Editing Covering space (section)

== Universal covering ==

=== Definition ===
Let <math>p: \tilde X \rightarrow X</math> be a [[Simply connected space|simply connected]] covering. If <math>\beta : E \rightarrow X</math> is another simply connected covering, then there exists a uniquely determined homeomorphism <math>\alpha : \tilde X \rightarrow E</math>, such that the diagram
[[File:Universelle_Überlagerung_2.0.png|center|frameless]]
commutes.{{r|Munkres|p=482}}

This means that <math>p</math> is, up to equivalence, uniquely determined and because of that [[universal property]] denoted as the '''universal covering''' of the space <math>X</math>.

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

Let <math>X</math> be a connected, [[Locally simply connected space|locally simply connected]] topological space. Then, there exists a universal covering <math>p:\tilde X \rightarrow X.</math>

The set <math>\tilde X</math> is defined as <math>\tilde X = \{\gamma:\gamma \text{ is a path in }X \text{ with }\gamma(0) = x_0 \}/\text{homotopy with fixed ends},</math> where <math>x_0 \in X</math> is any chosen base point. The map <math>p:\tilde X \rightarrow X</math> is defined by <math>p([\gamma])=\gamma(1).</math>{{r|Hatcher|p=64}}

The [[topology]] on <math>\tilde X</math> is constructed as follows: Let <math>\gamma:I \rightarrow X</math> be a path with <math>\gamma(0)=x_0.</math> Let <math>U</math> be a simply connected neighborhood of the endpoint <math>x=\gamma(1).</math> Then, for every <math>y \in U,</math> there is a [[path (topology)|path]] <math>\sigma_y</math> inside <math>U</math> from <math>x</math> to <math>y</math> that is unique up to [[homotopy]]. Now consider the set <math>\tilde U=\{\gamma\sigma_y:y \in U \}/\text{homotopy with fixed ends}.</math> The restriction <math>p|_{\tilde U}: \tilde U \rightarrow U</math> with <math>p([\gamma\sigma_y])=\gamma\sigma_y(1)=y</math> is a bijection and <math>\tilde U</math> can be equipped with the [[final topology]] of <math>p|_{\tilde U}.</math>{{explain|date=December 2024|reason=How do these topologies on the tilde-U combine into one on tilde-X?}}

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

=== Examples ===
[[File:Hawaiian_Earrings.svg|right|thumb|250x250px|The Hawaiian earring. Only the ten largest circles are shown.]]

* <math>r : \mathbb{R} \to S^1</math> with <math>r(t)=(\cos(2 \pi t), \sin(2 \pi t))</math> is the universal covering of the unit circle <math>S^1</math>.
* <math>p : S^n \to \mathbb{R}P^n \cong \{+1,-1\}\backslash S^n</math> with <math>p(x)=[x]</math> is the universal covering of the [[projective space]] <math>\mathbb{R}P^n</math> for <math>n>1</math>.
* <math>q : \mathrm{SU}(n) \ltimes \mathbb{R} \to U(n)</math> with <math display=block>q(A,t)= \begin{bmatrix}
 \exp(2 \pi i t) & 0\\
 0 & I_{n-1}
\end{bmatrix}_\vphantom{x} A </math> is the universal covering of the [[unitary group]] <math>U(n)</math>.<ref>{{Cite journal
|last1=Aguilar |first1=Marcelo Alberto
|last2=Socolovsky |first2=Miguel
|date=23 November 1999
|title=The Universal Covering Group of U(n) and Projective Representations
|journal=[[International Journal of Theoretical Physics]]
|publisher=Springer US
|publication-date=April 2000
|volume=39
|issue=4
|pages=997–1013
|arxiv=math-ph/9911028
|doi=10.1023/A:1003694206391
|bibcode=1999math.ph..11028A|s2cid=18686364
}}</ref>{{rp|p=5|at=Theorem 1}}
* Since <math>\mathrm{SU}(2) \cong S^3</math>, it follows that the [[quotient map (topology)|quotient map]] <math display=block>f : \mathrm{SU}(2) \rightarrow \mathrm{SU}(2) /  \mathbb{Z_2} \cong \mathrm{SO}(3)</math> is the universal covering of <math>\mathrm{SO}(3)</math>.
* A topological space which has no universal covering is the [[Hawaiian earring]]: <math display=block>
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\}
</math> One can show that no neighborhood of the origin <math>(0,0)</math> is simply connected.{{r|Munkres|p=487|at=Example 1}}