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 étale 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.^[1]^: 10

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).^[2]^: 29 Under certain conditions, covering spaces also exhibit a Galois correspondence with the subgroups of the fundamental group.

For every topological space $X$ , the $\operatorname {id} :X\rightarrow X$ 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 $X$ .

identity map

Properties[edit]

Local homeomorphism[edit]

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.

Let $n\in \mathbb {N}$ and $n\geq 2$ , then $f:\mathbb {C} \rightarrow \mathbb {C}$ with $f(z)=z^{n}$ is branched covering of degree $n$ , where by $z=0$ is a branch point.

Every non-constant, holomorphic map between compact Riemann surfaces $f:X\rightarrow Y$ of degree $d$ is a branched covering of degree $d$ .

Universal covering[edit]

Definition[edit]

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

G-coverings[edit]

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 ∘ 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.

Deck transformation[edit]

Definition[edit]

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

Galois correspondence[edit]

Let $X$ be a connected and locally simply connected space, then for every subgroup $H\subseteq \pi _{1}(X)$ there exists a path-connected covering $\alpha :X_{H}\rightarrow X$ with $\alpha _{\#}(\pi _{1}(X_{H}))=H$ .^[2]^: 66

Let $p_{1}:E\rightarrow X$ and $p_{2}:E'\rightarrow X$ be two path-connected coverings, then they are equivalent iff the subgroups $H=p_{1\#}(\pi _{1}(E))$ and $H'=p_{2\#}(\pi _{1}(E'))$ are conjugate to each other.^[6]^: 482

Let $X$ be a connected and locally simply connected space, then, up to equivalence between coverings, there is a bijection:

${\begin{matrix}\qquad \displaystyle \{{\text{Subgroup of }}\pi _{1}(X)\}&\longleftrightarrow &\displaystyle \{{\text{path-connected covering }}p:E\rightarrow X\}\\H&\longrightarrow &\alpha :X_{H}\rightarrow X\\p_{\#}(\pi _{1}(E))&\longleftarrow &p\\\displaystyle \{{\text{normal subgroup of }}\pi _{1}(X)\}&\longleftrightarrow &\displaystyle \{{\text{normal covering }}p:E\rightarrow X\}\end{matrix}}$

For a sequence of subgroups $\displaystyle \{{\text{e}}\}\subset H\subset G\subset \pi _{1}(X)$ one gets a sequence of coverings ${\tilde {X}}\longrightarrow X_{H}\cong H\backslash {\tilde {X}}\longrightarrow X_{G}\cong G\backslash {\tilde {X}}\longrightarrow X\cong \pi _{1}(X)\backslash {\tilde {X}}$ . For a subgroup $H\subset \pi _{1}(X)$ with index $\displaystyle [\pi _{1}(X):H]=d$ , the covering $\alpha :X_{H}\rightarrow X$ has degree $d$ .

is the universal cover of a Cayley graph

Bethe lattice

a covering space for an undirected graph, and its special case the bipartite double cover

Covering graph

Covering group

Galois connection

Quotient space (topology)

Hatcher, Allen (2002). Algebraic topology. Cambridge: Cambridge University Press. 0-521-79160-X. OCLC 45420394.

ISBN

Forster, Otto (1981). Lectures on Riemann surfaces. New York. 0-387-90617-7. OCLC 7596520.{{cite book}}: CS1 maint: location missing publisher (link)

ISBN

Munkres, James R. (2018). Topology. New York, NY. 978-0-13-468951-7. OCLC 964502066.{{cite book}}: CS1 maint: location missing publisher (link)

ISBN

Kühnel, Wolfgang (2011). Matrizen und Lie-Gruppen Eine geometrische Einführung (in German). Wiesbaden: Vieweg+Teubner Verlag. :10.1007/978-3-8348-9905-7. ISBN 978-3-8348-9905-7. OCLC 706962685.

Covering space

identity map

Properties[edit]

Local homeomorphism[edit]

Universal covering[edit]

Definition[edit]

G-coverings[edit]

Deck transformation[edit]

Definition[edit]

Galois correspondence[edit]

Bethe lattice

Covering graph

Covering group

Galois connection

Quotient space (topology)

ISBN

ISBN

ISBN

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