Abstract
Alain Connes introduced the use of Lie groupoids in noncommutative geometry in his pioneering work on the index theory of foliations. In the present paper, we recall the basic notion involved: groupoids, their C ∗-algebras, their pseudodifferential calculus, etc. We review several recent and older advances on the involvement of Lie groupoids in noncommutative geometry. We then propose some open questions and possible developments of the subject.
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Notes
- 1.
We may just assume that f satisfies the transversality condition.
- 2.
In [61] this element is just a morphism of K-groups since E-theory of Connes–Higson was defined later.
- 3.
The arrows are in fact E-theory elements or even KK-theory elements.
- 4.
We actually have to assume that the same holds also for the adjoint of Φ.
References
Johannes Aastrup, Severino T. Melo, Bertrand Monthubert, and Elmar Schrohe, Boutet de Monvel’s calculus and groupoids I., J. Noncommut. Geom. 4 no. 3 (2010), 313–329.
Rui Almeida and Pierre Molino, Suites d’Atiyah et feuilletages transversalement complets, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 1, 13–15. MR 778785
_________ , Flots riemanniens sur les 4-variétés compactes, Tohoku Math. J. (2) 38 (1986), no. 2, 313–326. MR 843815
C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique], vol. 36, L’Enseignement Mathématique, Geneva, 2000, With a foreword by Georges Skandalis and Appendix B by E. Germain. MR 1799683
Iakovos Androulidakis and Georges Skandalis, The holonomy groupoid of a singular foliation, J. Reine Angew. Math. 626 (2009), 1–37.
_________ , Pseudodifferential calculus on a singular foliation, J. Noncommut. Geom. 5 (2011), no. 1, 125–152.
Michael F. Atiyah, Elliptic operators discrete groups and von Neumann algebras, Astérisque 32–33 (1976), 43–72.
Michael F. Atiyah and Singer Isadore M., The index of elliptic operators. I, Ann. of Math. (2) 87 (1968), 484–530.
_________ , The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 0236952
_________ , The index of elliptic operators. IV, Ann. of Math. (2) 93 (1971), 119–138.
Paul Baum and Alain Connes, GeometricK-theory for Lie groups and foliations, Enseign. Math. (2) 46 (2000), no. 1-2, 3–42. MR 1769535
Paul Baum, Alain Connes, and Nigel Higson, Classifying space for proper actions andK-theory of groupC ∗-algebras, C ∗-algebras: 1943–1993 (San Antonio, TX, 1993), Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291. MR 1292018
Paul Baum, Erik Guentner, and Rufus Willett, Expanders, exact crossed products, and the Baum-Connes conjecture, Ann. K-Theory 1 (2016), no. 2, 155–208. MR 3514939
Richard Beals and Peter Greiner, Calculus on Heisenberg manifolds, Annals of Mathematics Studies, vol. 119, Princeton University Press, Princeton, NJ, 1988. MR 953082
Moulay Tahar Benameur and Indrava Roy, The Higson-Roe sequence for étale groupoids. I. dual algebras and compatibility with the BC map, ar**v:1801.06040.
Karsten Bohlen, Boutet de Monvel operators on singular manifolds, C. R. Math. Acad. Sci. Paris 354 (2016), no. 3, 239–243. MR 3463018
_________ , Boutet de Monvel operators on Lie manifolds with boundary, Adv. Math. 312 (2017), 234–285. MR 3635812
Louis Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), no. 1-2, 11–51.
_________ , A course on pseudo differential operators and their applications, Mathematics Department, Duke University, Durham, N.C., 1976, Duke University Mathematics Series, No. II.
Alcides Buss, Siegfried Echterhoff, and Rufus Willett, Exotic crossed products, Operator algebras and applications—the Abel Symposium 2015, Abel Symp., vol. 12, Springer, [Cham], 2017, pp. 67–114. MR 3837592
_________ , Exotic crossed products and the Baum-Connes conjecture, J. Reine Angew. Math. 740 (2018), 111–159. MR 3824785
P. Carrillo Rouse, J. M. Lescure, and B. Monthubert, A cohomological formula for the Atiyah-Patodi-Singer index on manifolds with boundary, J. Topol. Anal. 6 (2014), no. 1, 27–74. MR 3190137
Pierre Cartier, Groupoïdes de Lie et leurs algébroïdes, Astérisque (2009), no. 326, Exp. No. 987, viii, 165–196 (2010), Séminaire Bourbaki. Vol. 2007/2008. MR 2605322
Paul R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Analysis 12 (1973), 401–414. MR 0369890
Woocheol Choi and Raphael Ponge, Tangent maps and tangent groupoid for Carnot manifolds, ar**v:1510.05851, 2015.
_________ , Privileged coordinates and nilpotent approximation for Carnot manifolds, II. Carnot coordinates, ar**v:1703.05494, 2017.
_________ , Privileged coordinates and nilpotent approximation of Carnot manifolds, I. General results, ar**v:1709.09045, 2017.
Alain Connes, Sur la théorie non commutative de l’intégration, Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978), Lecture Notes in Math., vol. 725, Springer, Berlin, 1979, pp. 19–143.
_________ , An analogue of the Thom isomorphism for crossed products of aC ∗-algebra by an action ofR, Adv. in Math. 39 (1981), no. 1, 31–55.
_________ , A survey of foliations and operator algebras, Operator algebras and applications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 521–628.
_________ , Cyclic cohomology and the transverse fundamental class of a foliation, Geometric methods in operator algebras (Kyoto, 1983), Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech., Harlow, 1986, pp. 52–144. MR 866491 (88k:58149)
_________ , Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994.
Alain Connes and Nigel Higson, Déformations, morphismes asymptotiques etK-théorie bivariante, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 2, 101–106. MR 1065438
Alain Connes and Henri Moscovici, The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995), no. 2, 174–243. MR 1334867
_________ , Hopf algebras, cyclic cohomology and the transverse index theorem, Comm. Math. Phys. 198 (1998), no. 1, 199–246. MR 1657389
Alain Connes and Georges Skandalis, The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. 20 (1984), no. 6, 1139–1183. MR 775126
Marius Crainic and Rui Loja Fernandes, Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), no. 2, 575–620. MR 1973056
Claire Debord, Holonomy groupoids of singular foliations, J. Differential Geom. 58 (2001), no. 3, 467–500. MR 1906783
Claire Debord and Jean-Marie Lescure, In preparation.
_________ , K-duality for pseudomanifolds with isolated singularities, J. Funct. Anal. 219 (2005), no. 1, 109–133.
_________ , K-duality for stratified pseudomanifolds, Geom. Topol. 13 (2009), no. 1, 49–86. MR 2469513
_________ , Index theory and groupoids, Geometric and topological methods for quantum field theory, Cambridge Univ. Press, Cambridge, 2010, pp. 86–158.
Claire Debord, Jean-Marie Lescure, and Victor Nistor, Groupoids and an index theorem for conical pseudo-manifolds, J. Reine Angew. Math. 628 (2009), 1–35. MR 2503234
Claire Debord, Jean-Marie Lescure, and Frédéric Rochon, Pseudodifferential operators on manifolds with fibred corners, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 4, 1799–1880.
Claire Debord and Georges Skandalis, Adiabatic groupoid, crossed product by \(\mathbb {R}_+^\ast \)and pseudodifferential calculus, Adv. Math. 257 (2014), 66–91.
_________ , Blowup constructions for Lie groupoids and a Boutet de Monvel type calculus, ar**v:1705.09588, 2017.
_________ , Lie groupoids, exact sequences, Connes-Thom elements, connecting maps and index maps, Preprint (part of ar**v:1705.09588), 2017.
J. J. Duistermaat, Fourier integral operators, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2011, Reprint of the 1996 edition [MR1362544], based on the original lecture notes published in 1973 [MR0451313]. MR 2741911
J. J. Duistermaat and L. Hörmander, Fourier integral operators. II, Acta Math. 128 (1972), no. 3-4, 183–269. MR 0388464
George A. Elliott, Toshikazu Natsume, and Ryszard Nest, The Atiyah-Singer index theorem as passage to the classical limit in quantum mechanics, Comm. Math. Phys. 182 (1996), no. 3, 505–533. MR 1461941
Thierry Fack and Georges Skandalis, Connes’ analogue of the Thom isomorphism for the Kasparov groups, Invent. Math. 64 (1981), no. 1, 7–14.
Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 0578656
Damien Gaboriau, Orbit equivalence and measured group theory, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1501–1527. MR 2827853
Gerd Grubb, Functional calculus of pseudodifferential boundary problems, second ed., Progress in Mathematics, vol. 65, Birkhäuser Boston Inc., 1996.
André Haefliger, Groupoïdes d’holonomie et classifiants, Astérisque (1984), no. 116, 70–97, Transversal structure of foliations (Toulouse, 1982). MR 755163
Nigel Higson, Vincent Lafforgue, and Georges Skandalis, Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal. 12 (2002), no. 2, 330–354. MR 1911663 (2003g:19007)
Nigel Higson and John Roe, Map** surgery to analysis. I. Analytic signatures, K-Theory 33 (2005), no. 4, 277–299. MR 2220522
_________ , Map** surgery to analysis. II. Geometric signatures, K-Theory 33 (2005), no. 4, 301–324. MR 2220523
_________ , Map** surgery to analysis. III. Exact sequences, K-Theory 33 (2005), no. 4, 325–346. MR 2220524
_________ , K-homology, assembly and rigidity theorems for relative eta invariants, Pure and Applied Mathematics Quarterly 6 (2010), no. 2, 555–601.
Michel Hilsum and Georges Skandalis, MorphismesK-orientés d’espaces de feuilles et fonctorialité en théorie de Kasparov (d’après une conjecture d’A. Connes), Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 3, 325–390.
Lars Hörmander, Pseudo-differential operators and hypoelliptic equations, Singular integrals (Proc. Sympos. Pure Math., Vol. X, Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, pp. 138–183. MR 0383152
_________ , The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218. MR 0609014
_________ , The calculus of Fourier integral operators, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), Princeton Univ. Press, Princeton, N.J., 1971, pp. 33–57. Ann. of Math. Studies, No. 70. MR 0341193
_________ , Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 0388463
_________ , The analysis of linear partial differential operators. I, Classics in Mathematics, Springer-Verlag, Berlin, 2003, Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]. MR 1996773
_________ , The analysis of linear partial differential operators. II, Classics in Mathematics, Springer-Verlag, Berlin, 2005, Differential operators with constant coefficients, Reprint of the 1983 original. MR 2108588
_________ , The analysis of linear partial differential operators. III, Classics in Mathematics, Springer, Berlin, 2007, Pseudo-differential operators, Reprint of the 1994 edition. MR 2304165
_________ , The analysis of linear partial differential operators. IV, Classics in Mathematics, Springer-Verlag, Berlin, 2009, Fourier integral operators, Reprint of the 1994 edition. MR 2512677
Pierre Julg and Erik van Erp, The geometry of the osculating nilpotent group structures of the Heisenberg calculus, J. Lie Theory 28 (2018), no. 1, 107–138.
Gennadi G. Kasparov, Topological invariants of elliptic operators I: K-homology, Izv. Akad. Nauk. S.S.S.R. Ser. Mat. 39 (1975), 796–838.
_________ , The operatorK-functor and extensions ofC ∗-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 3, 571–636.
Gennadi G. Kasparov and Georges Skandalis, Groups acting on buildings, operator K-theory and Novikov’s conjecture, K-theory 4 (1991), 303–337.
Jean-Marie Lescure, Triplets spectraux pour les variétés à singularité conique isolé e, Bull. Soc. math. France 129 (2001), no. 4, 593–623.
Jean-Marie Lescure, Dominique Manchon, and Stéphane Vassout, About the convolution of distributions on groupoids, J. Noncommut. Geom. 11 (2017), no. 2, 757–789. MR 3669118
Jean-Marie Lescure and Stéphane Vassout, Fourier integral operators on Lie groupoids, Adv. Math. 320 (2017), 391–450. MR 3709110
Pedro T. P. Lopes and Severino T. Melo, K-theory of the Boutet de Monvel algebra with classical SG-symbols on the half space, Math. Nachr. 287 (2014), no. 16, 1804–1827. MR 3274491
Kirill C. H. Mackenzie, General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, vol. 213, Cambridge University Press, Cambridge, 2005.
Rafe Mazzeo, Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations 16 (1991), no. 10, 1615–1664. MR 1133743
Rafe R. Mazzeo and Richard B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), no. 2, 260–310.
_________ , Pseudodifferential operators on manifolds with fibred boundaries, Asian J. Math. 2 (1998), no. 4, 833–866, Mikio Sato: a great Japanese mathematician of the twentieth century.
Severino T. Melo, Thomas Schick, and Elmar Schrohe, Families index for Boutet de Monvel operators, Münster J. Math. 6 (2013), no. 2, 343–364. MR 3148215
Richard B. Melrose, Pseudodifferential operators, corners and singular limits, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, pp. 217–234. MR 1159214
_________ , Calculus of conormal distributions on manifolds with corners, Internat. Math. Res. Notices (1992), no. 3, 51–61. MR 1154213
_________ , The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, vol. 4, A K Peters Ltd., Wellesley, MA, 1993.
Richard B. Melrose and Paolo Piazza, AnalyticK-theory on manifolds with corners, Adv. Math. 92 (1992), no. 1, 1–26. MR 1153932
A. S. Mishchenko and A. T. Fomenko, The index of elliptic operators over C*-algebras, Izv. Akad. Nauk. S.S.S.R. Ser. Mat. 43 (1979), no. 4, 831–859.
Omar Mohsen, On the deformation groupoid of the inhomogeneous pseudo-differential calculus, ar**v:1806.08585, 2018.
Bertrand Monthubert, Pseudodifferential calculus on manifolds with corners and groupoids, Proc. Amer. Math. Soc. 127 (1999), no. 10, 2871–2881.
_________ , Groupoids and pseudodifferential calculus on manifolds with corners, J. Funct. Anal. 199 (2003), no. 1, 243–286.
Bertrand Monthubert and François Pierrot, Indice analytique et groupoïdes de Lie, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 2, 193–198.
Calvin C. Moore and Claude L. Schochet, Global analysis on foliated spaces, second ed., Mathematical Sciences Research Institute Publications, vol. 9, Cambridge University Press, New York, 2006. MR 2202625
Paul S. Muhly, Jean N. Renault, and Dana P. Williams, Equivalence and isomorphism for groupoidC ∗-algebras, J. Operator Theory 17 (1987), no. 1, 3–22.
F. J. Murray and J. Von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936), no. 1, 116–229. MR 1503275
Victor Nistor, Alan Weinstein, and ** Xu, Pseudodifferential operators on differential groupoids, Pacific J. Math. 189 (1999), no. 1, 117–152.
William L. Paschke, K-theory for commutants in the Calkin algebra, Pacific J. Math. 95 (1981), no. 2, 427–434. MR 632196
Alan L. T. Paterson, Continuous family groupoids, Homology Homotopy Appl. 2 (2000), 89–104. MR 1782594
Paolo Piazza and Thomas Schick, The surgery exact sequence, K-theory and the signature operator, Ann. K-Theory 1 (2016), no. 2, 109–154. MR 3514938
Paolo Piazza and Vito Felice Zenobi, Singular spaces, groupoids and metrics of positive scalar curvature, ar**v:1803.02697, 2018.
Raphaël Ponge, The tangent groupoid of a Heisenberg manifold, Pacific J. Math. 227 (2006), no. 1, 151–175.
Jean Pradines, Feuilletages: holonomie et graphes locaux, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 13, 297–300. MR 765427
Jean Renault, A groupoid approach toC ∗-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980.
_________ , C ⋆-algebras and dynamical systems, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2009, 27o Colóquio Brasileiro de Matemática. [27th Brazilian Mathematics Colloquium]. MR 2536186
John Roe, Lectures on coarse geometry, University Lecture Series, vol. 31, American Mathematical Society, 2003.
Anton Savin, Elmar Schrohe, and Sternin Boris, Elliptic operators associated with groups of quantized canonical transformations, ar**v:1612.02981.
Elmar Schrohe, A short introduction to Boutet de Monvel’s calculus, Approaches to singular analysis (Berlin, 1999), Oper. Theory Adv. Appl., vol. 125, Birkhäuser, Basel, 2001, pp. 85–116.
M. A. Shubin, Pseudodifferential operators and spectral theory, second ed., Springer-Verlag, Berlin, 2001, Translated from the 1978 Russian original by Stig I. Andersson. MR 1852334
Georges Skandalis, Jean-Louis Tu, and Guoliang Yu, The coarse Baum-Connes conjecture and groupoids, Topology 41 (2002), 807–834.
Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, vol. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463
François Trèves, Introduction to pseudodifferential and Fourier integral operators. Vol. 1, Plenum Press, New York-London, 1980, Pseudodifferential operators, The University Series in Mathematics. MR 597144
_________ , Introduction to pseudodifferential and Fourier integral operators. Vol. 2, Plenum Press, New York-London, 1980, Fourier integral operators, The University Series in Mathematics. MR 597145
Alain Valette, Introduction to the Baum-Connes conjecture, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2002, From notes taken by Indira Chatterji, With an appendix by Guido Mislin. MR 1907596
Erik van Erp, The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part I, Ann. of Math. (2) 171 (2010), no. 3, 1647–1681. MR 2680395
Erik van Erp and Robert Yuncken, A groupoid approach to pseudodifferential operators, J. Reine Agnew. Math. 2019 (2019), no. 756, 151–182.
_________ , On the tangent groupoid of a filtered manifold, Bull. Lond. Math. Soc. 49 (2017), no. 6, 1000–1012.
Stéphane Vassout, Unbounded pseudodifferential calculus on Lie groupoids, J. Funct. Anal. 236 (2006), no. 1, 161–200. MR 2227132
Dan Voiculescu, A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 1, 97–113. MR 0415338
H. E. Winkelnkemper, The graph of a foliation, Ann. Global Anal. Geom. 1 (1983), no. 3, 51–75. MR 739904
Mayuko Yamashita, A topological approach to indices of geometric operators on manifolds with fibered boundaries, ar**v:1902.03767.
Vito Felice Zenobi, The adiabatic groupoid and the Higson-Roe exact sequence, ar**v:1901.05081.
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The authors were partially supported by ANR-14-CE25-0012-01 (SINGSTAR).
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Debord, C., Skandalis, G. (2019). Lie groupoids, pseudodifferential calculus, and index theory. In: Chamseddine, A., Consani, C., Higson, N., Khalkhali, M., Moscovici, H., Yu, G. (eds) Advances in Noncommutative Geometry. Springer, Cham. https://doi.org/10.1007/978-3-030-29597-4_4
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