Pants Decompositions of Surfaces

TL;DR

This paper proves that all pants decompositions of a surface can be transformed into each other through simple moves, establishing the simply-connectedness of a complex formed by these decompositions and their relations.

Contribution

It introduces a 2-complex of pants decompositions and proves it is simply-connected, connecting different decompositions via elementary moves and relations.

Findings

Any two pants decompositions are connected by a sequence of simple moves.

Sequences of simple moves between the same decompositions are related by elementary relations.

The 2-complex of decompositions is simply-connected.

Abstract

We consider collections of disjoint simple closed curves in a compact orientable surface which decompose the surface into pairs of pants. The isotopy classes of such curve systems form the vertices of a 2-complex, whose edges correspond to certain simple moves in which only one curve changes, and whose 2-cells correspond to certain elementary cycles of simple moves. The main theorem is that this 2-complex is simply-connected. Thus any two pants decompositions of a surface are joinable by a sequence of simple moves, and any two such sequences of simple move are related by the elementary relations. The proof is similar to the proof, in a 1980 paper with W. Thurston, of an analogous result for curve systems with connected genus zero complement. [The present paper is essentially an excerpt from a joint paper with P. Lochak and L. Schneps which is to appear in Crelle's Journal.]