Geometry & Topology, Vol. 8 (2004) Paper no. 10, pages 413--474.

Extended Bloch group and the Cheeger-Chern-Simons class

Walter D Neumann

Abstract. We define an extended Bloch group and show it is naturally 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-Chern-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 3-manifolds conjectured by Neumann and Zagier [Topology 1985] and proved by Yoshida [Invent. Math. 1985] as well as effective formulae for the Chern-Simons invariant of a hyperbolic 3-manifold.

Keywords. Extended Bloch group, Cheeger-Chern-Simons class, hyperbolic, 3-manifold

AMS subject classification. Primary: 57M27. Secondary: 19E99, 57T99.

DOI: 10.2140/gt.2004.8.413

E-print: arXiv:math.GT/0307092

Submitted to GT on 23 July 2003. (Revised 17 January 2004.) Paper accepted 14 February 2004. Paper published 14 February 2004.

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Walter D Neumann
Department of Mathematics, Barnard College
Columbia University, New York, NY 10027, USA

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