Editing Covering space (section)

== Deck transformation ==
=== Definition ===
Let <math>p:E \rightarrow X</math> be a covering. A '''deck transformation''' is a homeomorphism <math>d:E \rightarrow E</math>, such that the diagram of continuous maps
[[File:Diagramm_Decktrafo.png|center|frameless]]
commutes. Together with the composition of maps, the set of deck transformation forms a [[Group (mathematics)|group]] <math>\operatorname{Deck}(p)</math>, which is the same as <math>\operatorname{Aut}(p)</math>.

Now suppose <math>p:C \to X</math> is a covering map and <math>C</math> (and therefore also <math>X</math>) is connected and locally path connected. The action of <math>\operatorname{Aut}(p)</math> on each fiber is [[Group action (mathematics)#Notable properties of actions|free]]. If this action is [[Group action (mathematics)#Remarkable properties of actions|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 bundle|principal {{nowrap|<math>G</math>-bundle}}]], where <math>G = \operatorname{Aut}(p)</math> is considered as a discrete topological group.

Every universal cover <math>p:D \to X </math> is regular, with deck transformation group being isomorphic to the [[fundamental group]] {{nowrap|<math>\pi_1(X)</math>.}}

=== Examples ===
* Let <math>q : S^1 \to S^1</math> be the covering <math>q(z)=z^{n}</math> for some <math>n \in \mathbb{N} </math>, then the map <math>d_k:S^1 \rightarrow S^1 : z \mapsto z \, e^{2\pi ik/n} </math> for <math>k \in \mathbb{Z}</math> is a deck transformation and <math>\operatorname{Deck}(q)\cong \mathbb{Z}/ n\mathbb{Z}</math>.
* Let <math>r : \mathbb{R} \to S^1</math> be the covering <math>r(t)=(\cos(2 \pi t), \sin(2 \pi t))</math>, then the map <math>d_k:\mathbb{R} \rightarrow \mathbb{R} : t \mapsto t + k</math> for <math>k \in \mathbb{Z}</math> is a deck transformation and <math>\operatorname{Deck}(r)\cong \mathbb{Z}</math>.
* As another important example, consider <math>\Complex</math> the complex plane and <math>\Complex^{\times}</math> the complex plane minus the origin.  Then the map <math>p: \Complex^{\times} \to \Complex^{\times}</math> with <math> p(z) = z^{n} </math> is a regular cover. The deck transformations are multiplications with <math>n</math>-th [[root of unity|roots of unity]] and the deck transformation group is therefore isomorphic to the [[cyclic group]] <math>\Z/n\Z</math>. Likewise, the map <math>\exp : \Complex \to \Complex^{\times}</math> with <math>\exp(z) = e^{z}</math> is the universal cover.

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

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

==== Properties ====
Let <math>X</math> be a path-connected space and <math>p:E \rightarrow X</math> be a connected covering. Let <math>H=p_{\#}(\pi_1(E))</math> be a [[subgroup]] of <math>\pi_1(X)</math>, then <math>p</math> is a normal covering iff <math>H</math> is a [[normal subgroup]] of <math>\pi_1(X)</math>.

If <math>p:E \rightarrow X</math> is a normal covering and <math>H=p_{\#}(\pi_1(E))</math>, then <math>\operatorname{Deck}(p) \cong \pi_1(X)/H</math>.

If <math>p:E \rightarrow X</math> is a path-connected covering and <math>H=p_{\#}(\pi_1(E))</math>, then <math>\operatorname{Deck}(p) \cong N(H)/H</math>, whereby <math>N(H)</math> is the [[normaliser]] of <math>H</math>.{{r|Hatcher|p=71}}

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

If a group <math>\Gamma</math> acts discontinuously on a topological space <math>E</math>, then the [[quotient map (topology)|quotient map]] <math>q: E \rightarrow \Gamma \backslash E </math> with <math>q(e)=\Gamma e</math> is a normal covering.{{r|Hatcher|p=72}} Hereby <math>\Gamma \backslash E = \{\Gamma e: e \in E\}</math> is the [[Quotient space (topology)|quotient space]] and <math>\Gamma e = \{\gamma(e):\gamma \in \Gamma\}</math> is the [[Orbit (group theory)|orbit]] of the group action.

==== Examples ====
* The covering <math>q : S^1 \to S^1 </math> with <math>q(z)=z^{n}</math> is a normal coverings for every <math>n \in \mathbb{N}</math>.
* Every simply connected covering is a normal covering.

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

* If <math>E</math> is path-connected, then <math>\operatorname{Deck}(q) \cong \Gamma</math>.{{r|Hatcher|p=72}}

* If <math>E</math> is simply connected, then <math>\operatorname{Deck}(q)\cong \pi_1(\Gamma \backslash E)</math>.{{r|Hatcher|p=71}}

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