Polynomial-valued constant hexagon cohomology

TL;DR

This paper introduces polynomial-valued cohomology for constant hexagon relations, providing new algebraic tools for 4-manifold invariants and exploring their connections to nonconstant relations.

Contribution

It constructs polynomial-valued cohomology for constant hexagon relations and demonstrates their relation to nonconstant relations, offering new insights into 4-manifold invariants.

Findings

Explicit calculations of polynomial mappings for examples

Constant relations as limits of nonconstant relations

Non-uniqueness of the limit suggests additional structure

Abstract

Hexagon relations are algebraic realizations of four-dimensional Pachner moves. `Constant' -- not depending on a 4-simplex in a triangulation of a 4-manifold -- hexagon relations are proposed, and their polynomial-valued cohomology is constructed. This cohomology yields polynomial mappings defined on the so called `coloring homology space', and these mappings can, in their turn, yield piecewise linear manifold invariants. These mappings are calculated explicitly for some examples. It is also shown that `constant' hexagon relations can be obtained as a limit case of already known `nonconstant' relations, and the way of taking the limit is not unique. This non-uniqueness suggests the existence of an additional structure on the `constant' coloring homology space.