Cocycles in Lie Groups, Cochains and Regularity Problem

After the fundamental work of Livschitz in [1; 2], various research directions emerged, among which the following stand out: (i) the study of cocycles with values in groups and semigroups beyond R, as well as the investigation of corresponding regularity results; (ii) the analysis of how a certain degree of regularity of the cocycle can confer corresponding regularity to the solution of the cohomological equation; and (iii) the study of higher-dimensional cohomology naturally associated with the action of groups other than Z or R. The aim of this article is to present, as self-contained as possible, a review of the natural generalizations of the notions of cocycles and cochains, as well as their corresponding results, in the study of cohomological equations.