Given an orientable complete hyperbolic 3-manifold of finite volume M we construct a canonical class α(M) in H3(B(SL2(C),T)) with B(SL2(C),T) the SL2(C)-orbit space of the classifying space for a certain family of isotropy subgroups. We prove that α(M) coincides with the Bloch invariant β(M) of M defined by Neumann and Yang in , giving with this a simpler proof that the Bloch invariant is independent of an ideal triangulation of M. We also give a new proof of the fact that the Bloch invariant lies in the Bloch group B(C).
Homology, Homotopy and Applications, Vol. 5(2003), No. 1, pp. 325-344
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