Hexagon relations, their cohomologies, and invariants of 4-dimensional PL manifolds

Pachner moves, also called bistellar moves, are elementary re-buildings of a manifold triangulation. A triangulation of a given piecewise linear (PL) manifold can be transformed into any other one by a finite sequence of Pachner moves. Hexagon relations are algebraic realizations of four-dimensional Pachner moves. It can be said that hexagon relations are in the same relationship with 4-manifolds and Pachner moves as quandles are with knots and Reidemeister moves, or as the same quandles are with 2-knots and Roseman moves.
We present an explicit hexagon relation in terms of vector spaces over a finite field. This allows us to define “permitted colorings” on triangulations of 4-manifolds, with a clear correspondence between such colorings before and after a Pachner move. We then define a “rough” invariant of a 4-manifold, based on the total number of permitted triangulation colorings.
And like in the quandle case, there are cohomologies that can be introduced for hexagon relations. Remarkably, nontrivial cohomologies do exist, and they give much more interesting invariants of PL 4-manifolds.