Fundamental group of a topological group

For loops based at the identity of a topological group, pointwise multiplication and concatenation obey the interchange law and have the same identity. The Eckmann-Hilton argument therefore makes the fundamental group abelian.

Suppose two binary operations $\ast$ and $\cdot$ on one set share an identity element $e$ and satisfy the interchange ruleThen the operations coincide, becauseThey are also commutative:These substitutions prove the Eckmann-Hilton argument without cancellation assumptions. In the fundamental group of a topological group, the two operations are loop concatenation and pointwise group multiplication. They share the constant loop as identity on homotopy classes, so the fundamental group is Abelian.

For $n\ge 1$, pointwise multiplication of based cube maps into a topological group induces the ordinary addition in its homotopy group ${\pi }_n(G,e)$. Concatenation and pointwise multiplication share an identity and satisfy the interchange rule, so the Eckmann-Hilton argument identifies the two operations. In particular, the $m$-th power map induces multiplication by $m$ on every homotopy group, for all integers $m$.