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The Bloch invariant as a characteristic class in B(SLn₂(C),$\mathfrak{T}$)

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 $\alpha(M)$ in $\homo[3](B(\SLn[2]{\C},\famit))$ with $B(\SLn[2]{\C},\famit)$ the $\SLn[2]{\C}$-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 \cite{Neumann/Yang:BIHM}, 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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