Extended Bloch group and the Cheeger–Chern–Simons class

Neumann, Walter D

doi:10.2140/gt.2004.8.413

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Extended Bloch group and the Cheeger–Chern–Simons class

Walter D Neumann

Geometry & Topology 8 (2004) 413–474

DOI: 10.2140/gt.2004.8.413

arXiv: math.GT/0307092

Abstract

We define an extended Bloch group and show it is naturally isomorphic to $H_{3} (PSL {(2, ℂ)}^{δ}; ℤ)$ . 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 and proved by Yoshida, 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

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 19E99, 57T99

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Publication

Received: 23 July 2003

Revised: 17 January 2004

Accepted: 14 February 2004

Published: 14 February 2004

Proposed: Robion Kirby

Seconded: Shigeyuki Morita, Benson Farb

Authors

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

Volume 8, issue 1 (2004)