Citation: Moriah, Yoav. "Heegaard splittings of Seifert fibered spaces." Inventiones mathematicae 91.3 (1988): 465-481.
Web: https://link.springer.com/article/10.1007/BF01388781
Tags: Mathematical
In this paper, Yoav Moriah demonstrates some exceptional isomorphisms between Heegaard splittings of Seifert fibered spaces. That is, homomorphic spaces are demonstrated with multiple Heegaard splittings whose connecting maps are non-isotopic. The takeaway from papers like this one is that is that Heegaard splittings are certainly not unique, and that there seems to be no easy rule for determining whether or not two Heegaard decompositions will give the same manifold.
I find this paper relevant in the context of modular functors and topological quantum field theory. A modular functor basically gives you a compatible family of mapping class group representations. Is there a nice way of knowing whether or not a given modular functor will lift?
There is a nice way of defining invariants of 3-manifolds given the data of a modular functor with just a bit of extra structure, namely you can take a Heegaard decomposition and define the 3-manifold invariant to be the expectation value of the operator associated to the gluing map. For this to well-defined, however, one must assign the same value for any Heegaard decomposition of the same manifold. In particular, this means that there is some condition which says that whenever two elements of the mapping class group induce the same manifold then they must have the same expectation value. However, seeing as the question of uniqueness of Heegaard splittings is deep and subtle, there seems to be no hope of turning this into a nice axiom on the level of 2D data. In other words, the question of lifting 2D modular functors to 3D TQFTs seems to use scary 3-manfifold topology which is not captured within the 2-dimensional world of modular functors.