Fundamental group, free product

Question

Let $r,q$ be positive integers. Can someone help me to find a topological space $X$ so that the fundamental group $\pi_1$ is isomorphic to the free product $\mathbb{Z}/r\mathbb{Z}\ast\mathbb{Z}/q\mathbb{Z}$?

Answer 1

Let $L_{p,1}$ be the Lens space forms by quotienting $S^3 \subset \mathbb{C}^2$ by $(z_{1},z_{2}) \sim e^{\frac{2 \pi i}{p}}(z_{1},z_{2}).$

Then $L(p,1)$ is a 3-manifold with $\pi_{1}(L_{p,1}) = \mathbb{Z}_p$.

Then let $X = L_{r,1} \# L_{q,1}$ (here $\#$ denotes connect sum). By Seifert-Van Kampen , the fundamental group is as desired.