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==== 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>.
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