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.
Categories: math.GT