Fundamental group of a bouquet of circles
= Fundamental group of a bouquet of circles
The <fundamental group> of a bouquet of $r$ circles is the <free group> on $r$ generators, represented by the oriented circles. For finite $r$, iterated <Seifert-van Kampen theorem> proves the claim by attaching one circle at a time. Nontrivial <reduced words in a free group> distinguish based loops, while <free homotopy> classes correspond to conjugacy classes.