# Free fermions on a piecewise linear four-manifold. II: Pachner moves

## TL;DR

This paper develops an invariant for four-dimensional piecewise linear manifolds with a specified cohomology class, analyzing its behavior under Pachner moves to determine a correction factor for its construction.

## Contribution

It introduces a method to compute the correction factor ensuring invariance of the constructed invariant under all Pachner moves in four dimensions.

## Key findings

- The correction factor can be expressed as a product over 2-faces and pentachora.
- The invariant is related to the square root of a chain complex torsion.
- The behavior under Pachner moves guides the invariant's normalization.

## Abstract

This is the second in a series of papers where we construct an invariant of a four-dimensional piecewise linear manifold $M$ with a given middle cohomology class $h\in H^2(M,\mathbb C)$. This invariant is the square root of the torsion of unusual chain complex introduced in Part I (arXiv:1605.06498) of our work, multiplied by a correcting factor. Here we find this factor by studying the behavior of our construction under all four-dimensional Pachner moves, and show that it can be represented in a multiplicative form: a product of same-type multipliers over all 2-faces, multiplied by a product of same-type multipliers over all pentachora.