Editing Fundamental group (section)

== Abstract results ==

As was mentioned above, computing the fundamental group of even relatively simple topological spaces tends to be not entirely trivial, but requires some methods of [[algebraic topology]].

=== Relationship to first homology group ===
The [[abelianization]] of the fundamental group can be identified with the first [[homology group]] of the space.

A special case of the [[Hurewicz theorem]] asserts that the first [[singular homology|singular homology group]] <math>H_1(X)</math> is, colloquially speaking, the closest approximation to the fundamental group by means of an abelian group. In more detail, mapping the homotopy class of each loop to the homology class of the loop gives a [[group homomorphism]]
:<math>\pi_1(X) \to H_1(X)</math>
from the fundamental group of a topological space ''X'' to its first singular homology group <math>H_1(X).</math> This homomorphism is not in general an isomorphism since the fundamental group may be non-abelian, but the homology group is, by definition, always abelian. This difference is, however, the only one: if ''X'' is path-connected, this homomorphism is [[surjective]] and its [[Kernel (algebra)|kernel]] is the [[commutator subgroup]] of the fundamental group, so that <math>H_1(X)</math> is isomorphic to the [[abelianization]] of the fundamental group.<ref>{{harvtxt|Fulton|1995|loc=Prop. 12.22}}</ref>

===Gluing topological spaces===
Generalizing the statement above, for a family of path connected spaces <math>X_i,</math> the fundamental group <math display="inline">\pi_1 \left(\bigvee_{i \in I} X_i\right)</math> is the [[free product]] of the fundamental groups of the <math>X_i.</math><ref>{{harvtxt|May|1999|loc=Ch. 2, §8, Proposition}}</ref> This fact is a special case of the [[Seifert–van Kampen theorem]], which allows to compute, more generally, fundamental groups of spaces that are glued together from other spaces. For example, the 2-sphere <math>S^2</math> can be obtained by gluing two copies of slightly overlapping half-spheres along a [[neighborhood (mathematics)|neighborhood]] of the [[equator]]. In this case the theorem yields <math>\pi_1(S^2)</math> is trivial, since the two half-spheres are contractible and therefore have trivial fundamental group. The fundamental groups of surfaces, as mentioned above, can also be computed using this theorem.

In the parlance of category theory, the theorem can be concisely stated by saying that the fundamental group functor takes [[Pushout (category theory)|pushouts]] (in the category of topological spaces) along inclusions to pushouts (in the category of groups).<ref>{{harvtxt|May|1999|loc=Ch. 2, §7}}</ref>

===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]].

===Fibrations===
''[[Fibrations]]'' provide a very powerful means to compute homotopy groups. A fibration ''f'' the so-called ''total space'', and the base space ''B'' has, in particular, the property that all its fibers <math>f^{-1}(b)</math> are homotopy equivalent and therefore can not be distinguished using fundamental groups (and higher homotopy groups), provided that ''B'' is path-connected.<ref>{{harvtxt|Hatcher|2002|loc=Prop. 4.61}}</ref> Therefore, the space ''E'' can be regarded as a "[[Dehn twist|twisted]] product" of the [[fibration|base space]] ''B'' and the [[Fiber (algebraic geometry)|fiber]] <math>F = f^{-1}(b).</math> The great importance of fibrations to the computation of homotopy groups stems from a [[Homotopy group#Long exact sequence of a fibration|long exact sequence]]
:<math>\dots \to \pi_2(B) \to \pi_1(F) \to \pi_1(E) \to \pi_1(B) \to \pi_0(F) \to \pi_0(E)</math>
provided that ''B'' is path-connected.<ref>{{harvtxt|Hatcher|2002|loc=Theorem 4.41}}</ref> The term <math>\pi_2(B)</math> is the second [[homotopy group]] of ''B'', which is defined to be the set of homotopy classes of maps from <math>S^2</math> to ''B'', in direct analogy with the definition of <math>\pi_1.</math>

If ''E'' happens to be path-connected and simply connected, this sequence reduces to an isomorphism
:<math>\pi_1(B) \cong \pi_0(F)</math>
which generalizes the above fact about the universal covering (which amounts to the case where the fiber ''F'' is also discrete). If instead ''F'' happens to be connected and simply connected, it reduces to an isomorphism
:<math>\pi_1(E) \cong \pi_1(B).</math>
What is more, the sequence can be continued at the left with the higher homotopy groups <math>\pi_n</math> of the three spaces, which gives some access to computing such groups in the same vein.

====Classical Lie groups====

Such fiber sequences can be used to inductively compute fundamental groups of compact [[classical Lie groups]] such as the [[special unitary group]] <math>\mathrm{SU}(n),</math> with <math>n \geq 2.</math> This group acts [[Group action#Remarkable properties of actions|transitively]] on the unit sphere <math>S^{2n-1}</math> inside <math>\mathbb C^n = \mathbb R^{2n}.</math> The [[stabilizer subgroup|stabilizer]] of a point in the sphere is isomorphic to <math>\mathrm{SU}(n-1).</math> It then can be shown<ref>{{harvtxt|Hall|2015|loc=Proposition 13.8}}</ref> that this yields a fiber sequence
:<math>\mathrm{SU}(n-1) \to \mathrm{SU}(n) \to S^{2n-1}.</math> 
Since <math>n \geq 2,</math> the sphere <math>S^{2n-1}</math> has dimension at least 3, which implies 
:<math>\pi_1(S^{2n-1}) \cong \pi_2(S^{2n-1}) = 1.</math>
The long exact sequence then shows an isomorphism
:<math>\pi_1(\mathrm{SU}(n)) \cong \pi_1(\mathrm{SU}(n - 1)).</math>
Since <math>\mathrm{SU}(1)</math> is a single point, so that <math>\pi_1(\mathrm{SU}(1))</math> is trivial, this shows that <math>\mathrm{SU}(n)</math> is simply connected for all <math>n.</math>

The fundamental group of noncompact Lie groups can be reduced to the compact case, since such a group is homotopic to its maximal compact subgroup.<ref>{{harvtxt|Hall|2015|loc=Section 13.3}}</ref> These methods give the following results:<ref>{{harvtxt|Hall|2015|loc=Proposition 13.10}}</ref>
{| class="wikitable"
|-
! Compact classical Lie group ''G'' !! Non-compact Lie group  !! <math>\pi_1</math> 
|-
| special unitary group <math>\mathrm{SU}(n)</math> || <math>\mathrm{SL}(n,\Complex)</math>  || 1 
|-
| [[unitary group]] <math>\mathrm{U}(n)</math> || <math>\mathrm{GL}(n,\Complex), \mathrm{Sp}(n, \R)</math> || <math>\Z</math>
|-
| [[special orthogonal group]] <math>\mathrm{SO}(n)</math> || <math>\mathrm{SO}(n, \C)</math> || <math>\Z/2</math> for <math>n\geq 3</math> and <math>\Z</math> for <math>n=2</math> 
|-
|  compact [[symplectic group]] <math>\mathrm{Sp}(n)</math> || <math>\mathrm{Sp}(n, \C)</math>|| 1 
|}

A second method of computing fundamental groups applies to all connected compact Lie groups and uses the machinery of the [[maximal torus]] and the associated [[root system]]. Specifically, let <math>T</math> be a maximal torus in a connected compact Lie group <math>K,</math> and let <math>\mathfrak t</math> be the [[Lie algebra]] of <math>T.</math> The [[exponential map (Lie theory)|exponential map]]
:<math>\exp : \mathfrak t \to T</math>
is a fibration and therefore its kernel <math>\Gamma \subset \mathfrak t</math> identifies with <math>\pi_1(T).</math> The map
:<math>\pi_1(T) \to \pi_1(K)</math>
can be shown to be surjective<ref>{{harvtxt|Bump|2013|loc=Prop. 23.7}}</ref> with kernel given by the set ''I'' of integer linear combination of [[coroot]]s. This leads to the computation 
:<math>\pi_1(K) \cong \Gamma / I.</math><ref>{{harvtxt|Hall|2015|loc=Corollary 13.18}}</ref> 
This method shows, for example, that any connected compact Lie group for which the associated root system is of [[G2 (mathematics)|type <math>G_2</math>]] is simply connected.<ref>{{harvtxt|Hall|2015|loc=Example 13.45}}</ref> Thus, there is (up to isomorphism) only one connected compact Lie group having Lie algebra of type <math>G_2</math>; this group is simply connected and has trivial center.