Homology, Homotopy and Applications
Volume 5 (2003)
The Bloch invariant as a characteristic class in $B(SL_2(C),T)$
Pages: 325 – 344
Given an orientable complete hyperbolic 3-manifold of finite volume $M$ we construct a canonical class $\alpha(M)$ in $H_3(B(SL_2(C),T))$ with $B(SL_2(C),T)$ the $SL_2(C)$-orbit space of the classifying space for a certain family of isotropy subgroups. We prove that $\alpha (M)$ coincides with the Bloch invariant $\beta(M)$ 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 $B(C)$.
hyperbolic 3-manifolds, Classifying space for families of isotropy subgroups, Bloch group, Bloch invariant, acyclic maps
2010 Mathematics Subject Classification
19Dxx, 19Exx, 53-xx, 57M27