Invariants and TQFT's for cut cellular surfaces from finite 2-groups

TL;DR

This paper introduces invariants for cut cellular surfaces using finite 2-groups, extending TQFTs and invariants from group theory to a broader class of surfaces with boundary.

Contribution

It generalizes Yetter's invariants and TQFT constructions to surfaces with boundary using finite 2-groups and a 'fake flatness' condition, expanding the mathematical framework.

Findings

Defined invariants under Pachner-like moves for CCSs.

Extended Yetter's invariants to surfaces with boundary.

Generalized the commuting fraction of a group to the 2-group context.

Abstract

In this brief sequel to a previous article, we recall 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- and 2-cells with elements of a finite 2-group, subject to a "fake flatness" condition for each 2-cell. These invariants, which extend Yetter's invariants to this class of surfaces, are also described in a TQFT setting. A result from the previous article concerning the commuting fraction of a group is generalized to the 2-group context.