Editing Fundamental group (section)

===Topological groups===

The fundamental group of a [[topological group]] ''X'' (with respect to the base point being the neutral element) is always commutative. In particular, the fundamental group of a [[Lie group]] is commutative. In fact, the group structure on ''X'' endows <math>\pi_1(X)</math> with another group structure: given two loops <math>\gamma</math> and <math>\gamma'</math> in ''X'', another loop <math>\gamma \star \gamma'</math> can defined by using the group multiplication in ''X'': 
:<math>(\gamma \star \gamma')(x) = \gamma(x) \cdot \gamma'(x).</math>
This binary operation <math>\star</math> on the set of all loops is ''a priori'' independent from the one described above. However, the [[Eckmann–Hilton argument]] shows that it does in fact agree with the above concatenation of loops, and moreover that the resulting group structure is abelian.<ref>{{harvtxt|Strom|2011|loc=Problem 9.30, 9.31}}, {{harvtxt|Hall|2015|loc=Exercise 13.7}}</ref><ref>Proof: Given two loops <math>\alpha, \beta: [0, 1] \to  G</math> in <math>\pi_1(G),</math> define the mapping <math>A\colon [0, 1] \times [0, 1] \to G</math> by <math>A(s, t) = \alpha(s)\cdot\beta(t),</math> multiplied pointwise in <math>G.</math> Consider the homotopy family of paths in the rectangle from <math>(s, t) = (0, 0)</math> to <math>(1, 1)</math> that starts with the horizontal-then-vertical path, moves through various diagonal paths, and ends with the vertical-then-horizontal path. Composing this family with <math>A</math> gives a homotopy <math>\alpha * \beta \sim \beta * \alpha,</math> which shows the fundamental group is abelian.</ref>

An inspection of the proof shows that, more generally, <math>\pi_1(X)</math> is abelian for any [[H-space]] ''X'', i.e., the multiplication need not have an inverse, nor does it have to be associative. For example, this shows that the fundamental group of a [[loop space]] of another topological space ''Y'', <math>X = \Omega(Y),</math> is abelian. Related ideas lead to [[Heinz Hopf]]'s computation of the [[Hopf algebra#Cohomology of Lie groups|cohomology of a Lie group]].