Heptagon relations from a simplicial 3-cocycle, and their cohomology

TL;DR

This paper introduces algebraic structures called heptagon relations, parameterized by simplicial 3-cocycles, and explores their applications in constructing invariants for piecewise linear 5-manifolds using cohomology.

Contribution

It develops new algebraic structures related to heptagon relations and links them to simplicial 3-cocycles, extending the framework of Pachner move analogues.

Findings

Heptagon relations are parameterized by simplicial 3-cocycles.

Application to invariants of 5-manifolds with cohomology classes.

Extension of algebraic structures from pentagon to heptagon relations.

Abstract

We introduce new algebraic structures associated with heptagon relations -- higher analogue of the well-known pentagon. The main points we deal with are: (i) polygon relations as algebraic imitations of Pachner moves, on the example of heptagon, (ii) parameterization of heptagon relations by simplicial 3-cocycles, (iii) applications to invariants of pairs "piecewise linear 5-manifold, a 3rd cohomology class on it".