information:extraction
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 S U (n), {\displaystyle \mathrm {SU} (n),} with n ≥ 2. {\displaystyle n\geq 2.} This group acts transitively on the unit sphere S 2 n − 1 {\displaystyle S^{2n-1}} inside C n = R 2 n. {\displaystyle \mathbb {C} ^{n}=\mathbb {R} ^{2n}.} The stabilizer of a point in the sphere is isomorphic to S U (n − 1). {\displaystyle \mathrm {SU} (n-1).} It then can be shown that this yields a fiber sequence

Since n ≥ 2, {\displaystyle n\geq 2,} the sphere S 2 n − 1 {\displaystyle S^{2n-1}} has dimension at least 3, which implies

The long exact sequence then shows an isomorphism

Since S U (1) {\displaystyle \mathrm {SU} (1)} is a single point, so that π 1 (S U (1)) {\displaystyle \pi _{1}(\mathrm {SU} (1))} is trivial, this shows that S U (n) {\displaystyle \mathrm {SU} (n)} is simply connected for all n. {\displaystyle n.}

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. These methods give the following results:

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 T {\displaystyle T} be a maximal torus in a connected compact Lie group K, {\displaystyle K,} and let t {\displaystyle {\mathfrak {t}}} be the Lie algebra of T. {\displaystyle T.} The exponential map

is a fibration and therefore its kernel Γ ⊂ t {\displaystyle \Gamma \subset {\mathfrak {t}}} identifies with π 1 (T). {\displaystyle \pi _{1}(T).} The map

can be shown to be surjective with kernel given by the set I of integer linear combination of coroots. This leads to the computation

This method shows, for example, that any connected compact Lie group for which the associated root system is of type G 2 {\displaystyle G_{2}} is simply connected. Thus, there is (up to isomorphism) only one connected compact Lie group having Lie algebra of type G 2 {\displaystyle G_{2}}; this group is simply connected and has trivial center.