# Nonconstant hexagon relations and their cohomology

## TL;DR

This paper introduces a novel algebraic framework for four-dimensional Pachner moves using nonconstant colorings of 3-faces in pentachora, and develops a cohomology theory demonstrating nontrivial structures.

## Contribution

It presents a new construction of hexagon relations with variable permitted colorings and formulates a cohomology theory for these relations, showing their nontriviality.

## Key findings

- Constructed algebraic realizations of four-dimensional Pachner moves.
- Developed a cohomology theory for nonconstant hexagon relations.
- Demonstrated nontrivial cohomology examples.

## Abstract

A construction of hexagon relations - algebraic realizations of four-dimensional Pachner moves - is proposed. It goes in terms of "permitted colorings" of 3-faces of pentachora (4-simplices), and its main feature is that the set of permitted colorings is nonconstant - varies from pentachoron to pentachoron. Further, a cohomology theory is formulated for these hexagon relations, and its nontriviality is demonstrated on explicit examples.