I know the quotient definition of Lens space: given complex coordinate $z_1,z_2$, the Lens $L(p,q)$ manifold is defined by the identification $$ (z_1,z_2) \sim (e^{\frac{2\pi i}{p}} z_1, e^{\frac{2\pi qi}{p}} z_1) $$ I also know the definition like in Heegaard decomposition we need to glue the $(1,0)$ circle with $(p,q)$ circle of two $D_2\times S^1$ manifold to construct Lens $L(p,q)$. But it is just not very clear to me what is the meaning of such statement.
I roughly understand what is the meaning of $p$ parameter, because we can calculate the first fundamental group of Lens space $L(p,q)$ which will be $\mathbb{Z}_p$. But I want to know what is the effect of $q$ on the manifold if $q>1$. Is it related to any geometric quantities like torsion or something else, such that we might have more intuitive picture to tell the difference between $L(p,1)$ and $L(p,q)$.
