Let $L(p,q)$ be the lens space, that is $L(p,q)=S^3/\mathbb{Z}_p$.
Here, $\mathbb{Z}_p$ acts on $S^3$ by $(z_1,z_2)\mapsto (\rho z_1,\rho^q z_2)$, $ \rho=e^{\frac{2\pi i}{p}}$.
It is well known that
$L(p,q)$ and $L(p',q')$ are diffeomorphic if and only if $p'=p, q'=\pm q^{\pm1}$ (mod $p$).
In A. Hatcher's note page 39-42, there is a proof of the above classification theorem of lens space using the uniqueness of Heegaard torus in Lens space up to isotopy. But I have some misunderstandings with his argument when I following it line by line.
Where can I find the original proof of classification of Lens space by using uniqueness of Heegaard torus up to isotopy?
Note : I know that there is a proof that uses whitehead torsion of lens space and its invariance under the homeomorphism which I already familiar sufficiently.
The proof that Hatcher presents is due to Bonahon and Otal. The reference is here:
MR0663085 (83f:57008) Bonahon, Francis; Otal, Jean-Pierre Scindements de Heegaard des espaces lenticulaires. (French. English summary) [Heegaard splittings of lens spaces] C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 17, 585–587. 57N10
If I recall, I believe Hatcher's write-up is similarly detailed to Bonahon and Otal's argument. And there are steps missing in both presentations but they're readily filled.