Extended Bloch group and the Cheeger–Chern–Simons class

Neumann, Walter D¹

¹ Department of Mathematics, Barnard College, Columbia University, New York, New York 10027, USA

Geometry & topology, Tome 8 (2004) no. 1, pp. 413-474.

Résumé

We define an extended Bloch group and show it is naturally isomorphic to $H_3\left(PSL{\left(2,ℂ\right)}^{\delta };\mathbb{Z}\right)$. 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.

Détail

DOI : 10.2140/gt.2004.8.413

Keywords: extended Bloch group, Cheeger–Chern–Simons class, hyperbolic, 3–manifold

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