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

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