Editing Fundamental group (section)

====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.