Duality and invariants of representations of fundamental groups of 3-manifolds into PGL(3,C)

TL;DR

This paper studies the duality transformations of flag configurations in PGL(3,C) and proves the invariance of a related Bloch group invariant derived from 3-manifold triangulations.

Contribution

It explicitly characterizes the duality action on configurations and establishes the invariance of a Bloch group invariant under this duality.

Findings

Explicit duality transformation formulas for flag configurations.

Proof of invariance of a Bloch group invariant under duality.

Application to decorated triangulations of 3-manifolds.

Abstract

We determine the explicit transformation under duality of generic configurations of four flags in $\PGL(3,\bC)$ in cross-ratio coordinates. As an application we prove invariance under duality of an invariant in the Bloch group obtained from decorated triangulations of 3-manifolds.