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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
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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References

ABS64.Atiyah, M. F., Bott, R., and Shapiro, A.. Clifford modules. >Topology 3 (suppl. 1):3–38, 1964.Google Scholar
Ada66.Adams, J. F.. On the groups J.(X). IV. >Topology 5:21–71, 1966.CrossRefGoogle Scholar
Ada74.Adams, J. F.. Stable homotopy and generalised homology. University of Chicago Press, Chicago, Ill., 1974. Chicago Lectures in Mathematics.Google Scholar
Bor77.Borel, A.. Cohomologie de SLn et valeurs de fonctions zeta aux points entiers. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4(4):613–636, 1977.Google Scholar
CMJ03.Cisneros-Molina, J. L. and Jones, J. D. S.. The Bloch invariant as a characteristic class in B(SL2(ℂ), T). >Homology Homotopy Appl. 5(1):325–344 (electronic), 2003.CrossRefGoogle Scholar
CS83.Culler, M. and Shalen, P. B.. Varieties of group representations and splittings of 3-manifolds. Ann. of Math. (2) 117(1):109–146, 1983.CrossRefGoogle Scholar
DS82.Dupont, J. L. and Sah, C. H.. Scissors congruences. II. J. Pure Appl. Algebra 25(2):159–195, 1982.CrossRefGoogle Scholar
Dup01.Dupont, J. L.. Scissors congruences, group homology and characteristic classes, volume 1 of Nankai Tracts in Mathematics. World Scientific Publishing Co. Inc., River Edge, NJ, 2001.Google Scholar
DZ06.Dupont, J. L. and Zickert, C. K.. A dilogarithmic formula for the Cheeger-Chern- Simons class. >Geom. Topol. 10:1347–1372 (electronic), 2006.CrossRefGoogle Scholar
FJ93.Farrell, F. T. and Jones, L. E.. Isomorphism conjectures in algebraic K-theory. >J. Amer. Math. Soc. 6(2):249–297, 1993.Google Scholar
GZ07.Goette, S. and Zickert, C. K.. The extended Bloch group and the Cheeger-Chern- Simons class. >Geom. Topol. 11:1623–1635 (electronic), 2007.CrossRefGoogle Scholar
Gon99.Goncharov, A.. Volumes of hyperbolic manifolds and mixed Tate motives. J. Amer. Math. Soc. 12(2):569–618, 1999.Google Scholar
Lod76.Loday, J.-L.. K-théorie algébrique et représentations de groupes. Ann. Sci. École Norm. Sup. (4) 9(3):309–377, 1976.CrossRefGoogle Scholar
May80.May, J. P.. Pairings of categories and spectra. >J. Pure Appl. Algebra 19:299–346, 1980.Google Scholar
MO02.Matthey, M. and Oyono-Oyono, H.. Algebraic K-theory in low degree and the Novikov assembly map. Proc. London Math. Soc. (3) 85(1):43–61, 2002.Google Scholar
MS60.Milnor, J. and Spanier, E.. Two remarks on fiber homotopy type. Pacific J. Math. 10:585–590, 1960.CrossRefGoogle Scholar
MS74.Milnor, J. W. and Stasheff, J. D.. Characteristic classes. Princeton University Press, Princeton, N. J., 1974. Annals of Mathematics Studies 76.Google Scholar
Neu04.Neumann, W. D.. Extended Bloch group and the Cheeger-Chern-Simons class. >Geom. Topol. 8:413–474.(electronic), 2004.Google Scholar
NY99.Neumann, W. D. and Yang, J.. Bloch invariants of hyperbolic 3-manifolds. >Duke Math. J. 96(1):29–59, 1999.Google Scholar
Rat94.Ratcliffe, J. G.. Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149. Springer-Verlag, New York, 1994.CrossRefGoogle Scholar
Rud98.Rudyak, Y. B.. On Thom spectra, orientability, and cobordism. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. With a foreword by Haynes Miller.Google Scholar
Sus84.Suslin, A. A.. Homology of GLn, characteristic classes and Milnor K-theory in Algebraic K-theory, number theory, geometry and analysis (Bielefeld, 1982), Lecture Notes in Mathematics 1046. Springer-Verlag, Berlin, 1984.Google Scholar
Sus90.Suslin, A. A.. K 3 of a field, and the Bloch group. Trudy Mat. Inst. Steklov. 183:180–199, 229, 1990. Translated in Proc. Steklov Inst. Math. 1991, no. 4, 217–239, Galois theory, rings, algebraic groups and their applications (Russian).Google Scholar
Swi02.Switzer, R. M.. Algebraic topology—homotopy and homology. Classics in Mathematics. Springer-Verlag, Berlin, 2002. Reprint of the 1975 original.Google Scholar
Thu97.Thurston, W. P.. Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series 35. Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy.CrossRefGoogle Scholar