Extended Bloch group and the Chern-Simons class (Incomplete working version)

TL;DR

This paper introduces an extended Bloch group linked to a specific homology group, providing new formulas for the Chern-Simons class and confirming conjectured relationships between volume and invariants of hyperbolic manifolds.

Contribution

It defines an extended Bloch group, establishes its isomorphism with a homology group, and derives formulas for the Chern-Simons class and related invariants.

Findings

Isomorphism between extended Bloch group and H_3(PSL(2,C)),

Exact formula for the universal Cheeger-Simons class,

Proof of the volume-Chern-Simons invariant relationship.

Abstract

We define an extended Bloch group and show it is isomorphic to $H_3(PSL(2,C)^\delta;Z)$. Using the Rogers dilogarithm function this leads to an exact simplicial formula for the universal Cheeger-Simons class on this homology group. It also leads to an independent proof of the analytic relationship between volume and Chern-Simons invariant of hyperbolic manifolds conjectured in \cite{neumann-zagier} and proved in \cite{yoshida}, as well as an effective formula for the Chern-Simons invariant of a hyperbolic manifold.