Invariants and TQFT's for cut cellular surfaces from finite groups

TL;DR

This paper introduces cut cellular surfaces (CCS), defines invariants based on finite group colorings, and explores their properties within a novel TQFT framework, extending topological invariants to new surface decompositions.

Contribution

It develops a new class of invariants for CCS's using finite group colorings and formulates a related TQFT that differs from traditional 2D TQFTs.

Findings

Invariants are preserved under Pachner-like moves.

Functions associated to basic surfaces satisfy topological properties.

The framework extends invariants to new surface decompositions.

Abstract

We introduce the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of 0-, 1- and 2-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by counting colourings of the 1-cells with elements of a finite group, subject to a "flatness" condition for each 2-cell. These invariants are also described in a TQFT setting, which is not the same as the usual 2-dimensional TQFT framework. We study the properties of functions which arise in this context, associated to the disk, the cylinder and the pants surface, and derive general properties of these functions from topology, including properties which come from invariance under the Hatcher-Thurston moves on pants decompositions.