Editing Fundamental group (section)

====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.