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

Nonconstant hexagon relations and their cohomology