Two-Cocycles Give a Full Nonlinear Parameterization of the Simplest 3–3 Relation

A parameterization of Grassmann-algebraic relations corresponding to the Pachner move 3–3 is proposed. In these relations, each 4-simplex is assigned a Grassmann weight depending on five anticommuting variables associated with its 3-faces. The weights are chosen to have the “simplest” form—a Grassmann–Gaussian exponent or its analogue (satisfying a similar system of differential equations). Our parameterization works for a Zariski open set of such relations, looks relevant from the algebraic-topological viewpoint, and reveals intriguing nonlinear relations between objects associated with simplices of different dimensions.