Connected components of the topological surgery graph of a unicellular collection

TL;DR

This paper investigates the connected components of the topological surgery graph of unicellular collections on surfaces, revealing they are classified by a homological invariant through a group-theoretic approach involving the mapping class group.

Contribution

It identifies the connected components of the surgery graph using a homological invariant and analyzes the mapping class group's action, including stabilizer generators for mod-2 homology classes.

Findings

Connected components are classified by a homological invariant.

The action of the mapping class group on unicellular collections is characterized.

Simple generating sets for stabilizers of mod-2 homology classes are determined.

Abstract

A unicellular collection on a surface is a collection of curves whose complement is a single disk. There is a natural surgery operation on unicellular collections, endowing the set of such with a graph structure where the edge relation is given by surgery. Here we determine the connected components of this graph, showing that they are enumerated by a certain homological "surgery invariant". Our approach is group-theoretic and proceeds by understanding the action of the mapping class group on unicellular collections. In the course of our arguments, we determine simple generating sets for the stabilizer in the mapping class group of a mod-$2$ homology class, which may be of independent interest.