cocycles

Cocycles are mathematical objects that encode a compatibility condition arising from gluing local data or tracking evolution in a system. In cohomology theories and related areas, a cocycle is a cochain whose coboundary is zero, placing it in the kernel of a coboundary operator. This condition makes cocycles suitable for representing global, consistent structures assembled from local information.

In group cohomology, let G act on an abelian group A. A k-cochain is a function from G^k to A, and the coboundary operator d maps k-cochains to (k+1)-cochains. A cochain is a cocycle if dφ = 0, and cocycles modulo coboundaries define cohomology groups H^k(G,A) = Z^k(G,A)/B^k(G,A). In degree one, a 1-cocycle is a function f with f(gh) = g·f(h) + f(g); such objects are crossed homomorphisms and classify certain group extensions. In degree two, cocycles satisfy a higher associativity condition that governs projective representations and more intricate extensions. Coboundaries are cocycles arising as dψ from lower-degree data.

More generally, the framework of cochains, coboundaries, and cocycles provides a unifying language for various constructions. Čech cocycles, defined on overlaps of a cover, describe transition data for fiber bundles; de Rham cocycles are closed differential forms representing topological features through cohomology classes.

In dynamics, a cocycle over a map T is a rule c(x,n) that satisfies a cocycle identity c(x, n+m) = c(T^n x, m) c(x, n). Such cocycles yield skew-product systems and time-dependent behavior, with measurable, continuous, or smooth variants.

Examples include additive cocycles for abelian targets and cocycles arising from group extensions, reflecting how local data assemble into global structure.