INVARIANTS FROM TRIANGULATIONS OF HYPERBOLIC 3 - MANIFOLDS 3 Theorem

For any nite volume hyperbolic 3-manifold M we use ideal tri-angulation to deene an invariant (M) in the Bloch group B(C). It actually lies in the subgroup of B(C) determined by the invariant trace eld of M. The Chern-Simons invariant of M is determined modulo rationals by (M). This implies rationality and | assuming the Ramakrishnan conjecture | ir-rationality results for Chern Simons invariants. The pre-Bloch group P(k) of a eld k is the quotient of the free Z-module Z(k ? f0; 1g) by all instances of the following relations: Misprint in (1) in the published paper corrected here 03/96 (2) The Bloch group B(k) is the kernel of the map : P(k) ! k ^ Z k given by ((z]) = 2(z ^ (1 ? z)): (There are several variants of this deenition in the literature. Dupont and Sah 6] show that the various deenitions diier at most by torsion and that they agree with each other for algebraically closed elds. See also the discussion in 15].) By results of Borel, Bloch, and Suslin 2, 1, 20] (see Theorem 4.1) the Bloch group B(k) of a number eld is isomorphic modulo torsion to Z r 2 , where r 2 is the number of complex embeddings of k (a complex embedding is an embedding k ! C with image not in R). Thus B(k) Q = Q r 2. Moreover, if a speciic complex embedding is chosen, that is, k is given as a subbeld of C , then the induced map B(k) Q ! B(C) Q is injective, so we may write B(k) Q B(C) Q. Let M = H 3 =? be an oriented complete hyperbolic manifold of nite volume (brieey just \hyperbolic 3-manifold" from now on). The invariant trace eld k(M) = k(?) is the eld generated over Q by squares of traces of elements of ?. It is the smallest eld among trace elds of nite index subgroups of ? ((19], see also 13]). It is a number eld and comes with a speciic embedding in C .