Every construction I know of 3-manifold invariants from Heegaard splittings factors through the Reidemeister-Singer Theorem:

**Reidemeister-Singer Theorem**: For any two Heegaard splittings and of a 3-manifold , there exists a third Heegaard splitting which is a stabilization of both.

This theorem is definitely part of the big story in 3-manifold topology, and is usually proven in the PL category, as for example in Nikolai Saveliev’s Lectures on the Topology of 3-manifolds. There is another nice PL proof due to Craggs, Proc. Amer. Math. Soc. 57, n 1 (1976), 143-147.

I think of a Heegaard splitting as being intrinsically a smooth topology construction (a level set of a Morse function), and so I would really like the proof of Reidemeister-Singer to live in the smooth category. I think that there should be consistent smooth and PL stories of 3-manifold topology living side by side. In the 1970′s, Bonahon wrote a smooth proof of Reidemeister-Singer, which uses Cerf Theory (naturally, because we’re investigating paths between Morse functions). Unfortunately, Bonahon’s proof was never published, and it is lost.

A year ago (but I only saw it this morning), François Laudenbach posted a smooth proof of Reidemeister-Singer to arXiv: http://arxiv.org/abs/1202.1130. I think that this is wonderful! There are too few papers like this- there is insufficient incentive to streamline the storylines of foundations. I am very happy to have found this proof, and I want such a proof to be a part of my smooth 3-manifold topology foundations.

**Edit**: Thanks to George Mossessian and to Ryan Budney, who point out in the comments that Jesse Johnson proved Reidemeister-Singer using Rubinstein and Scharlemann’s sweep-outs, which involves singularity theory which is much less sophisticated that Cerf Theory: http://front.math.ucdavis.edu/0705.3712

Perhaps that should be the “smooth proof from The Book” (or the “proof from The Smooth Book”)!

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I imagine Jesse can confirm, but I thought this also follows from Rubinstein-Scharlemann sweep-out technology. That also uses a singularity theory but it can use less-sophisticated singularity theory than Cerf theory.

Comment by Ryan Budney — July 11, 2013 @ 11:24 pm |

That sounds really nice! An optimal smooth proof of Reidemeister-Singer would make me very happy.

Comment by dmoskovich — July 12, 2013 @ 3:56 am |

Daniel, it’s at http://front.math.ucdavis.edu/0705.3712; the machinery it uses is indeed far less sophisticated than Cerf theory.

The note of Laudenbach that you bring up is interesting, thank you for making a post about it. I’ll look at it more carefully, particularly the “Elementary swallow-tail lemma” — if you look at Cerf’s original work, it’s only proven for Cerf graphics on cobordisms of n-folds with n >= 5, for paths of index <= n-4, only in the case that the attaching surface is simply connected… very restrictive.

Comment by George Mossessian — July 12, 2013 @ 10:10 am

Thank you for this! I didn’t know about Jesse’s paper- I will now look at it.

Comment by dmoskovich — July 14, 2013 @ 11:38 pm

I agree – I’ve always thought of Reidemeister-Singer as a more natural statement from the perspective of Morse theory, following roughly the same lines as Bonahon’s unpublished proof. In fact, the basic thin position machinery is really just a combinatorial description of some hidden underlying Morse theory. But unfortunately, this isn’t usually explained in papers. In fact, it doesn’t seem to be very common for papers (including my own) to explicitly state whether they’re working in the smooth or pl category.

Comment by Jesse Johnson — July 22, 2013 @ 11:33 am |