Free fermions on a piecewise linear four-manifold. I: Exotic chain complex

TL;DR

This paper introduces an exotic chain complex related to Grassmann-Gaussian exponentials and 2-cocycles, providing a foundation for algebraic realizations of all four-dimensional Pachner moves in triangulated manifolds.

Contribution

It defines a new exotic chain complex linked to 2-cocycles, advancing algebraic methods for four-dimensional Pachner move realizations.

Findings

Established a new chain complex structure for 4D manifolds.

Linked the complex to Grassmann-Gaussian exponentials and 2-cocycles.

Provided a basis for algebraic realization of Pachner moves.

Abstract

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.