Angle structures on $3$-manifolds

TL;DR

This paper introduces a general framework called angle structures for labeled triangulations of 3-manifolds, inspired by hyperbolic volume, applicable to any abelian group and emphasizing the importance of smoothness.

Contribution

It defines a new non-trivial condition for labelings of tetrahedra in 3-manifolds using abelian groups, extending concepts related to hyperbolic volume.

Findings

Defines angle structures on triangulated 3-manifolds.

Establishes conditions for labelings using abelian groups.

Highlights the role of smoothness in the construction.

Abstract

Given a compact oriented triangulated $3$-manifold we find a non-trivial condition satisfied by certain labelings of the tetrahedra by elements of an arbitrary abelian group which we call angle structures. Smoothness of the manifold is used in an essential way. This is inspired by the notion of the volume of hyperbolic manifolds, which would correspond to the case when the abelian group is the multiplicative group of $\mathbb{C}$, but the construction here seems to be more general, in particular it only uses the abelian group structure.