I know two definitions of the Lens Space $L(p,q)$:
Take $S^3\subset\mathbb{C}^2$, and consider the $\mathbb{Z}_p$ action $T(z_1,z_2)\mapsto (\omega z_1, \omega^q z_2)$, where $\omega = e^{\frac{2\pi i}{p}}$ and $T$ is the generator of $\mathbb{Z}_p$. Then $S^3/\mathbb{Z}_p$ is $L(p,q)$.
Take two solid tori $S^1\times D^2$, and glue them along a diffeomorphism of their boundaries $h:S^1\times S^1\to S^1\times S^1$, mapping a meridian $\mu_1$ of the first to $q\mu_2 + p\lambda_2$ of the second.
(In definition 2, the meridian $\mu_i$ is $\{1\}\times S^1$, a curve that bounds a disk in the interior. Longitude is $S^1\times \{1\}$, a generator of the fundamental group of the solid torus).
I have been trying to show that these definitions are equivalent, but have not been successful yet. Would someone have a suggestion?
