Heegaard splittings and singularities of the product map of Morse functions

TL;DR

This paper establishes an upper bound on the Reidemeister-Singer distance between Heegaard splittings using the singularity properties of the product map of Morse functions, linking topology and singularity theory.

Contribution

It introduces a new bound for the Reidemeister-Singer distance based on singularity theory, connecting Morse functions' product maps with Heegaard splitting complexity.

Findings

Upper bound relates genus and cusp points of product maps

Suggests singularity theory can optimize bounds for splitting distances

Links Morse function singularities to topological invariants

Abstract

We give an upper bound for the Reidemeister-Singer distance between two Heegaard splittings in terms of the genera and the number of cusp points of the product map of Morse functions for the splittings. It suggests that a certain development in singularity theory may lead to the best possible bound for the Reidemeister-Singer distance.