Double pants decompositions of 2-surfaces

TL;DR

This paper introduces a groupoid acting on double pants decompositions of 2-surfaces, demonstrating its transitive action on decompositions related to the 3-sphere and linking it to the mapping class group.

Contribution

It defines a new groupoid generated by flips and handle twists and proves its transitivity on certain double pants decompositions, connecting to 3-manifold topology.

Findings

FT acts transitively on double pants decompositions of the 3-sphere

The mapping class group is contained within FT

The groupoid is generated by simple local transformations

Abstract

We consider a union of two pants decompositions of the same orientable 2-dimensional surface of any genus g. Each pants decomposition corresponds to some handlebody bounded by this surface, so two pants decompositions correspond to a Heegaard splitting of a 3-manifold. We introduce a groupoid FT acting on double pants decompositions. This groupoid is generated by two simple transformations (called flips and handle twists), each transformation affecting only one curve of the double pants decomposition. We prove that FT acts transitively on all double pants decompositions corresponding to Heegaard splittings of a 3-dimensional sphere. As a corollary, the mapping class group is contained in FT.