Covering space

In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms. If \(p : \tilde X \to X\) is a covering, \((\tilde X, p)\) is said to be a covering space or cover of \(X\), and \(X\) is said to be the base of the covering, or simply the base. By abuse of terminology, \(\tilde X\) and \(p\) may sometimes be called covering spaces as well. Since coverings are local homeomorphisms, a covering space is a special kind of étalé space.

Covering spaces first arose in the context of complex analysis (specifically, the technique of analytic continuation), where they were introduced by Riemann as domains on which naturally multivalued complex functions become single-valued. These spaces are now called Riemann surfaces.

Covering spaces are an important tool in several areas of mathematics. In modern geometry, covering spaces (or branched coverings, which have slightly weaker conditions) are used in the construction of manifolds, orbifolds, and the morphisms between them. In algebraic topology, covering spaces are closely related to the fundamental group: for one, since all coverings have the homotopy lifting property, covering spaces are an important tool in the calculation of homotopy groups. A standard example in this vein is the calculation of the fundamental group of the circle by means of the covering of \(S^1\) by \(\mathbb{R}\) (see below). Under certain conditions, covering spaces also exhibit a Galois correspondence with the subgroups of the fundamental group.

Let \(X\) be a topological space. A covering of \(X\) is a continuous map

\(\pi : \tilde X \rightarrow X\)

such that for every \(x \in X\) there exists an open neighborhood \(U_x\) of \(x\) and a discrete space \(D_x\) such that \(\pi^{-1}(U_x)\) is the disjoint union \(\displaystyle \bigsqcup_{d \in D_x} V_d\) and \(\pi|_{V_d}:V_d \rightarrow U_x\) is a homeomorphism for every \(d \in D_x\). The open sets \(V_{d}\) are called sheets, which are uniquely determined up to homeomorphism if \(U_x\) is connected. For each \(x \in X\) the discrete set \(\pi^{-1}(x)\) is called the fiber of \(x\). If \(X\) is connected (and \(\tilde X\) is non-empty), it can be shown that \(\pi\) is surjective, and the cardinality of \(D_x\) is the same for all \(x \in X\); this value is called the degree of the covering. If \(\tilde X\) is path-connected, then the covering \(\pi : \tilde X \rightarrow X\) is called a path-connected covering. This definition is equivalent to the statement that \(\pi\) is a locally trivial fiber bundle.

Some authors also require that \(\pi\) be surjective in the case that \(X\) is not connected.

• For every topological space \(X\), the identity map \(\operatorname{id}:X \rightarrow X\) is a covering. Likewise for any discrete space \(D\) the projection \(\pi:X \times D \rightarrow X\) taking \((x, i) \mapsto x\) is a covering. Coverings of this type are called trivial coverings; if \(D\) has finitely many (say \(k\)) elements, the covering is called the trivial \(k\)-sheeted covering of \(X\).

• The map \(r : \mathbb{R} \to S^1\) with \(r(t)=(\cos(2 \pi t), \sin(2 \pi t))\) is a covering of the unit circle \(S^1\). The base of the covering is \(S^1\) and the covering space is \(\mathbb{R}\). For any point \(x = (x_1, x_2) \in S^1\) such that \(x_1 > 0\), the set \(U := \{(x_1, x_2) \in S^1 \mid x_1 > 0 \}\) is an open neighborhood of \(x\). The preimage of \(U\) under \(r\) is

  \(r^{-1}(U)=\displaystyle\bigsqcup_{n \in \mathbb{Z}} \left( n - \frac 1 4, n + \frac 1 4\right)\)

and the sheets of the covering are \(V_n = (n - 1/4, n+1/4)\) for \(n \in \mathbb{Z}.\) The fiber of \(x\) is

\(r^{-1}(x) = \{t \in \mathbb{R} \mid (\cos(2 \pi t), \sin(2 \pi t)) = x\}.\)

• Another covering of the unit circle is the map \(q : S^1 \to S^1\) with \(q(z)=z^{n}\) for some positive \(n \in \mathbb{N}.\) For an open neighborhood \(U\) of an \(x \in S^1\), one has:

\(q^{-1}(U)=\displaystyle\bigsqcup_{i=1}^{n} U\).

• A map which is a local homeomorphism but not a covering of the unit circle is \(p : \mathbb{R_{+}} \to S^1\) with \(p(t)=(\cos(2 \pi t), \sin(2 \pi t))\). There is a sheet of an open neighborhood of \((1,0)\), which is not mapped homeomorphically onto \(U\).

• Let \(n \geq 1\) be odd. The map \(p : \mathrm{O}(n) \to \mathrm{SO}(n)\) defined by \(p(Q)=(\det Q) Q\) is a homomorphic double covering.

Since a covering \(\pi:E \rightarrow X\) maps each of the disjoint open sets of \(\pi^{-1}(U)\) homeomorphically onto \(U\) it is a local homeomorphism, i.e. \(\pi\) is a continuous map and for every \(e \in E\) there exists an open neighborhood \(V \subset E\) of \(e\), such that \(\pi|_V : V \rightarrow \pi(V)\) is a homeomorphism.

It follows that the covering space \(E\) and the base space \(X\) locally share the same properties.

• If \(X\) is a connected and non-orientable manifold, then there is a covering \(\pi:\tilde X \rightarrow X\) of degree \(2\), whereby \(\tilde X\) is a connected and orientable manifold.
• If \(X\) is a connected Lie group, then there is a covering \(\pi:\tilde X \rightarrow X\) which is also a Lie group homomorphism and \(\tilde X := \{\gamma:\gamma \text{ is a path in X with }\gamma(0)= \boldsymbol{1_X} \text{ modulo homotopy with fixed ends}\}\) is a Lie group.
• If \(X\) is a graph, then it follows for a covering \(\pi:E \rightarrow X\) that \(E\) is also a graph.
• If \(X\) is a connected manifold, then there is a covering \(\pi:\tilde X \rightarrow X\), whereby \(\tilde X\) is a connected and simply connected manifold.
• If \(X\) is a connected Riemann surface, then there is a covering \(\pi:\tilde X \rightarrow X\) which is also a holomorphic map and \(\tilde X\) is a connected and simply connected Riemann surface.

Let \(X, Y\) and \(E\) be path-connected, locally path-connected spaces, and \(p,q\) and \(r\) be continuous maps, such that the diagram

commutes.

• If \(p\) and \(q\) are coverings, so is \(r\).
• If \(p\) and \(r\) are coverings, so is \(q\).

Let \(X\) and \(X'\) be topological spaces and \(p:E \rightarrow X\) and \(p':E' \rightarrow X'\) be coverings, then \(p \times p':E \times E' \rightarrow X \times X'\) with \((p \times p')(e, e') = (p(e), p'(e'))\) is a covering. However, coverings of \(X\times X'\) are not all of this form in general.

Let \(X\) be a topological space and \(p:E \rightarrow X\) and \(p':E' \rightarrow X\) be coverings. Both coverings are called equivalent, if there exists a homeomorphism \(h:E \rightarrow E'\), such that the diagram

commutes. If such a homeomorphism exists, then one calls the covering spaces \(E\) and \(E'\) isomorphic.

All coverings satisfy the lifting property, i.e.:

Let \(I\) be the unit interval and \(p:E \rightarrow X\) be a covering. Let \(F:Y \times I \rightarrow X\) be a continuous map and \(\tilde F_0:Y \times \{0\} \rightarrow E\) be a lift of \(F|_{Y \times \{0\}}\), i.e. a continuous map such that \(p \circ \tilde F_0 = F|_{Y \times \{0\}}\). Then there is a uniquely determined, continuous map \(\tilde F:Y \times I \rightarrow E\) for which \(\tilde F(y,0) = \tilde F_0\) and which is a lift of \(F\), i.e. \(p \circ \tilde F = F\).

If \(X\) is a path-connected space, then for \(Y=\{0\}\) it follows that the map \(\tilde F\) is a lift of a path in \(X\) and for \(Y=I\) it is a lift of a homotopy of paths in \(X\).

As a consequence, one can show that the fundamental group \(\pi_{1}(S^1)\) of the unit circle is an infinite cyclic group, which is generated by the homotopy classes of the loop \(\gamma: I \rightarrow S^1\) with \(\gamma (t) = (\cos(2 \pi t), \sin(2 \pi t))\).

Let \(X\) be a path-connected space and \(p:E \rightarrow X\) be a connected covering. Let \(x,y \in X\) be any two points, which are connected by a path \(\gamma\), i.e. \(\gamma(0)= x\) and \(\gamma(1)= y\). Let \(\tilde \gamma\) be the unique lift of \(\gamma\), then the map

\(L_{\gamma}:p^{-1}(x) \rightarrow p^{-1}(y)\) with \(L_{\gamma}(\tilde \gamma (0))=\tilde \gamma (1)\)

is bijective.

If \(X\) is a path-connected space and \(p: E \rightarrow X\) a connected covering, then the induced group homomorphism

\(p_{\#}: \pi_{1}(E) \rightarrow \pi_{1}(X)\) with \(p_{\#}([\gamma])=[p \circ \gamma]\),

Condensed: the full section is in Wikipedia.

Let \(p: \tilde X \rightarrow X\) be a simply connected covering. If \(\beta : E \rightarrow X\) is another simply connected covering, then there exists a uniquely determined homeomorphism \(\alpha : \tilde X \rightarrow E\), such that the diagram

commutes.

This means that \(p\) is, up to equivalence, uniquely determined and because of that universal property denoted as the universal covering of the space \(X\).

A universal covering does not always exist. The following theorem guarantees its existence for a certain class of base spaces.

Let \(X\) be a connected, locally simply connected topological space. Then, there exists a universal covering \(p:\tilde X \rightarrow X.\)

The set \(\tilde X\) is defined as \(\tilde X = \{\gamma:\gamma \text{ is a path in }X \text{ with }\gamma(0) = x_0 \}/\text{homotopy with fixed ends},\) where \(x_0 \in X\) is any chosen base point. The map \(p:\tilde X \rightarrow X\) is defined by \(p([\gamma])=\gamma(1).\)

The topology on \(\tilde X\) is constructed as follows: Let \(\gamma:I \rightarrow X\) be a path with \(\gamma(0)=x_0.\) Let \(U\) be a simply connected neighborhood of the endpoint \(x=\gamma(1).\) Then, for every \(y \in U,\) there is a path \(\sigma_y\) inside \(U\) from \(x\) to \(y\) that is unique up to homotopy. Now consider the set \(\tilde U=\{\gamma\sigma_y:y \in U \}/\text{homotopy with fixed ends}.\) The restriction \(p|_{\tilde U}: \tilde U \rightarrow U\) with \(p([\gamma\sigma_y])=\gamma\sigma_y(1)=y\) is a bijection and \(\tilde U\) can be equipped with the final topology of \(p|_{\tilde U}.\)

The fundamental group \(\pi_{1}(X,x_0) = \Gamma\) acts freely on \(\tilde X\) by \(([\gamma],[\tilde x]) \mapsto [\gamma\tilde x],\) and the orbit space \(\Gamma \backslash \tilde X\) is homeomorphic to \(X\) through the map \([\Gamma \tilde x]\mapsto\tilde x(1).\)

• \(p : \mathbb{R} \to S^1\) with \(p(t)=(\cos(2 \pi t), \sin(2 \pi t))\) is the universal covering of the unit circle \(S^1\).
• \(p : S^n \to \mathbb{R}P^n \cong \{+1,-1\}\backslash S^n\) with \(p(x)=[x]\) is the universal covering of the projective space \(\mathbb{R}P^n\) for \(n>1\).

• \(p : \mathrm{SU}(n) \times \mathbb{R} \to U(n)\) with \[p(A,t)= \exp(2 \pi i t) A\] is the universal covering of the unitary group \(U(n)\).

• Since \(\mathrm{SU}(2) \cong S^3\), it follows that the quotient map \[p : \mathrm{SU}(2) \rightarrow \mathrm{SU}(2) / \mathbb{Z_2} \cong \mathrm{SO}(3)\] is the universal covering of \(\mathrm{SO}(3)\).

• A topological space which has no universal covering is the Hawaiian earring: \[X = \bigcup_{n\in \N}\left\{(x_1,x_2)\in\R^{2} : \Bigl(x_1-\frac{1}{n}\Bigr)^2+x_2^2=\frac{1}{n^2}\right\}\] One can show that no neighborhood of the origin \((0,0)\) is simply connected.

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 H_g of X onto itself, in such a way that H_{g h} is always equal to H_g \(\circ\) H_h 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.

Let E and M be smooth manifolds with or without boundary. A covering \(\pi : E \to M\) is called a smooth covering if it is a smooth map and the sheets are mapped diffeomorphically onto the corresponding open subset of M. (This is in contrast to the definition of a covering, which merely requires that the sheets are mapped homeomorphically onto the corresponding open subset.)

Let \(p:E \rightarrow X\) be a covering. A deck transformation is a homeomorphism \(d:E \rightarrow E\), such that the diagram of continuous maps

commutes. Together with the composition of maps, the set of deck transformation forms a group \(\operatorname{Deck}(p)\), which is the same as \(\operatorname{Aut}(p)\).

Now suppose \(p:C \to X\) is a covering map and \(C\) (and therefore also \(X\)) is connected and locally path connected. The action of \(\operatorname{Aut}(p)\) 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 \(G = \operatorname{Aut}(p)\) is considered as a discrete topological group.

Every universal cover \(p:D \to X\) is regular, with deck transformation group being isomorphic to the fundamental group \(\pi_1(X)\).