The Bloch invariant as a characteristic class in $B(SL_2(\mathbb {C}), \mathfrak{ T})$.

Abstract

Given an orientable complete hyperbolic $3$-manifold of finite volume $M$ we construct a canonical class $\alpha(M)$ in $H_3(B(SL_2(\mathbb{C}),\mathfrak{T}))$ with $B(SL_2(\mathbb{C}),\mathfrak{T})$ the $SL_2(\mathbb{C})$-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 $B(\mathbb{C})$.