An integral bilinear form and related forms on abelian groups as hexagon cocycles

TL;DR

This paper explores how certain nontrivial cohomologies related to hexagon relations, which are algebraic models of 4D Pachner moves, can be derived from an integral bilinear form analogous to the intersection form of manifolds.

Contribution

It demonstrates that some known nontrivial cohomologies can be obtained from a single integral bilinear form using Frobenius homomorphisms, linking algebraic structures to PL 4-manifold invariants.

Findings

Some nontrivial cohomologies are derived from the bilinear form.

The form acts as an analogue of the intersection form.

Not all known cohomologies can be obtained this way.

Abstract

Hexagon relations are algebraic realizations of four-dimensional Pachner moves, and there are hexagon relations admitting nontrivial cohomologies and leading thus to piecewise linear (PL) 4-manifold invariants. We show that some - but not all! - of the known nontrivial cohomologies can be obtained from a single integral bilinear form corresponding to a PL 4-manifold by using a Frobenius homomorphism for a half of `color' variables (or different Frobenius homomorphisms for both halves). This form can be regarded as a sophisticated analogue of the manifold's intersection form.