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# Homology, Homotopy and Applications

## Volume 5 (2003)

### Number 1

### The Bloch invariant as a characteristic class in $B(SL_2(C),T)$

Pages: 325 – 344

DOI: http://dx.doi.org/10.4310/HHA.2003.v5.n1.a14

#### Authors

#### Abstract

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 [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 $B(C)$.

#### Keywords

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