Deformation of a 3 -> 3 Pachner move relation capturing exotic second homologies

TL;DR

This paper introduces a new Grassmann algebra relation linked to the 3 -> 3 Pachner move in 4D, derived by deforming an exotic chain complex, revealing connections to second homologies.

Contribution

It presents a novel nonlinear relation in Grassmann algebra associated with 4D Pachner moves, incorporating deformations tied to exotic second homologies.

Findings

New Grassmann algebra relation for 4D Pachner move 3 -> 3

Deformation terms correspond to exotic second homologies

Relation involves nonlinear terms of degrees 0, 1, and 2

Abstract

A new relation in Grassmann algebra is presented, corresponding naturally to the four-dimensional Pachner move 3 -> 3. This relation is obtained by deforming a known relation associated with an exotic chain complex built for a triangulated four-manifold. The possible deformation terms turn out to correspond to second (middle) homologies of one more exotic complex. Both sides of the new relation are nonlinear in the deformation: they contain terms of degrees 0, 1 and 2.