamental group on one fiber. Connected coverings correspond to transitive actions, since any two points of the total space can be joined exactly when monodromy carries their fiber representatives within a single orbit. Under this description, choosing a point in the fiber identifies the stabilizer of that point with the subgroup associated with the based covering.

This categorical form extends the subgroup classification without requiring a preferred point in every component. Coverings of (B) correspond to set-valued functors on the fundamental groupoid of (B), provided the standard local hypotheses hold. Disjoint unions of connected coverings correspond to actions with multiple orbits, while regular connected coverings correspond to transitive actions whose point stabilizers are normal.