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An interesting example for spectral invariants

Published online by Cambridge University Press:  03 April 2014

Moulay-Tahar Benameur ,
James L. Heitsch  and
Charlotte Wahl
Moulay-Tahar Benameur
Affiliation:
I3M UMR 5149 du CNRS, Université de Montpellier 2, 34095 Montpellier, France, moulay.benameur@univ–montp2.fr
James L. Heitsch
Affiliation:
Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Mathematics, Northwestern University, USA, heitsch@math.uic.edu
Charlotte Wahl
Affiliation:
Leibniz-Archiv, Waterloostr. 8, 30169 Hannover, Germany, wahlcharlotte@gmail.com
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Abstract

In [HL99], the heat operator of a Bismut superconnection for a family of generalized Dirac operators is defined along the leaves of a foliation with Hausdorff groupoid. The Novikov-Shubin invariants of the Dirac operators were assumed greater than three times the codimension of the foliation. It was then shown that the associated heat operator converges to the Chern character of the index bundle of the operator. In [BH08], this result was improved by reducing the requirement on the Novikov-Shubin invariants to one half of the codimension. In this paper, we construct examples which show that this is the best possible result.

Keywords

spectral invariantsfoliationsDirac operatorsindex theory58J2219K5646L8058J42
Type
Research Article
Information
Journal of K-Theory , Volume 13 , Issue 2 , April 2014 , pp. 305 - 311
Copyright
Copyright © ISOPP 2014 

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References

REFERENCES

A76.Atiyah, M. F., Elliptic operators, discrete groups and von Neumann algebras. Astérisque 32/33 (1976), 43–72.Google Scholar
BH04.Benameur, M-T. and Heitsch, J. L., Index theory and non-commutative geometry, I: Higher families index theory. K-Theory 33 (2004), 151–183.Google Scholar
BH08.Benameur, M-T. and Heitsch, J. L., Index theory and non-commutative geometry, II: Dirac operators and index bundles. J. K-Theory 1 (2008), 305–356.Google Scholar
BH11.Benameur, M-T. and Heitsch, J. L., The twisted higher harmonic signature for foliations. J. Diff. Geo. 87 (2011), 389–468.Google Scholar
C86.Connes, A., Cyclic cohomology and the transverse fundamental class of a foliation. Research Notes in Mathematics 123, Pitman, London, 1986.Google Scholar
C87.Connes, A., Cyclic cohomology and noncommutative differential geometry. Proc. Int. Congress, Berkeley 1986, 879–889. Amer. Math. Soc., Providence, R.I., 1987.Google Scholar
CS84.Connes, A. and Skandalis, G., The longitudinal index theorem for foliations. Publ. Res. Inst. Math. Sci. Kyoto 20 (1984), 1139–1183.Google Scholar
EH86.Kacimi-Alaoui, A. El and Hector, G., Decomposition de Hodge basique pour un feuilletage riemannien. Ann. Inst. Fourier (Grenoble) 36 (1986), 207–227.Google Scholar
EHS85.Kacimi-Alaoui, A. El, Hector, G. and Sergiescu, V., La cohomologie basique d'un feuilletage riemannien est de dimension finie. Math. Z. 188 (1985), 593–599.Google Scholar
HL94.Heitsch, J. L. and Lazarov, C., Spectral asymptotics of foliated manifolds. Illinois J. of Math. 38 (1994), 653–678.Google Scholar
HL99.Heitsch, J. L. and Lazarov, C., A general families index theorem. K-Theory 18 (1999), 181–202.Google Scholar
NS86a.Novikov, S. P. and Shubin, M. A., Morse theory and von Neumann II1 factors. Doklady Akad. Nauk SSSR 289 (1986), 289–292.Google Scholar
NS86b.Novikov, S. P. and Shubin, M. A., Morse theory and von Neumann invariants on non-simply connected manifolds. Uspekhi Nauk SSSR 41 (1986), 222–223.Google Scholar