The complex of pant decompositions of a surface

TL;DR

This paper constructs a simply connected complex of pant decompositions on a surface, introducing moves and relations that are independent of the mapping class group, and illustrating the Grothendieck principle.

Contribution

It provides a new construction of the complex of pant decompositions with independent structural transformations, differing from previous approaches.

Findings

The complex is simply connected.

Moves and relations are supported in small subsurfaces.

The construction exemplifies the Grothendieck principle.

Abstract

We exhibit a set of edges (moves) and 2-cells (relations) making the complex of pant decompositions on a surface a simply connected complex. Our construction, unlike the previous ones, keeps the arguments concerning the structural transformations independent from those deriving from the action of the mapping class group. The moves and the relations turn out to be supported in subsurfaces with 3g-3+n=1,2 (where g is the genus and n is the number of boundary components), illustrating in this way the so called Grothendieck principle.