P-moves between pants-block decompositions of 3-manifolds

TL;DR

This paper introduces P-moves, a set of transformations connecting any two pants-block decompositions of 3-manifolds, analogous to Pachner moves for triangulations, using Morse 2-functions, Reeb complexes, and P-complexes.

Contribution

It establishes that all pants-block decompositions of a 3-manifold are related by finitely many P-moves and provides a classification of these moves.

Findings

P-moves connect any two pants-block decompositions

A finite list of P-move types is provided

Uses Morse 2-functions, Reeb complexes, and P-complexes as tools

Abstract

A pants-block decomposition of a 3-manifold is similar to a triangulation of a 3-manifold in many aspects. In this paper we show that any two pants-block decompositions of a 3-manifold are related by a finite sequence of moves which are called P-moves. The P-moves between pants-block decompositions are similar to the Pachner moves between triangulations. Moreover, we also give a list of types of P-moves. The main tools we used in this paper are the Morse 2-functions, Reeb complexes and a new 2-dimensional complex called P-complex.