  We consider relations in Grassmann algebra corresponding to the
four-dimensional Pachner move 3-3, assuming that there is just one Grassmann
variable on each 3-face, and a 4-simplex weight is a Grassmann-Gaussian
exponent depending on these variables on its five 3-faces. We show that there
exists a large family of such relations; the problem is in finding their
algebraic-topologically meaningful parameterization. We solve this problem in
part, providing two nicely parameterized subfamilies of such relations. For the
second of them, we further investigate the nature of some of its parameters:
they turn out to correspond to an exotic analogue of middle homologies. In
passing, we also provide the 2-4 Pachner move relation for this second case.
