Favourite classifications of the 3-dimensional lens spaces

With the proof of the Poincare conjecture, it's a great time to look at the whole of 3-manifold theory and poke at it. Some things are "more proven" than others. For example: We only have one proof of the Poincare conjecture. Similarly, there's only one proof of the Smale conjecture. Some things are "super proven", like Dehn's Lemma, the Loop Theorem and the Sphere Theorem -- there's even more than one proof of their equivariant versions.

Take a look at the classification of Seifert fibred manifolds. The classification of the "sufficiently large" ones is a rather elegant demonstration of incompressible surfaces. But the small Seifert fibred manifolds -- manifolds that fibre over S3 with 3 or less singular fibers, their classification is rather fussy and involves a bunch of special cases.

At the moment my favorite proof of the classification of lens spaces is due to:

Przytycki, Yasuhara. Symmetry of Links and Classification of Lens Spaces. Geom. Ded. Vol 98. No. 1. (2003)

Actually, there's aspects of their proof I'd change if I was presenting it myself. But the general idea is what I like. Take a knot K in a lens space Lp,q such that its lift to S3 is a knot with a trivial Alexander polynomial. Now consider K to be a generator of H1(Lp,q). Compute the torsion linking form on (K,K). This is an element of Q/Z. The classification boils down to computing this number and checking that it has little dependance on the choice of K.

The first proof of the classification that I read (and liked) is due to Francis Bonahon.

F. Bonahon, Difféotopies des espaces lenticulaires, Topology 22 (1983), 305--314.

His technique is to show that lens spaces have a single genus 1 Heegaard splitting up to isotopy. He uses a variant of a standard incompressible surface argument, but adapted to the slightly more awkward situation where the surface is the 2-skeleton of the "standard" CW-decomposition of the lens space.

Paolo Salvatore and Riccardo Longoni had a beautiful related insight into lens spaces recently. They showed that even though the lens spaces L7,1 and L7,2 are homotopy-equivalent, their configuration spaces are not homotopy-equivalent. Moreover, they only need the homotopy type of C2(Lp,q) -- the configuration space of two points in the lens space. This could potentially give a new and rather elegant classification of the lens spaces if this were true:

Two lens spaces Lp,q and La,b are diffeomorphic if and only if C2(Lp,q) and C2(La,b) are homotopy-equivalent.

Shortly after Paolo and Riccardo put their paper on the arXiv I ran into several different groups who said they were thinking about extending the Salvatore-Longoni result, but I haven't heard much positive or negative from any of them since, and it's been almost 4 years now.

I do not know of much work towards simplifying the classification of Seifert-fibered manifolds that fibre over S2 with 3 singular fibres.