Consider the space \(X\) obtained from a seven-sided polygonal region by means of the labelling scheme \(abaaab^{-1}a^{-1}\). Show that the fundamental group of \(X\) is the free product of two cyclic groups. [Hint: See Theorem 68.7.]
Solution
By the polygon presentation theorem, all vertices are identified and \[ \pi_1(X)\cong \langle a,b\mid ab a^3 b^{-1}a^{-1}=1\rangle. \] The relator is equivalent to \(a^3=1\): indeed \[ ab a^3b^{-1}a^{-1}=1 \iff ba^3b^{-1}=1 \iff a^3=1. \] Thus \[ \pi_1(X)\cong\langle a,b\mid a^3=1\rangle. \] There is no relation involving \(b\). Therefore, by the normal-form theorem for free products, \[ \boxed{\pi_1(X)\cong C_3*\mathbb Z,} \] the free product of the cyclic group of order \(3\) and an infinite cyclic group.