Generalized orientations and the Bloch invariant

For compact hyperbolic 3-manifolds we lift the Bloch invariant defined by Neumann and Yang to an integral class in K3(ℂ). Applying the Borel and the Bloch regulators, one gets back the volume and the Chern-Simons invariant of the manifold. We perform our constructions in stable homotopy theory, pushing a generalized orientation of the manifold directly into K-theory. On the way we give a purely homotopical construction of the Bloch-Wigner exact sequence which allows us to explain the ℚ/ℤ ambiguity that appears in the non-compact case. Copyright © ISOPP 2009.