Elliptic operators and higher signatures

Leichtnam, Éric; Piazza, Paolo

doi:10.5802/aif.2049

Elliptic operators and higher signatures
[Opérateurs elliptiques et hautes signatures]

Leichtnam, Éric ; Piazza, Paolo

Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1197-1277.

Résumé
Abstract

En s'appuyant sur la théorie des opérateurs elliptiques, nous donnons une approche unifiée des sujets suivants : - le problème de l'invariance par homotopie des hautes signatures de Novikov des variétés compactes orientées sans bord, - le problème de l'invariance par coupure et collage des hautes signatures de Novikov des variétés compactes orientées sans bord, - le problème de définir les hautes signatures de variétés à bord et de prouver leur invariance par homotopie.

Building on the theory of elliptic operators, we give a unified treatment of the following topics: - the problem of homotopy invariance of Novikov's higher signatures on closed manifolds, - the problem of cut-and-paste invariance of Novikov's higher signatures on closed manifolds, - the problem of defining higher signatures on manifolds with boundary and proving their homotopy invariance.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.2049

Classification : 19E20, 53C05, 58J05, 58J28

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Leichtnam, Éric; Piazza, Paolo. Elliptic operators and higher signatures. Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1197-1277. doi : 10.5802/aif.2049. https://www.numdam.org/articles/10.5802/aif.2049/

Bibliographie
Cité par

[1] M. Atiyah, Global theory of elliptic operators, Univ. of Tokyo Press, 1970, p. 21-30 | Zbl

[2] M.F. Atiyah & R. Bott, The index theorem for manifolds with boundary, 1964, p. 175-186 | Zbl

[3] M.F. Atiyah, H. Donnelly & I.M. Singer, Eta invariants, signature defects of cusps and values of L-functions, Ann. of Math. 118 (1983) p. 131-177 | MR | Zbl

[4] M.F. Atiyah, H. Donnelly & I.M. Singer, Signature defects of cusps and values of L-functions: the non-split case., Ann. of Math 119 (1984) p. 635-637 | MR | Zbl

[5] M.F. Atiyah, V.K. Patodi & I.M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Phil. Soc 77 (1975) p. 43-69 | MR | Zbl

[6] M.F. Atiyah, V.K. Patodi & I.M. Singer, Spectral asymmetry and Riemannian geometry. II, Math. Proc. Camb. Phil. Soc 78 (1975) p. 405-432 | MR | Zbl

[7] M.F. Atiyah, V.K. Patodi & I.M. Singer, Spectral asymmetry and Riemannian geometry. III, Math. Proc. Camb. Phil. Soc 79 (1976) p. 71-99 | MR | Zbl

[8] M. Atiyah & I. Singer, The index of elliptic operators. IV., Annals of Math. (2) 93 (1971) p. 119-138 | MR | Zbl

[9] M. Atiyah & I. Singer, Index theory of skew-adjoint Fredholm operators, Inst. Hautes Études Sci. Publ. Math 37 (1969) p. 305-326 | Numdam | MR | Zbl

[10] P. Baum & A. Connes, Leafwise homotopy equivalence and rational Pontrjagin classes, Foliations, Advances Studies in Pure Math. 5, North-Holland, 1985, p. 1-14 | Zbl

[11] P. Baum, A. Connes & N. Higson, Classifying Space for proper actions and $K -$ Theory of group $C^{*} -$ algebras, Contemporary Mathematics 167 (1994) p. 241-291 | MR | Zbl

[12] P. Baum & R. Douglas, $K$ homology and index theory, Proc. Sympos. Pure Math. Soc. 38, Amer. Math. Soc., 1982, p. 117-173 | Zbl

[13] N. Berline, E. Getzler & M. Vergne, Heat kernels and Dirac operators, 298, Springer Verlag, 1992 | MR | Zbl

[14] J.-M. Bismut, The Atiyah-Singer index theorem for families of Dirac operators: two heat-equation proofs, Inv. Math 83 (1986) p. 91-151 | MR | Zbl

[15] J.-M. Bismut & J. Cheeger, $η$ -Invariants and their adiabatic limits, Jour. of the Amer. Math. Soc 2 (1989) p. 33-70 | MR | Zbl

[16] J.-M. Bismut & J. Cheeger, Families index for manifolds with boundary, superconnections and cones I, Jour. Funct. Anal. 89 (1990) p. 313-363 | MR | Zbl

[17] J.-M. Bismut & D.S. Freed, The analysis of elliptic families: Metrics and connections on determinant bundles, Comm. Math. Phys 106 (1986) p. 159-176 | MR | Zbl

[18] J.-M. Bismut & D.S. Freed, The analysis of elliptic families: Dirac operators, eta invariants and the holonomy theorem of Witten, Comm. Math. Phys. 107 (1986) p. 103-163 | MR | Zbl

[19] B. Booss-Bavnbek & K. Wojciechowski, Elliptic boundary problems for Dirac operators, Mathematics : theory and applications, Birkhäuser, 1993 | Zbl

[20] L. Boutet De Monvel, Boundary problems for pseudodifferential operators, Acta Math. 126 (1971) p. 11-51 | MR | Zbl

[21] P. Brown, R. Douglas & P. Fillmore, Unitary equivalence modulo the compact operators and extensions of $C^{*} -$ algebras, Lecture Notes in Math 345, Springer Verlag, 1973, p. 58-128 | Zbl

[22] J. Brüning & R. Seeley, An index theorem for first order regular singular operators, Amer. J. Math 110 (1988) p. 659-714 | MR | Zbl

[23] U. Bunke, On the gluing problem for the $η -$ invariant, Journal of Differential Geometry 41 (1995) p. 397-448 | MR | Zbl

[24] J. Cheeger, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A 76 (1979) no.5 p. 2103-2106 | MR | Zbl

[25] J. Cheeger, Spectral geometry of singular Riemann spaces, J. Differential Geom. 18 (1983) p. 575-657 | MR | Zbl

[26] A. Chou, The Dirac operator on spaces with conical singularities and positive scalar curvatures, Trans. Amer. Math. Soc. 289 (1985) no.1 p. 1-40 | MR | Zbl

[27] A. Connes, Noncommutative Geometry. Part I: The Chern character in $K -$ Homology. Part II de Rham homology and noncommutative algebras., Preprint I.H.E.S, 1983

[28] A. Connes, Noncommutative Geometry, Academic Press, 1994 | MR | Zbl

[29] A. Connes & H. Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29 (1990) p. 345-388 | MR | Zbl

[30] A. Connes, M. Gromov & H. Moscovici, Conjecture de Novikov et fibrés presque plats, C.R. Acad. Sci. Paris Sér. I Math 310 (1990) no.5 p. 273-277 | MR | Zbl

[31] A. Connes, M. Gromov & H. Moscovici, Group cohomology with Lipschitz control and higher signatures, Geom. Funct. Anal 3 (1993) no.1 p. 1-78 | MR | Zbl

[32] J. Cuntz, Noncommutative simplicial complexes and the Baum-Connes conjecture, Geom. Funct. Anal 12 (2002) no.2 p. 307-329 | MR | Zbl

[33] X. Dai & W. Zhang, Splitting the family index, Comm. Math. Phys 182 (1996) p. 303-318 | MR | Zbl

[34] X. Dai & W. Zhang, Higher spectral flow, Journal of Funct. Analysis 157 (1998) p. 432-469 | MR | Zbl

[35] X. Dai & W. Zhang, Real embeddings and the Atiyah-Patodi-Singer index theorem for Dirac operators, Loo-Keng Hua: a great mathematician of the twentieth century, Asian J. Math 4 (2000) no.4 p. 775-794 | MR | Zbl

[36] M. Farber & S. Weinberger, On the zero-in-the-spectrum conjecture, Ann. of Math. (2) 154 (2001) no.1 p. 139-154 | MR | Zbl

[37] S. Ferry, A. Ranicki & J. Rosenberg, A history and survey of the Novikov conjecture., Lecture Note Ser. 226, Cambridge Univ. Press, 1995, p. 7-66 | Zbl

[38] E. Getzler, Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem, Comm. Math. Phys 92 (1983) no.2 p. 163-178 | MR | Zbl

[39] E. Ghys, Les groupes hyperboliques, Astérisque 189-190, 1990, p. 203-238 | Numdam | Zbl

[40] E. Ghys & P. De La Harpe, Hyperbolic groups in the theory of Mikhael Gromov, Birkhäuser, 1990

[41] D. Grieser, Basics of the $b$ -calculus, Oper. Theory Adv. Appl 125, Birkhäuser, 1999, p. 30-84 | Zbl

[42] A. Gorokhovsky & J. Lott, Local index theory over etale groupoids, J. Reine angew. Math 560 (2003) p. 151-198 | MR | Zbl

[43] M. Gromov, Hyperbolic groups, Math. Sci. Res. Inst. Publ 8, Springer, 1987, p. 75-263 | Zbl

[44] M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, Progress in Mathematics, Birkäuser, 1995 | Zbl

[45] M. Gromov & M. Shubin, Von Neumann spectra near zero, Geom. Funct. Anal 1 (1991) no.4 p. 375-404 | MR | Zbl

[46] E. Guentner, N. Higson & S. Weinberger, The Novikov Conjecture for linear groups, Preprint, 2003 | Numdam | Zbl

[47] N. Higson, A Primer in KK-Theory, Proc. Sympos. Pure Math. 51, 1990, p. 239-283 | Zbl

[48] N. Higson & G. Kasparov, Operator $K$ -theory for groups which act properly and isometrically on Hilbert space, Electron. Res. Announc. Amer. Math. Soc 3 (1997) p. 131-142 | MR | Zbl

[49] N. Higson & G. Kasparov, $E$ -theory and $K K$ -theory for groups which act properly and isometrically on Hilbert space, Invent. Math. 144 (2001) no.1 p. 23-74 | MR | Zbl

[50] N. Higson & J. Roe, John Analytic $K$ -homology, Oxford University Press, Oxford Mathematical Monographs, Oxford Science Publications, 2000 | MR | Zbl

[51] N. Higson, J. Roe & T. Schick, Spaces with vanishing $l^{2}$ -homology and their fundamental groups (after Farber and Weinberger), Geom. Dedicata 87 (2001) no.1-3 p. 335-343 | MR | Zbl

[52] M. Hilsum, Index classes of Hilbert modules with boundary, Preprint Paris 6, March 2001 | MR

[53] M. Hilsum & G. Skandalis, Invariance de la signature à coefficients dans un fibré presque plat, J. Reine Angew. math 423 (1990) p. 73-99 | MR | Zbl

[54] M. Hirsch, Differential topology, Graduate texts in mathematics 33, Springer-Verlag, 1976 | Zbl

[55] F. Hirzebruch, The signature theorem: reminiscences and recreation, Ann. of Math. Studies 70, Princeton Univ. Press, 1970, p. 3-31 | Zbl

[56] R. Ji, Smooth dense subalgebras of reduced group $C^{*}$ -algebras, Schwartz cohomology of groups, and cyclic cohomology, J. Funct. Anal. 107 (1992) no.1 p. 1-33 | MR | Zbl

[57] M. Joachim & T. Schick, Positive and negative results concerning the Gromov-Lawson-Rosenberg conjecture, Contemp. Math 258, Amer. Math. Soc., 2000, p. 213-226 | Zbl

[58] P. Jolissaint, Rapidly decreasing functions in reduced $C^{*}$ -algebras of groups, Trans. Amer. Math. Soc 317 (1990) no.1 p. 167-196 | MR | Zbl

[59] M. Karoubi, Homologie cyclique et $K$ -théorie, Astérisque 149 (1987) | Numdam | MR | Zbl

[60] J. Kaminker & J. Miller, Homotopy invariance of the analytic index of signature operators over $C^{*} -$ algebras, J. Operator Theory 14 (1985) p. 113-127 | MR | Zbl

[61] U. Karras, M. Kreck, W. Neumann & E. Ossa, Cutting and pasting of manifolds; SK-groups, Publish or Perish, 1973 | MR | Zbl

[62] G. Kasparov, Topological invariants of elliptic operators $K$ -homology (Russian), Math. USSR-Izv 9 (1975) no.4 p. 751-792 | MR | Zbl

[64] G. Kasparov, Equivariant $K K$ -theory and the Novikov conjecture, Invent. Math 91 (1988) no.1 p. 147-201 | MR | Zbl

[65] G. Kasparov, Novikov’s conjecture on higher signatures: The operator $K$ -theory approach, Contemporary Math 145 (1993) p. 79-99 | Zbl

[66] G. Kasparov & G. Skandalis, Groups acting on buildings, operator $K$ -theory, and Novikov’s conjecture, $K$ -Theory 4 (1991) no.4 p. 303-337 | MR | Zbl

[67] G. Kasparov & G. Skandalis, Groups acting properly on bolic spaces and the Novikov conjecture, Ann. of Math 158 (2003) p. 165-206 | MR | Zbl

[68] N. Keswani, Geometric $K$ -homology and controlled paths, New York J. Math 5 (1999) p. 53-81 | MR | Zbl

[69] V. Lafforgue, $K$ -théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes, Invent. Math 149 (2002) no.1 p. 1-95 | MR | Zbl

[70] M. Lesch, Operators of Fuchs type, conical singularities, and asymptotic methods, Teubner-Texte zur Mathematik 136, B. G. Teubner Verlagsgesellschaft mbH, 1997 | Zbl

[71] B. Lawson & M-L. Michelsohn, Spin Geometry, Princeton mathematical series 38, Princeton University Press, 1989 | MR | Zbl

[72] E. Leichtnam, J. Lott & P. Piazza, On the homotopy invariance of higher signatures for manifolds with boundary, Journal of Differential Geometry 54 (2000) p. 561-633 | MR | Zbl

[73] E. Leichtnam, W. Lück & M. Kreck, On the cut-and-paste property of higher signatures on a closed oriented manifold, Topology 41 (2002) p. 725-744 | MR | Zbl

[74] E. Leichtnam & P. Piazza, The $b$ -pseudo-differential calculus on Galois coverings and a higher Atiyah-Patodi-Singer index theorem, Mémoires de la Société Mathématiques de France 68 (1997) | Numdam | Zbl

[75] E. Leichtnam & P. Piazza, Spectral sections and higher Atiyah-Patodi-Singer index theory on Galois coverings, GAFA 8 (1998) p. 17-58 | MR | Zbl

[76] E. Leichtnam & P. Piazza, A Higher Atiyah-Patodi-Singer Index theorem for the signature operator on Galois Coverings, Annals of Global Analysis and Geometry 18 (2000) p. 171-189 | MR | Zbl

[77] E. Leichtnam & P. Piazza, Homotopy invariance of twisted higher signatures on manifolds with boundary, Bull. Soc. Math. France 127 (1999) p. 307-331 | Numdam | MR | Zbl

[78] E. Leichtnam & P. Piazza, On higher eta invariants and metrics of positive scalar curvature, $K$ -Theory 24 (2001) p. 341-359 | MR | Zbl

[79] E. Leichtnam & P. Piazza, Dirac index classes and the noncommutative spectral flow, Jour. Funct. Anal 200 (2003) p. 348-400 | MR | Zbl

[80] E. Leichtnam & P. Piazza, Etale Groupoids, eta invariants and index theory, e-print. To appear in J. Reine Angew. Math, math.DG/0308184, August 2003 | Zbl

[81] E. Leichtnam & P. Piazza, Cut-and-Paste on Foliated Bundles, e-print. To appear in the Contemporary Mathematics. Volume Spectral Geometry of Manifolds with boundary (ed. B. Booss-Bavnbek, G. Grubb, K. Wojciechowski), math.DG/0407401, July 2004 | MR | Zbl

[82] J. Lott, Superconnections and higher index theory, GAFA 2 p. 421-454 | MR | Zbl

[83] J. Lott, Higher eta invariants, $K$ -Theory 6 (1992) p. 191-233 | MR | Zbl

[84] J. Lott, Diffeomorphisms and noncommutative analytic torsion, Memoirs American Math. Soc 141 (1999) | MR | Zbl

[85] J. Lott, The zero-in-the-spectrum question, Enseign. Math (2) 42 (1996) no.3-4 p. 341-376 | MR | Zbl

[86] J. Lott, Signatures and higher signatures on $S^{1}$ -quotients, Math. Annalen 316 (2000) p. 617-657 | MR | Zbl

[87] J. Lott & W. Lück, $L^{2}$ -topological invariants of $3$ -manifolds, Inventiones Math 120 p. 15-60 | MR | Zbl

[88] W. Lueck & H. Reich, The Baum-Connes and the Farrell-Jones Conjectures in $K -$ and $L -$ Theory, To appear in $K -$ Theory handbook | Zbl

[89] G. Lusztig, Novikov's higher signature and families of elliptic operators, J. Differential Geometry 7 (1972) p. 229-256 | MR | Zbl

[90] V. Mathai, The Novikov conjecture for low degree cohomology classes, Geometriae Dedicata 99 (2003) p. 1-15 | MR | Zbl

[91] R. Mazzeo & P. Piazza, Dirac operators, heat kernels and microlocal analysis. II: Analytic surgery., Rend. Mat. Appl. (7) 18 (1998) no.2 p. 221-288 | MR | Zbl

[92] R. Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics 4, A.K. Peters, 1993 | MR | Zbl

[93] R. Melrose & V. Nistor, Homology of pseudodifferential operators I (manifolds with boundary), to appear in Amer. J. Math, 2003

[94] R. Melrose & P. Piazza, Families of Dirac operators, boundaries and the $b$ -calculus, Journal of Differential Geometry 46 (1997) p. 99-180 | MR | Zbl

[95] R. Melrose & P. Piazza, An index theorem for families of Dirac operators on odd-dimensional manifolds with boundary, Journal of Differential Geometry 46 (1997) p. 287-334 | MR | Zbl

[96] J. Milnor & J. Stasheff, Characteristic Classes, Annals of Math. Studies 76 (1974) | MR | Zbl

[97] I. Mineyev & G. Yu, The Baum-Connes conjecture for hyperbolic groups, Invent. Math (2002) p. 97-122 | MR | Zbl

[98] A. Mishchenko, Homotopy invariants of non-simply connected manifolds. I: Rational Invariants, Math. USSR-Izvestija 4 (1970) p. 509-519 | Zbl

[99] A. Mishchenko, $C^{*} -$ algebras and $K -$ theory, Lecture notes in Math 763, Springer, 1979, p. 262-274 | Zbl

[100] A. Mishchenko & A. Fomenko, The index of elliptic operators over $C^{*}$ -algebras (Russian), Izv. Akad. Nauk SSSR Ser. Mat 43 (1979) no.4 p. 831-859 | MR | Zbl

[101] W. Müller, Signature defects of cusps of Hilbert modular varieties and values of L-series at $s = 1$ , J. Diff. Geometry 20 (1984) p. 55-119 | MR | Zbl

[102] W. Müller, $L^{2}$ -index theory, eta invariants and values of L-functions, Contemporary Mathematics 105 (1990) p. 145-187 | Zbl

[103] W. Neumann, Manifold cutting and pasting groups, Topology 14 (1975) p. 237-244 | MR | Zbl

[104] C. Ogle, Assembly maps, $K$ -theory, and hyperbolic groups, K-Theory 6 (1992) no.3 p. 235-265 | MR | Zbl

[105] P. Piazza, On the index of elliptic operators on manifolds with boundary, J. Funct. Anal 117 (1993) no.2 p. 308-359 | MR | Zbl

[106] P. Piazza & T. Schick, Bordism, rho-invariants and the Baum-Connes conjecture, e-print, math.KT/0407388, July 2004

[107] D. Quillen, Superconnections and the Chern character, Topology 24 (1985) p. 89-95 | MR | Zbl

[108] M. Ramachandran, Von Neumann index theorems for manifolds with boundary, Journal of Differential Geometry 38 (1993) no.2 p. 315-349 | MR | Zbl

[109] A. Ranicki, Exact sequences in the algebraic theory of surgery, Princeton University Press, 1981 | MR | Zbl

[110] J. Rosenberg, $C^{*}$ -algebras, positive scalar curvature and the Novikov conjecture, Publ. Math. IHES 58 (1983) p. 197-212 | Numdam | MR | Zbl

[111] J. Rosenberg, $C^{*}$ -algebras, positive scalar curvature and the Novikov conjecture II, Pitman Res. Notes Math 123, 1986, p. 342-374 | Zbl

[112] J. Rosenberg, $C^{*}$ -algebras, positive scalar curvature and the Novikov conjecture III, Topology 25 (1986) p. 319-336 | MR | Zbl

[113] T. Schick, Operator algebra and Topology, ICTP Lect. Notes Volume 9, 2002, p. 571-660 | Zbl

[114] T. Schick, Index, KK and connections, Preprint, 2003

[115] Ju. P. Solov'Ev, Discrete subgroups, Bruhat-Tits buildings and homotopy invariance of higher signatures (Russian), Uspehi Mat. Nauk 31 (1976) no.1(187) p. 261-262 | MR | Zbl

[116] Yu. P. Solovyov & E.V. Troitsky, $C^{*}$ -algebras and elliptic operators in differential topology. Translated from the 1996 Russian original by Troitsky., Mathematical Monographs 192, American Mathematical Society, 2001 | Zbl

[117] S. Stolz, Positive Scalar Curvature Metrics. Existence and Classification Questions, Birkhäuser Verlag, 1994 | Zbl

[118] B. Tsygan, Homology of matrix Lie algebras over rings and Hochschild homology, Russian Math. Surv 38 (1983) no.2 p. 198-199 | MR | Zbl

[119] A. Valette, Introduction to the Baum-Connes conjecture. From notes taken by Indira Chatterji. With an appendix by Guido Mislin., Lectures in Mathematics ETH Zürich., Birkhäuser Verlag, 2002 | MR | Zbl

[120] N. Wegge-Olsen, $K -$ Theory and $C^{*} -$ algebras, Oxford University Press, 1993 | MR | Zbl

[121] S. Weinberger, Aspects of the Novikov conjecture., Contemp. Math. 105, Amer. Math. Soc., 1990, p. 281-297 | Zbl

[122] S. Weinberger, Higher $ρ$ -invariants, Contemporary Mathematics 231, 1999 | Zbl

[123] G. Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math 139 (2000) no.1 p. 201-240 | MR | Zbl

[124] F. Wu, The noncommutative spectral flow, unpublished preprint, 1997

[125] F. Wu, The Higher $Γ -$ index for coverings of manifolds with boundaries, Fields Institute Communications 17 (1997) p. 169-183 | MR | Zbl

[<L>16</L>] J.-M. Bismut & J. Cheeger, Families index for manifolds with boundary, superconnections and cones. II, J. Funct. Anal. 90 (1990) no.2 p. 306-354 | MR | Zbl

[<L>27</L>] A. Connes, Noncommutative geometry. Part I: the Chern character in $K$ -homology. Part II: de Rham homology and noncommutative algebras, Publications I.H.E.S. 62 (1985) p. 257-360 | MR

[<L>62</L>] G. Kasparov, Topological invariants of elliptic operators $K$ -homology, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975) no.4 p. 796-838 | MR | Zbl

[63] G. Kasparov, $K$ -Theory, group $C^{*}$ -algebras and higher signatures (Conspectus), London Math. Soc. Lecture Note Ser. 226, Cambridge Univ. Press, 1995, p. 101-146 | Zbl

[<L>98</L>] A. Mishchenko, Homotopy invariants of non-simply connected manifolds. I: Rational Invariants, Math. USSR Izv. 15 (1980) p. 87-112 | Zbl

[<L>115</L>] Ju. P. Solov'Ev, Discrete subgroups, Bruhat-Tits buildings and homotopy invariance of higher signatures (English translation), Russian Math. Survey 31 (1976) no.1 | MR | Zbl

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