Editing Fundamental group (section)

===Coverings===

[[File:Covering_map.svg|thumb|The map <math>\mathbb{R} \times [0,1] \to S^1 \times [0,1]</math> is a covering: the preimage of ''U'' (highlighted in gray) is a disjoint union of copies of ''U''. Moreover, it is a universal covering since <math>\mathbb{R} \times [0,1]</math> is contractible and therefore simply connected.]]
Given a topological space ''B'', a [[continuous function (topology)|continuous map]] 
:<math>f: E \to B</math>
is called a ''covering'' or ''E'' is called a ''[[covering space]]'' of ''B'' if every point ''b'' in ''B'' admits an [[open neighborhood]] ''U'' such that there is a [[homeomorphism]] between the [[preimage]] of ''U'' and a [[disjoint union]] of copies of ''U'' (indexed by some set ''I''),
:<math>\varphi: \bigsqcup_{i \in I} U \to f^{-1}(U)</math>
in such a way that <math>\pi \circ \varphi</math> is the standard projection map <math>\bigsqcup_{i \in I} U \to U.</math><ref>{{harvtxt|Hatcher|2002|loc=§1.3}}</ref>

====Universal covering====
A covering is called a [[universal covering]] if ''E'' is, in addition to the preceding condition, simply connected.<ref>{{harvtxt|Hatcher|2002|loc=p. 65}}</ref> It is universal in the sense that all other coverings can be constructed by suitably identifying points in ''E''. Knowing a universal covering 
:<math>p: \widetilde{X} \to X</math> 
of a topological space ''X'' is helpful in understanding its fundamental group in several ways: first, <math>\pi_1(X)</math> identifies with the group of [[deck transformations]], i.e., the group of [[homeomorphism]]s <math>\varphi : \widetilde{X} \to \widetilde{X}</math> that commute with the map to ''X'', i.e., <math>p \circ \varphi = p.</math>
Another relation to the fundamental group is that <math>\pi_1(X, x)</math> can be identified with the fiber <math>p^{-1}(x).</math> For example, the map
:<math>p: \mathbb{R} \to S^1,\, t \mapsto \exp(2 \pi i t)</math>
(or, equivalently, <math>\pi: \mathbb{R} \to \mathbb{R} / \mathbb{Z},\ t \mapsto [t]</math>) is a universal covering. The deck transformations are the maps <math>t \mapsto t + n</math> for <math>n \in \mathbb{Z}.</math> This is in line with the identification <math>p^{-1}(1) = \mathbb{Z},</math> in particular this proves the above claim <math>\pi_1(S^1) \cong \mathbb{Z}.</math>

Any path connected, [[Locally_connected_space#Definitions|locally path connected]] and [[locally simply connected]] topological space ''X'' admits a universal covering.<ref>{{harvtxt|Hatcher|2002|loc=Proposition 1.36}}</ref> An abstract construction proceeds analogously to the fundamental group by taking pairs (''x'',&nbsp;γ), where ''x'' is a point in ''X'' and γ is a homotopy class of paths from ''x''<sub>0</sub> to ''x''. The passage from a topological space to its universal covering can be used in understanding the geometry of ''X''. For example, the [[uniformization theorem]] shows that any simply connected [[Riemann surface]] is (isomorphic to) either <math>S^2,</math> <math>\mathbb{C},</math> or the [[upper half-plane]].<ref>{{harvtxt|Forster|1981|loc=Theorem 27.9}}</ref> General Riemann surfaces then arise as quotients of [[group action]]s on these three surfaces.

The [[quotient topology|quotient]] of a [[Group action#Remarkable properties of actions|free action]] of a [[discrete topology|discrete]] group ''G'' on a simply connected space ''Y'' has fundamental group 
:<math>\pi_1(Y/G) \cong G.</math>
As an example, the real ''n''-dimensional real [[projective space]] <math>\mathbb{R}\mathrm{P}^n</math> is obtained as the quotient of the ''n''-dimensional unit sphere <math>S^n</math> by the antipodal action of the group <math>\mathbb{Z}/2</math> sending <math>x \in S^n</math> to <math>-x.</math> As <math>S^n</math> is simply connected for ''n'' ≥ 2, it is a universal cover of <math>\mathbb{R}\mathrm{P}^n</math> in these cases, which implies <math>\pi_1(\mathbb{R}\mathrm{P}^n) \cong \mathbb{Z}/2</math> for ''n'' ≥ 2.

====Lie groups====
Let ''G'' be a connected, simply connected [[compact Lie group]], for example, the [[special unitary group]] SU(''n''), and let Γ be a finite subgroup of ''G''. Then the [[homogeneous space]] ''X''&nbsp;=&nbsp;''G''/Γ has fundamental group Γ, which acts by right multiplication on the universal covering space ''G''. Among the many variants of this construction, one of the most important is given by [[locally symmetric space]]s ''X''&nbsp;=&nbsp;Γ{{hairsp}}\''G''/''K'', where

*''G'' is a non-compact simply connected, connected [[Lie group]] (often [[semisimple Lie group|semisimple]]),
*''K'' is a maximal compact subgroup of ''G''
*  Γ is a discrete [[countable set|countable]] [[torsion-free group|torsion-free]] subgroup of ''G''.

In this case the fundamental group is Γ and the universal covering space ''G''/''K'' is actually [[contractible]] (by the [[Cartan decomposition]] for Lie groups).

As an example take ''G''&nbsp;=&nbsp;SL(2, '''R'''), ''K''&nbsp;=&nbsp;SO(2) and Γ any torsion-free [[congruence subgroup]] of the [[modular group]] SL(2, '''Z''').

From the explicit realization, it also follows that the universal covering space of a path connected [[topological group]] ''H'' is again a path connected topological group ''G''.  Moreover, the covering map is a continuous [[open map|open]] homomorphism of ''G'' onto ''H'' with kernel Γ, a closed discrete [[normal subgroup]] of ''G'':

:<math>1 \to \Gamma \to G \to H \to 1.</math>

Since ''G'' is a connected group with a continuous action by conjugation on a discrete group Γ, it must act trivially, so that Γ has to be a subgroup of the [[center (group theory)|center]] of ''G''.  In particular π<sub>1</sub>(''H'') =  Γ is an [[abelian group]]; this can also easily be seen directly without using covering spaces.  The group ''G'' is called the ''[[universal covering group]]'' of&nbsp;''H''.

As the universal covering group suggests, there is an analogy between the fundamental group of a topological group and the center of a group; this is elaborated at [[Covering group#Lattice of covering groups|Lattice of covering groups]].