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.

中文翻译：

---

非常数六边形关系及其上同调

提出了六边形关系的构造——四维 Pachner 移动的代数实现。它是根据五角星（4 个单纯形）的 3 个面的“允许着色”而定的，其主要特征是允许的着色集是非常规的——从五角星到五角星各不相同。此外，还为这些六边形关系制定了上同调理论，并在显式示例中证明了其非平凡性。