On product of cocycles in a polyhedral complex

TL;DR

This paper introduces a new product operation for cochains in polyhedral complexes, generalizing known algorithms and exploring algebraic structures relevant to convex geometry and polyhedral theory.

Contribution

It develops a parameter-dependent multiplication algorithm for cochains in polyhedral complexes, extending existing methods like Čech cohomology to more general settings.

Findings

Defines a product of cochains in polyhedral complexes

Shows cocycles form a subring and cobounds form an ideal

Provides applications in convex geometry

Abstract

A product of cochains in a polyhedral complex is constructed. The multiplication algorithm depends on the choice of a parameter. The parameter is a linear functional on the ambient space. Cocycles form a subring of the ring of cochains, cobounds form an ideal of the ring of cocycles, and the quotient ring is a ring of cohomologies. If the complex is simplicial, then the algorithm reduces to the well known algorithm of ${\rm\breve{C}}$ech. Similar algorithms are used in geometry of polyhedra for multiplying of cocycles taking values in the exterior algebra of the ambient space. Therefore, we assume the ring of values to be supercommutative. This text contains the basic definitions, the statements of theorems and some of their applications in convex geometry. The full text will be offered to "Izvestiya: Mathematics".