| Extended Bloch Group And The Chern-Simons Class (2007) | |||||||||||||||
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| . We define an extended Bloch group and show it is isomorphic to H 3 (PSL(2; C ) ffi ; 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 [14] and proved in [17], as well as an effective formula for the Chern-Simons invariant of a hyperbolic manifold. 1. Introduction There are several variations of the definition of the Bloch group in the literature; by [7] they differ at most by torsion and they agree with each other for algebraically closed fields. In this paper we shall use the following. Definition 1.1. Let k be a field. The pre-Bloch group P(k) is the quotient of the free Z-module Z(k \Gamma f0; 1g) by all instances of the following relation: [x] \Gamma [y] + [ y x ] \Gamma [ 1 \Gamma x \Gamma1 1 \Gamma y \Gamma1 ] + [ 1 \Gamma x ... | |||||||||||||||
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