The Bloch invariant as a characteristic class in B(SL₂(C),T)

Jose Luis Cisneros-Molina and John D. S. Jones

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 [13], 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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