Paper detail

Double pants decompositions of 2-surfaces

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.