(Co)homology of racks and multiple group racks for compact oriented surfaces in the 3-sphere (Q6653376)

A multiple group rack is a structure formed by the disjoint union of groups, generalizing the concept of racks, which are known for producing invariants of framed links. The paper under review develops a (co)homology theory for multiple group racks and constructs cocycle invariants of compact oriented surfaces in the 3-sphere using their 2-cocycles. The authors present a method for deriving new rack cocycles and multiple group rack cocycles from existing rack cocycles. By analyzing these cocycle invariants, they identify various symmetry types of compact oriented surfaces in the 3-sphere, influenced by chirality and invertibility.