Parameterizing the Simplest Grassmann-Gaussian Relations for Pachner Move 3-3

TL;DR

This paper explores algebraic relations associated with the 3-3 Pachner move in four-dimensional topology, identifying parameterized families of Grassmann-Gaussian relations and revealing connections to exotic homological structures.

Contribution

It introduces two parameterized subfamilies of Grassmann-Gaussian relations for the 3-3 Pachner move, with one revealing links to exotic middle homologies.

Findings

Existence of large families of Grassmann-Gaussian relations for Pachner move 3-3.

Two well-parameterized subfamilies of these relations are identified.

One subfamily's parameters relate to exotic analogues of middle homologies.

Abstract

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.