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Generalized orientations and the Bloch invariant

Published online by Cambridge University Press:  18 November 2009

Michel Matthey ,
Wolfgang Pitsch  and
Jérôme Scherer
Wolfgang Pitsch
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain, pitsch@mat.uab.es
Jérôme Scherer
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain, jscherer@mat.uab.es
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Abstract

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.

Keywords

Hyperbolic manifoldgeneralized orientationscissors congruencehyperbolic volumeBloch invariant

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Type
Research Article
Information
Journal of K-Theory , Volume 6 , Issue 2 , October 2010 , pp. 241 - 261
Copyright
Copyright © ISOPP 2009

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