User:Moritz Metzger/covering

A covering of a topological space $$X$$ is a continuous map $$\pi : E \rightarrow X$$ with special properties.

Definition
Let $$X$$ be a topological space. A covering of $$X$$ is a continuous map


 * $$ \pi : E \rightarrow X$$

s.t. there exists a discrete space $$D$$ and for every $$x \in X$$ an open neighborhood $$U \subset X$$, s.t. $$\pi^{-1}(U)= \displaystyle \bigsqcup_{d \in D} V_d $$ and $$\pi|_{V_d}:V_d \rightarrow U $$ is a homeomorphism for every $$d \in D $$. Often, the notion of a covering is used for the covering space $$E$$ as well as for the map $$ \pi : E \rightarrow X$$. The open sets $$V_{d}$$ are called sheets, which are uniquely determined if $$U$$ is connected. $$\,^{p.56}$$ For a $$x \in X$$ the discrete subset $$\pi^{-1}(x)$$ is called the fiber of $$x$$. The degree of a covering is the cardinality of the space $$D$$. If $$E$$ is path-connected, then the covering $$ \pi : E \rightarrow X$$ is denoted as a path-connected covering.

Examples

 * For every topological space $$X$$ there exists the covering $$\pi:X \rightarrow X$$ with $$\pi(x)=x$$, which is denoted as the trivial covering of $$X.$$
 * The map $$r \colon \mathbb{R} \to S^1$$ with $$r(t)=(\cos(2 \pi t), \sin(2 \pi t))$$ is a covering of the unit circle $$S^1$$. For an open neighborhood $$U \subset X$$ of an $$x \in S^1$$, which has positiv $$\cos(2 \pi t)$$-value, one has: $$r^{-1}(U)=\displaystyle\bigsqcup_{n \in \mathbb{Z}} ( n - \frac 1 4, n + \frac 1 4)$$.
 * Another covering of the unit circle is the map $$q \colon S^1 \to S^1$$ with $$q(z)=z^{n}$$ for some $$n \in \mathbb{N}$$. For an open neighborhood $$U \subset X$$ 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 $$p \colon \mathbb{R_{+}} \to S^1$$ 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 $$U$$.

Local homeomorphism
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 $$e$$, s.t. $$\pi|_V : V \rightarrow \pi(V)$$ 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. $$\,^{p.234}$$
 * 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. $$^{p.174}$$
 * If $$X$$ is a graph, then it follows for a covering $$\pi:E \rightarrow X$$ that $$E$$ is also a graph. $$^{p.85}$$
 * 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. $$^{p.32}$$
 * If $$X$$ is a connected Riemann surface, then there is a covering $$\pi:\tilde X \rightarrow X$$ which is also a holomorphic map $$^{p.22}$$ and $$\tilde X$$ is a connected and simply connected Riemann surface. $$^{p.32}$$

Factorisation
Let $$p,q$$ and $$r$$ be continuous maps, s.t. the diagram commutes.


 * If $$p$$ and $$q$$ are coverings, so is $$r$$. $$^{p.485}$$
 * If $$p$$ and $$r$$ are coverings, so is $$q$$. $$^{p.485}$$

Product of coverings
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. $$^{p.339}$$

Equivalence of coverings
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 $$h:E \rightarrow E'$$, s.t. the diagram commutes. If such a homeomorphism exists, then one calls the covering spaces $$E$$ and $$E'$$ isomorphic.

Lifting property
An important property of the covering is, that it satisfies 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 $$\tilde F:Y \times I \rightarrow E$$, which is a lift of $$F$$, i.e. $$p \circ \tilde F = F$$. $$^{p.60}$$

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 $$X$$.

Because of that property 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 $$\gamma (t) = (\cos(2 \pi t), \sin(2 \pi t))$$. $$^{p.29}$$

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. $$^{p.69 }$$

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 $$ p_{\#}([\gamma])=[p \circ \gamma]$$,

is injective and the subgroup $$p_{\#}(\pi_1(E))$$ of $$\pi_1(X)$$ consists of the homotopy classes of loops in $$X$$, whose lifts are loops in $$E$$. $$^{p.61}$$

Holomorphic maps between Riemann surfaces
Let $$X$$ and $$Y$$ be Riemann surfaces, i.e. one dimensional complex manifolds, and let $$f: X \rightarrow Y$$ be a continuous map. $$f$$ is holomorphic in a point $$x \in X$$, if for any charts $$\phi _x:U_1 \rightarrow V_1$$ of $$x$$ and $$\phi_{f(x)}:U_2 \rightarrow V_2$$ of $$f(x)$$, with $$\phi_x(U_1) \subset U_2$$, the map $$\phi _{f(x)} \circ f \circ \phi^{-1} _x: \mathbb{C} \rightarrow \mathbb{C}$$ is holomorphic.

If $$f$$ is for all $$x \in X$$ holomorphic, we say $$f$$ is holomorphic.

The map $$F =\phi _{f(x)} \circ f \circ \phi^{-1} _x$$ is called the local expression of $$f$$ in $$x \in X$$.

If $$f: X \rightarrow Y$$ is a non-constant, holomorphic map between compact Riemann surfaces, then $$f$$ is surjective $$^{p.11}$$ and an open map $$^{p.11}$$, i.e. for every open set $$U \subset X$$ the image $$f(U) \subset Y$$ is also open.

Ramification point and branch point
Let $$f: X \rightarrow Y$$ be a non-constant, holomorphic map between compact Riemann surfaces. For every $$x \in X$$ there exist charts for $$x$$ and $$f(x)$$ and there exists a uniquely determined $$k_x \in \mathbb{N_{>0}}$$, s.t. the local expression $$F$$ of $$f$$ in $$x$$ is of the form $$z \mapsto z^{k_{x}}$$. $$^{p.10 }$$ The number $$k_x$$ is called the ramification index of $$f$$ in $$x$$ and the point $$x \in X$$ is called a ramification point if $$k_x \geq 2$$. If $$k_x =1$$ for an $$x \in X$$, then $$x$$ is unramified. The image point $$y=f(x) \in Y$$ of a ramification point is called a branch point.

Degree of a holomorphic map
Let $$f: X \rightarrow Y$$ be a non-constant, holomorphic map between compact Riemann surfaces. The degree $$deg(f)$$ of $$f$$ is the cardinality of the fiber of an unramified point $$y=f(x) \in Y$$, i.e. $$deg(f):=|f^{-1}(y)|$$.

This number is well-defined, since for every $$y \in Y$$ the fiber $$f^{-1}(y)$$ is discrete $$^{p.20}$$ and for any two unramified points $$y_1,y_2 \in Y$$, it is: $$|f^{-1}(y_1)|=|f^{-1}(y_2)|.$$ $$^{p.29}$$

It can be calculated by:


 * $$\sum_{x \in f^{-1}(y)} k_x = deg(f)$$ $$^{p.29}$$

Definition
A continuous map $$f: X \rightarrow Y$$ is called a branched covering, if there exists a closed set with dense complement $$E \subset Y$$, s.t. $$f_{|X \smallsetminus f^{-1}(E)}:X \smallsetminus f^{-1}(E) \rightarrow Y \smallsetminus E$$ is a covering.

Examples

 * 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 $$n$$, whereby $$z=0$$ 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
Let $$p: \tilde X \rightarrow X$$ be a simply connected covering and $$\beta : E \rightarrow X$$ be a covering, then there exists a uniquely determined covering $$\alpha : \tilde X \rightarrow E$$, s.t. the diagram

commutes. $$^{p.486 }$$

Definition
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 $$\alpha : \tilde X \rightarrow E$$, s.t. the diagram commutes. $$^{p.482}$$

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

Existence
A universal covering does not always exists, but the following properties guarantee the existence:

Let $$X$$ be a connected, locally simply connected, then there exists a universal covering $$p:\tilde X \rightarrow X$$.

$$\tilde X$$ is defined as $$\tilde X := \{\gamma:\gamma \text{ is a path in }X \text{ with }\gamma(0) = x_0 \}/\text{ homotopy with fixed ends} $$ and $$p:\tilde X \rightarrow X$$ by $$p([\gamma]):=\gamma(1)$$. $$^{p.64}$$

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 $$y \in U$$ the paths $$\sigma_y$$ inside $$U $$ from $$x$$ to $$y$$ are uniquely determined up to homotopy. Now consider $$\tilde U:=\{\gamma.\sigma_y:y \in U \}/\text{ homotopy with fixed ends}$$, then $$p_{|\tilde U}: \tilde U \rightarrow U$$ 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 through $$([\gamma],[\tilde x]) \mapsto [\gamma.\tilde x]$$ on $$\tilde X$$ and $$\psi:\Gamma \backslash \tilde X \rightarrow X$$ with $$\psi([\Gamma \tilde x])=\tilde x(1)$$ is a homeomorphism, i.e. $$\Gamma \backslash \tilde X \cong X $$.

Examples
\exp(2 \pi i t) & 0\\ 0 & I_{n-1} \end{bmatrix} A$$ is the universal covering of the unitary group $$U(n)$$.
 * $$r \colon \mathbb{R} \to S^1$$ with $$r(t)=(\cos(2 \pi t), \sin(2 \pi t))$$ is the universal covering of the unit circle $$S^1$$.
 * $$p \colon 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$$.
 * $$q \colon SU(n) \ltimes \mathbb{R} \to U(n)$$ with $$q(A,t)= \begin{bmatrix}
 * Since $$SU(2) \cong S^3$$, it follows that the quotient map $$f:SU(2) \rightarrow \mathbb{Z_2} \backslash SU(2) \cong SO(3)$$ is the universal covering of the $$SO(3)$$.Hawaiian_Earrings.svg
 * 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. $$^{p.487 \, Example \, 1 }$$

Definition
Let $$p:E \rightarrow X$$ be a covering. A deck transformation is a homeomorphism $$d:E \rightarrow E$$, s.t. the diagram of continuous maps commutes. Together with the composition of maps, the set of deck transformation forms a group $$Deck(p)$$, which is the same as $$Aut(p)$$.

Examples

 * Let $$q \colon S^1 \to S^1$$ be the covering $$q(z)=z^{n}$$ for some $$n \in \mathbb{N} $$, then the map $$d:S^1 \rightarrow S^1 : z \mapsto z \, e^{2\pi i/n} $$ is a deck transformation and $$Deck(q)\cong \mathbb{Z}/ \mathbb{nZ}$$.
 * Let $$r \colon \mathbb{R} \to S^1$$ be the covering $$r(t)=(\cos(2 \pi t), \sin(2 \pi t))$$, then the map $$d_k:\mathbb{R} \rightarrow \mathbb{R} : t \mapsto t + k$$ with $$k \in \mathbb{Z}$$ is a deck transformation and $$Deck(r)\cong \mathbb{Z}$$.

Properties
Let $$X$$ be a path-connected space and $$p:E \rightarrow X$$ be a connected covering. Since a deck transformation $$d:E \rightarrow E$$ is bijective, it permutes the elements of a fiber $$p^{-1}(x)$$ with $$x \in X$$ and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber. $$^{p.70 }$$Because of this property every deck transformation defines a group action on $$E$$, i.e. let $$U \subset X$$ be an open neighborhood of a $$x \in X$$ and $$\tilde U \subset E$$ an open neighborhood of an $$e \in p^{-1}(x)$$, then $$Deck(p) \times E \rightarrow E: (d,\tilde U)\mapsto d(\tilde U)$$ is a group action.

Definition
A covering $$p:E \rightarrow X$$ is called normal, if $$Deck(p) \backslash X \cong E$$. This means, that for every $$x \in X$$ and any two $$e_0,e_1 \in p^{-1}(x)$$ there exists a deck transformation $$d:E \rightarrow E$$, s.t. $$d(e_0)=e_1$$.

Properties
Let $$X$$ be a path-connected space and $$p:E \rightarrow X$$ be a connected covering. Let $$H=p_{\#}(\pi_1(E))$$ be a subgroup of $$\pi_1(X)$$, then $$p$$ is a normal covering iff $$H$$ is a normal subgroup of $$\pi_1(X)$$. $$^{p.71}$$

If $$p:E \rightarrow X$$ is a normal covering and $$H=p_{\#}(\pi_1(E))$$, then $$Deck(p) \cong \pi_1(X)/H$$. $$^{p.71}$$

If $$p:E \rightarrow X$$ is a path-connected covering and $$H=p_{\#}(\pi_1(E))$$, then $$Deck(p) \cong N(H)/H$$, whereby $$N(H)$$ is the normaliser of $$H$$. $$^{p.71}$$

Let $$E$$ be a topological space. A group $$\Gamma$$ acts discontinuously on $$E$$, if every $$e \in E$$ has an open neighborhood $$V \subset E$$ with $$V \neq \empty$$, such that for every $$\gamma \in \Gamma$$ with $$\gamma V \cap V \neq \empty$$ one has $$d_1 = d_2$$.

If a group $$\Gamma$$ acts discontinuously on a topological space $$E$$, then the quotient map $$q: E \rightarrow \Gamma \backslash E $$ with $$q(e)=\Gamma e$$ is a normal covering. $$^{p.72}$$ Hereby $$\Gamma \backslash E = \{\Gamma e: e \in E\}$$ is the quotient space and $$\Gamma e = \{\gamma(e):\gamma \in \Gamma\}$$ is the orbit of the group action.

Examples

 * The covering $$q \colon S^1 \to S^1 $$ with $$q(z)=z^{n}$$ is a normal coverings for every $$n \in \mathbb{N}$$.
 * Every simply connected covering is a normal covering.

Calculation
Let $$\Gamma$$ be a group, which acts discontinuously on a topological space $$E$$ and let $$q: E \rightarrow \Gamma \backslash E $$ be the normal covering.


 * If $$E$$ is path-connected, then $$Deck(q) \cong \Gamma$$. $$^{p.72}$$


 * If $$E$$ is simply connected, then $$Deck(q)\cong \pi_1(X)$$. $$^{p.71}$$

Examples

 * Let $$n \in \mathbb{N}$$. The antipodal map $$g:S^n \rightarrow S^n$$ with $$g(x)=-x$$ generates, together with the composition of maps, a group $$D(g) \cong \mathbb{Z/2Z}$$ and induces a group action $$D(g) \times S^n \rightarrow S^n, (g,x)\mapsto g(x)$$, which acts discontinuously on $$S^n$$. Because of $$\mathbb{Z_2} \backslash S^n \cong \mathbb{R}P^n$$ it follows, that the quotient map $$q \colon S^n \rightarrow \mathbb{Z_2}\backslash S^n \cong \mathbb{R}P^n$$ is a normal covering and for $$n > 1$$ a universal covering, hence $$Deck(q)\cong \mathbb{Z/2Z}\cong \pi_1({\mathbb{R}P^n})$$ for $$n > 1$$.
 * Let $$SO(3)$$ be the special orthogonal group, then the map $$f:SU(2) \rightarrow SO(3) \cong \mathbb{Z_2} \backslash SU(2)$$ is a normal covering and because of $$SU(2) \cong S^3$$, it is the universal covering, hence $$Deck(f) \cong \mathbb{Z/2Z} \cong \pi_1(SO(3))$$.
 * With the group action $$(z_1,z_2)*(x,y)=(z_1+(-1)^{z_2}x,z_2+y)$$ of $$\mathbb{Z^2}$$ on $$\mathbb{R^2}$$, whereby $$(\mathbb{Z^2},*)$$ is the semidirect product $$\mathbb{Z} \rtimes \mathbb{Z} $$, one gets the universal covering $$f: \mathbb{R^2} \rightarrow (\mathbb{Z} \rtimes \mathbb{Z}) \backslash \mathbb{R^2} \cong K $$ of the klein bottle $$K$$, hence $$Deck(f) \cong \mathbb{Z} \rtimes \mathbb{Z} \cong \pi_1(K)$$.
 * Let $$T = S^1 \times S^1$$ be the torus which is embedded in the $$\mathbb{C^2}$$. Then one gets a homeomorphism $$\alpha: T \rightarrow T: (e^{ix},e^{iy}) \mapsto (e^{i(x+\pi)},e^{-iy})$$, which induces a discontinuous group action $$G_{\alpha} \times T \rightarrow T$$, whereby $$G_{\alpha} \cong \mathbb{Z/2Z}$$. It follows, that the map $$f: T \rightarrow G_{\alpha} \backslash T \cong K$$ is a normal covering of the klein bottle, hence $$Deck(f) \cong \mathbb{Z/2Z}$$.
 * Let $$S^3$$ be embedded in the $$\mathbb{C^2}$$. Since the group action $$S^3 \times \mathbb{Z/pZ} \rightarrow S^3: ((z_1,z_2),[k]) \mapsto (e^{2 \pi i k/p}z_1,e^{2 \pi i k q/p}z_2)$$ is discontinuously, whereby $$p,q \in \mathbb{N}$$ are coprime, the map $$f:S^3 \rightarrow \mathbb{Z_p} \backslash S^3 =: L_{p,q}$$ is the universal covering of the lens space $$L_{p,q}$$, hence $$Deck(f) \cong \mathbb{Z/pZ} \cong \pi_1(L_{p,q})$$.

Galois correspondence
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 $$\alpha_{\#}(\pi_1(X_H))=H$$. $$^{p.66}$$

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. $$^{p.482}$$

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

$$ \begin{matrix} \qquad \displaystyle \{\text{Subgroup of }\pi_1(X)\} & \longleftrightarrow & \displaystyle \{\text{path-connected covering } p:E \rightarrow X\} \\ H & \longrightarrow & \alpha:X_H \rightarrow X \\ p_\#(\pi_1(E))&\longleftarrow & p \\ \displaystyle \{\text{normal subgroup of }\pi_1(X)\} & \longleftrightarrow & \displaystyle \{\text{normal covering } p:E \rightarrow X\} \\ H & \longrightarrow & \alpha:X_H \rightarrow X \\ p_\#(\pi_1(E))&\longleftarrow & p  \end{matrix} $$

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 $$d$$.

Category of coverings
Let $$X$$ be a topological space. The objects of the category $$\boldsymbol{Cov(X)}$$ are the coverings $$p:E \rightarrow X$$ of $$X$$ and the morphisms between two coverings $$p:E \rightarrow X$$ and $$q:F\rightarrow X$$ are continuous maps $$f:E \rightarrow F$$, s.t. the diagram commutes.

G-Set
Let $$G$$ be a topological group. The category $$\boldsymbol{G-Set}$$ is the category of sets which are G-sets. The morphisms are G-maps $$\phi:X \rightarrow Y$$ between G-sets. They satisfy the condition $$\phi(gx)=g \, \phi(x)$$ for every $$g \in G$$.

Equivalence
Let $$X$$ be a connected and locally simply connected space, $$x \in X$$ and $$G = \pi_1(X,x)$$ be the fundamental group of $$X$$. Since $$G$$ defines, by lifting of paths and evaluating at the endpoint of the lift, a group action on the fiber of a covering, the functor $$F:\boldsymbol{Cov(X)} \longrightarrow \boldsymbol{G-Set}: p \mapsto p^{-1}(x)$$ is an equivalence of categories. $$^{p.68-70 }$$

Literatur

 * Allen Hatcher: Algebraic Topology. Cambridge Univ. Press, Cambridge, ISBN 0-521-79160-X
 * Otto Forster: Lectures on Riemann surfaces. Springer Berlin, München 1991, ISBN 978-3-540-90617-9
 * James Munkres: Topology. Upper Saddle River, NJ: Prentice Hall, Inc., ©2000, ISBN 978-0-13-468951-7
 * Wolfgang Kühnel: Matrizen und Lie-Gruppen. Springer Fachmedien Wiesbaden GmbH, Stuttgart, ISBN 978-3-8348-9905-7
 * Maximiliano Aguilar and Miguel Socolovsky: The Universal Covering Group of U(n) and Projective Representations. Hrsg.: International Journal of Theoretical Physics. Dezember 1999