  Recently, an algebraic realization of the four-dimensional Pachner move 3--3
was found in terms of Grassmann--Gaussian exponentials, and a remarkable
nonlinear parameterization for it, going in terms of a $\mathbb C$-valued
2-cocycle. Here we define, for a given triangulated four-dimensional manifold
and a 2-cocycle on it, an `exotic' chain complex intimately related to the
mentioned parameterization, thus providing a basis for algebraic realizations
of all four-dimensional Pachner moves.
