# Explicit cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf-Witten invariants

## TL;DR

This paper derives explicit formulas for cocycles on finite abelian groups, enabling the classification of braided linear Gr-categories and the computation of Dijkgraaf-Witten invariants for tori.

## Contribution

It introduces a unified method to explicitly compute cocycles on finite abelian groups and applies this to classify braided categories and calculate topological invariants.

## Key findings

- Explicit cocycle formulas for all degrees on finite abelian groups.
- Complete classification of braided linear Gr-categories for these groups.
- Calculation of Dijkgraaf-Witten invariants for n-tori.

## Abstract

We provide explicit and unified formulas for the cocycles of all degrees on the normalized bar resolutions of finite abelian groups. This is achieved by constructing a chain map from the normalized bar resolution to a Koszul-like resolution for any given finite abelian group. With a help of the obtained cocycle formulas, we determine all the braided linear Gr-categories and compute the Dijkgraaf-Witten Invariants of the $n$-torus for all $n$.