  Hexagon relations are algebraic realizations of four-dimensional Pachner
moves. `Constant' -- not depending on a 4-simplex in a triangulation of a
4-manifold -- hexagon relations are proposed, and their polynomial-valued
cohomology is constructed. This cohomology yields polynomial mappings defined
on the so called `coloring homology space', and these mappings can, in their
turn, yield piecewise linear manifold invariants. These mappings are calculated
explicitly for some examples. It is also shown that `constant' hexagon
relations can be obtained as a limit case of already known `nonconstant'
relations, and the way of taking the limit is not unique. This non-uniqueness
suggests the existence of an additional structure on the `constant' coloring
homology space.
