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Elliptic symbols, elliptic operators and Poincaré duality on conical pseudomanifolds

Published online by Cambridge University Press:  01 December 2008

Jean-Marie Lescure
Jean-Marie Lescure
Affiliation:
[email protected] de Mathématiques UMR6620Université Blaise PascalCompexe Universtaire des Cézeaux63177 AubiéreFrance
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Abstract

In [7], a notion of noncommutative tangent space is associated with a conical pseudomanifold and Poincaré duality in K-theory is proved between this space and the pseudomanifold. The present paper continues this line of work. We show that an appropriate presentation of the notion of symbol on a manifold generalizes right away to conical pseudomanifolds and that it enables us to interpret Poincaré duality in the singular setting as a noncommutative symbol map.

Keywords

Manifolds with isolated conical singularitiesbivariant K-theorygroupoidspseudodifferential calculus
Type
Research Article
Information
Journal of K-Theory , Volume 4 , Issue 2 , October 2009 , pp. 263 - 297
Copyright
Copyright © ISOPP 2009

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References

1.Anantharaman-Delaroche, C. and Renault, J.. Amenable groupoids, Contemp. Math. 282 Amer. Math. Soc., Providence, RI, 2001Google Scholar
2.Baaj, S. and Julg, P.. Théorie bivariante de Kasparov et opérateurs non bornés dans les C*-modules hilbertiens. C.R. Acad Sc. Paris Série I, 296:875, 878Google Scholar
3.Blackadar, B.. K-theory for operator algebras, Mathematical Sciences Research Institute Publications 5. Cambridge University Press, Cambridge, second edition, 1998Google Scholar
4.Brasselet, J.P., Hector, G., and Saralegi, M.. Théorème de de Rham pour les variétés stratifiées. Ann. Global Analy. Geom. 9 (3) (1991):211–243CrossRefGoogle Scholar
5.Connes, A.. Noncommutative Geometry. Academic Press, 1994Google Scholar
6.Connes, A. and Skandalis, G.. The longitudinal index theorem for foliations. Publ. R.I.M.S. Kyoto Univ. 20 (1984):1139–1183CrossRefGoogle Scholar
7.Debord, C. and Lescure, J.M.. K-duality for pseudomanifolds with isolated singularities. J. Funct. Anal. 219 (1) (2005):109–133CrossRefGoogle Scholar
8.Debord, C., M, J.M., and Nistor, V.. Groupoids and an index theorem for conical pseudo-manifolds. Preprint, http://arxiv.org/abs/math/0609438v1, 2006.Google Scholar
9.Elliott, G., Natsume, T., and Nest, R.. The Atiyah-Singer index theorem as passage to the classical limit in quantum mechanics. Comm. Math. Phys. 182 (3) (1996):505–533CrossRefGoogle Scholar
10.Getzler, E.. Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem. Commun. Math. Phys. 92 (1983):163–178CrossRefGoogle Scholar
11.Kasparov, G.G.. The operator K-functor and extensions of C*-algebras. Izv. Akad. Nauk SSSR, Ser. Math. 44 (1980):571–636Google Scholar
12.Kasparov, G.G.. Equivariant KK-theory and the Novikov conjecture. Invent. math. 91 (1988):147–201CrossRefGoogle Scholar
13.Lauter, R., Monthubert, B., and Nistor, V.. Spectral invariance for certain algebras of pseudodifferential operators. J. Inst. Math. Jussieu 4 (3) (2005):405–442CrossRefGoogle Scholar
14.Melrose, R.B.. The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics 4. A. K. Peters, Massachusetts, 1993Google Scholar
15.Melrose, R.B.. The eta invariant and families of pseudodifferential operators. Math. Res. Lett. 2 (5) (1995):541–561CrossRefGoogle Scholar
16.Monthubert, B.. Groupoids of manifolds with corners and index theory. In Groupoids in analysis, geometry, and physics (Boulder, CO, 1999), pages 147–157. Amer. Math. Soc., Providence, RI, 2001CrossRefGoogle Scholar
17.Monthubert, B.. Groupoids and pseudodifferential calculus on manifolds with corners. J. Funct. Anal. 199 (1) (2003):243–286CrossRefGoogle Scholar
18.Monthubert, B. and Pierrot, F.. Indice analytique et groupoïde de Lie. C.R.A.S Série 1 325 (1997):193–198Google Scholar
19.Nazaikinskii, V. E., Savin, A. Yu., and Sternin, B. Yu.. On the homotopy classification of elliptic operators on stratified manifolds. Preprint, math.KT/0608332, 2006CrossRefGoogle Scholar
20.Nazaikinskii, V. E., Savin, A. Yu., and Sternin, B. Yu.. Pseudodifferential operators on stratified manifolds. Preprint, math.AP/0512025, 2006Google Scholar
21.Nistor, V., Weinstein, A., and Xu, P.. Pseudodifferential operators on differential groupoids. Pacific J. of Math. 181 (1) (1999):117–152CrossRefGoogle Scholar
22.Savin, A.. Elliptic operators on manifolds with singularities and K-homology. K-Theory 34 (1) (2005):71–98CrossRefGoogle Scholar
23.Schulze, B.W.. Pseudodifferential boundary value problems, conical singularities, and asymptotics, mathematical topics 4. Akademie Verlag, 1994Google Scholar
24.Schulze, B.W.. Boundary value problems and singular pseudodifferential operators. Wiley-Intersciences, 1998Google Scholar
25.Shubin, M.A.. Pseudodifferential operators and spectral theory. Springer Verlag, 1980Google Scholar
26.Skandalis, G.. Kasparov's bivariant K-theory and applications. Expositiones mathematicae 9 (1991):193–250Google Scholar
27.Vassout, S.. Unbounded pseudodifferential calculus on Lie groupoids. Journal of Functional analysis 236 (1) (2006):161–200CrossRefGoogle Scholar
28.Vassout, S.. Feuilletages et Résidu non Commutatif Longitudinal. PhD thesis, Université Paris VI, 2001Google Scholar
29.Wegge-Olsen, N. E.. K-theory and C*-algebras. Oxford Science Publications. The Clarendon Press Oxford University Press, New York, 1993. A friendly approachCrossRefGoogle Scholar
30.Widom, H.. A complete symbolic calculus for pseudodifferential operators. Bull. Sci. Math. (2) 104 (1) (1980):19–63Google Scholar