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Positive Curvature, Macroscopic Dimension, Spectral Gaps and Higher Signatures

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Functional Analysis on the Eve of the 21st Century Volume II

Part of the book series: Progress in Mathematics ((PM,volume 132))

Abstract

Our journey starts with a macroscopic view of Riemannian manifolds with positive scalar curvature and terminates with a glimpse of the proof of the homotopy invariance of some Novikov higher signatures of non-simply connected manifolds. Our approach focuses on the spectra of geometric differential operators on compact and non-compact manifolds V where the link with the macroscopic geometry and topology is established with suitable index theorems for our operators twisted with almost flat bundles over V. Our perspective mainly comes from the asymptotic geometry of infinite groups and foliations.

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References

  1. N. Anghel, An abstract index theorem on non-compact Riemannian manifolds, Houston J. Math. 19:2 (1993), 223–237.

    MathSciNet  MATH  Google Scholar 

  2. M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry, Part I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43–69

    Article  MathSciNet  MATH  Google Scholar 

  3. M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry, Part II, Math. Proc. Camb. Phil. Soc. 78 (1975), 405–432

    Article  MathSciNet  MATH  Google Scholar 

  4. M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry, Part III, Math. Proc. Camb. Phil. Soc. 79 (1976), 71–99.

    Article  MathSciNet  MATH  Google Scholar 

  5. S.C. de Almeida, The geometry of manifolds of nonnegative scalar curvature, dissertation, Stony Brook, 1982.

    Google Scholar 

  6. M.F. Atiyah, Eigenvalues of the Dirac operator, In Proceedings of the 25th Math. Arbeitstagung, Bonn 1984, 251–260; Lecture Notes in Math., (1985) Vol. 1111.

    Google Scholar 

  7. M.F. Atiyah, Global aspects of the theory of elliptic differential operators, Proc. ICM-1966 (1968), 57–66, Moscow.

    Google Scholar 

  8. M.F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Asterisque (1976) no. 32–33, 43–72.

    Google Scholar 

  9. M.F. Atiyah, The signature of fibre bundles, Global Analysis, papers in honor of K. Kodaira, Princeton Univ. Press, 1969, 73–84.

    Google Scholar 

  10. P. Baum and R. Douglas K-theory and index theory, Operator algebras and applications, Proc. Symp. Pure Math., 38 (I) (1982), 117–173.

    MathSciNet  Google Scholar 

  11. C. Bavard, Courbure presque négative en dimension 3, Comp. Mathematica 63 (1987), 223–236.

    MathSciNet  MATH  Google Scholar 

  12. G. Besson, G. Courtois, S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, preprint Institut Fourier, (1994) no. 28.

    Google Scholar 

  13. G. Besson, G. Courtois, S. Gallot, Le volume et l’entropie minimal des espaces localement symétriques, Invent. Math. 103 (1991), 417–445.

    Article  MathSciNet  MATH  Google Scholar 

  14. M.E.B. Bekka, P.-A. Cherix and A. Valette, Proper affine isometric actions of amenable groups, preprint.

    Google Scholar 

  15. P.H. Berard, The Bochner technique revisited, Bull. AMS, Vol. 19 (1988), no. 2, 371–406.

    Article  MathSciNet  MATH  Google Scholar 

  16. L. Bérard-Bergery, La courbure scalaire des variétés riemanniennes, Séminaire Bourbaki, 32e année, no. 556, 1979–80.

    Google Scholar 

  17. Y. Burago, M. Gromov and G. Perelman, A. D. Alexandrov’s spaces with curvature bounded below, Uspekhi Mat. Nauk 47, 3–51 (1992); English transi, in Russian Math. Surveys 47 (1992).

    MathSciNet  Google Scholar 

  18. D.E. Blair, The “total scalar curvature” as a symplectic invariant and related results, preprint.

    Google Scholar 

  19. P. Buser and D. Gromoll, Gromov’s examples of almost negatively curved metrics on \( \mathbb{S} \) 3, notes.

    Google Scholar 

  20. A. Connes, M. Gromov and H. Moscovici, Conjecture de Novikov et fibrés prsque plats, C.R. Acad. Sci. Paris, t. 310, Série I, 273–277.

    Google Scholar 

  21. A. Connes, M. Gromov and H. Moscovici, Group cohomology with Lipschitz control and higher signatures, Geometric and Functional analysis, 3 (1990), 1–78.

    Article  MathSciNet  Google Scholar 

  22. G. Carlson and E. Pedersen, Controlled algebra and the Novikov conjectures for K- and L-theory, preprint, 1993.

    Google Scholar 

  23. S. Cappell, On homotopy invariance of higher signatures, Inventions Math. 33 (1976), 171–179.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Cheeger, M. Gromov, On the characteristic numbers of complete manifolds of bounded curvature and finite volume, in Diff. Geom. and Complex Analysis, Rauch Memorial Volume, I. Chavel and H.M. Farkas, eds., Springer, Berlin (1985).

    Google Scholar 

  25. J. Cheeger, M. Gromov, Bounds on the Von Neumann dimension of L 2-cohomology and the Gauss-Bonnet theorem for open manifolds, J. Diff. Geom. 21 (1985), 1–34.

    MathSciNet  MATH  Google Scholar 

  26. J. Cheeger, M. Gromov, L 2-cohomology and group cohomology, Topology, vol. 25, no 2 (1986), 189–215.

    Article  MathSciNet  MATH  Google Scholar 

  27. A. W. Chou, Remarks on the Dirac operator of pseudomanifolds, preprint.

    Google Scholar 

  28. A.W. Chou, The Dirac operator on spaces with conical singularities and positive scalar curvatures, preprint.

    Google Scholar 

  29. Y. Colin de Verdière, Spectres de variétés riemanniennes et spectres de graphes, Proc. ICM-1986 in Berkeley, A.M.S. Vol. 1 (1987), 522–530.

    Google Scholar 

  30. A. Connes and N. Higson, Déformations, morphismes asymptotiques et K-théorie bivariante, preprint.

    Google Scholar 

  31. A. Connes and N. Higson, Almost homomorphisms and KK-theory, preprint.

    Google Scholar 

  32. A. Connes and H. Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology, 29, 345 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Connes and G. Skandalis, The longitudinal index theorem for foliations, Pubi. Res. Inst. Math. Sci. Kyoto Univ., 20 (1984), 1139–1183.

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Connes, Non Commutative Geometry, Academic Press, 1994.

    Google Scholar 

  35. A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, in Geometric methods in Operator Algebras, H. Araki and E.G. Effros, eds., Pitman, Research Notes in Math. Series 123, 1986, 52–144.

    Google Scholar 

  36. J.-P. Demailly, Champs magnétiques et inégalités de Morsepour la d 11-cohomology, Ann. Inst. Fourier 35, 4 (1985), 189–229.

    Article  MathSciNet  MATH  Google Scholar 

  37. F.T. Farrell and W.C. Hsiang, On Novikov’s conjecture for non-positively curved manifolds, Ann. Math. 113 (1981), 197–209.

    Article  MathSciNet  Google Scholar 

  38. F.T. Farrell and L.E. Jones, Rigidity in Geometry and Topology, Proc. 1990 I.C.M., Kyoto, 1991, 653–663.

    Google Scholar 

  39. D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure and Appl. Math 33 (1980), 199–211.

    Article  MathSciNet  MATH  Google Scholar 

  40. S.C. Ferry and S. Weinberger, A coarse approach to the Novikov conjecture, preprint, 1993.

    Google Scholar 

  41. S.C. Ferry and S. Weinberger, A flexible uniformly aspherical manifold, preprint.

    Google Scholar 

  42. S. Gallot, Inégalités isopérimétriques, courbure de Ricci et invariants géométriques, I and II, C.R. Acad. Sci. Paris 296 (1983), 333–336 and 365–368.

    MathSciNet  MATH  Google Scholar 

  43. I.M. Gelfand and A.S. Miščenko, Quadratic forms over commutative group rings and K-theory, Funkcional. Anal, i Priložen 3 (1969), 28–33 (Russian).

    Google Scholar 

  44. M. Gromov, J. Lafontaine and P. Pansu, Structures métriques pour les variétés riemanniennes, Cedic-Fernand Nathan, Paris, 1981.

    MATH  Google Scholar 

  45. M. Gromov and H.B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. I.H.E.S., 58 (1983), 83–196.

    MathSciNet  MATH  Google Scholar 

  46. M. Gromov and H.B. Lawson, Spin and scalar curvature in the presence of a fundamental group, Ann. of Math. 111 (1980), 209–230.

    Article  MathSciNet  MATH  Google Scholar 

  47. W. Goldman, Representations of fundamental groups of surfaces, Geometry and Topology, Proceedings, Univ. of Maryland 1983–84, J. Alexander and J. Harer eds., Lecture Notes in Math. 1167 (1985), 95–117.

    Google Scholar 

  48. M. Gromov and M. Shubin, Von Neumann spectra near zero, GAFA 1:4(1991), 375–404.

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Gromov, Almost flat manifolds, J. Diff. Geom. 13 (1978), 231–241.

    MathSciNet  MATH  Google Scholar 

  50. M. Gromov, Asymptotic invariants of infinite groups, Lond. Math. Soc. Lecture notes 182, ed. Niblo and Koller, Cambridge Univ. Press 1993.

    Google Scholar 

  51. M. Gromov, Carnot-caratheodory spaces seen from within, preprint IHES (1994).

    Google Scholar 

  52. M. Gromov, Foliated plateau problem, parts I, II, Geometric and Functional Analysis 1:1, 14–79 (1991); 253–320 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  53. M. Gromov, Filling Riemannian manifolds, Journal of diff. Geometry 18, 1–147 (1983).

    MathSciNet  MATH  Google Scholar 

  54. M. Gromov, Hyperbolic groups, in Essays in group theory, S.M. Gersten, ed. Math. Sciences Research Institute Publication, 8, Springer-Verlag, New York, Inc., 1987, 75–264.

    Google Scholar 

  55. M. Gromov, Kähler hyperbolicity and L 2 -Hodge theory, J. Diff. Geom. 33 (1991), 263–292.

    MathSciNet  MATH  Google Scholar 

  56. M. Gromov, Large Riemannian manifolds, Lecture Notes in Math., no 1201 (1985), 108–122.

    Google Scholar 

  57. M. Gromov, Metric invariants of Kahler manifolds, preprint IHES.

    Google Scholar 

  58. M. Gromov, Partial differential relations, Springer-Verlag (1986).

    MATH  Google Scholar 

  59. M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent Math. 82 (1985), 307–347.

    Article  MathSciNet  MATH  Google Scholar 

  60. M. Gromov, Paul Levy’s isoperimetric inequality, Preprint IHES (1980).

    Google Scholar 

  61. M. Gromov, Rigid transformation groups, in Géométrie Différentielle, Paris (1986), 65–139; (Bernard, Choquet-Bruhat), Travaux en cours, 33 (1988), Hermann, Paris.

    Google Scholar 

  62. M. Gromov and M. Shubin, Von Neumann spectra near zero, GAFA 1:4 (1991), 375–404.

    Article  MathSciNet  MATH  Google Scholar 

  63. M. Gromov, Sign and geometric meaning of curvature, Sem. Mat. e Fis. di Milano, LXI (1991), 10–123, Pavia 9 (1994).

    Google Scholar 

  64. M. Gromov, Volume and bounded cohomology, Publ. Math. IHES 56, 5–100 (1982).

    MathSciNet  MATH  Google Scholar 

  65. M. Gromov, Width and related invariants of Riemannian manifolds, Astérisque 163–164, 93–109 (1988).

    Google Scholar 

  66. J. Hersch, Quatre propriétés isopérimétriques des membranes sphériques homogènes, C.R. Acad. Sci. Paris 270 (1970), 1645–1648.

    Google Scholar 

  67. N. Hitchin, Harmonie spinors, Adv. in Math. 14 (1974), 1–55.

    Article  MathSciNet  MATH  Google Scholar 

  68. N. Higson and J. Roe, A homotopy invariance theorem in coarse cohomology and K-theory, preprint (1993).

    Google Scholar 

  69. M. Hilsum and G. Skandalis, Invariance par homotopie de la signature à coeffcients dans un fibré presque plat, Jour. Reine Angew. Math., 423 (1992), 73–99.

    Article  MathSciNet  Google Scholar 

  70. N. Higson, K-homology and operators on non-compact manifolds, preprint.

    Google Scholar 

  71. O. Hijazi, A conformai lower bound for the smallest eigenvalue of the Dirac operator and killing spinors, Comm. Math. Phys. 104 (1986), 151–162.

    Article  MathSciNet  MATH  Google Scholar 

  72. H. Hess, R. Schrader, D. Uhlenbrock, Kato’s inequality and the spectral distribution of Laplacians on compact Riemannian manifold, J. Diff. Geom., 15 (1980), 27–39.

    MathSciNet  MATH  Google Scholar 

  73. S. Hurder, Coarse geometry of foliations, preprint IHES, 1994.

    Google Scholar 

  74. S. Hurder, Eta invariants and the odd index theorem for coverings, Cont. Math., (1989).

    Google Scholar 

  75. S. Hurder, Exotic index theory and the Novikov conjecture, preprint.

    Google Scholar 

  76. S. Hurder, Exotic index theory for foliations, preprint, Univ. of Illinois (1993).

    Google Scholar 

  77. S. Hurder, Topology of covers and spectral theory of geometric operators, In Proc. Conference on K-homology and index theory, J. Fox, ed., Providence, Amer. Math. Soc. Contemp. Math. Vol. 148 (1993).

    Google Scholar 

  78. J. Kaminker and J.G. Miller, Homotopy invariance of the analytic index of signature operators over C*-algebras, J. Operator Theory, 14 (1985), 113–127.

    MathSciNet  MATH  Google Scholar 

  79. G. Kasparov, Novikov’s conjecture on higher signatures: The operator K-theory approach, Contemporary Math., Vol. 145 (1993).

    Google Scholar 

  80. G. Kasparov and G. Skandalis, Groupes “boliques” et conjecturede Novikou, preprint.

    Google Scholar 

  81. M. Katz, The first diameter of 3-manifolds of positive scalar curvature, Proc. of the Amer. Math. Soc., vol. 104 (1988), no 2.

    Google Scholar 

  82. J. Kazdan and F. Warner, Prescribing curvatures, Proc. Symp. in Pure Math. 27 (1975), 309–319.

    MathSciNet  Google Scholar 

  83. J.L. Kazdan, Positive energy in general relativity, Séminaire Bourbaki, 34e année, no 593, Astérisque (1982).

    Google Scholar 

  84. R. Kirby and L. Siebenmann, Foundational essays on topological manifolds smoothings and triangulations, Ann. Math. Stud. 88 (1977), Princeton.

    Google Scholar 

  85. F. Lalonde and D. McDuff, The geometry of symplectic energy, to appear in Annals of Math.

    Google Scholar 

  86. H.B. Lawson, Jr. and M.-L. Michelsohn, Spin Geometry, Vol.38, Princeton Math. Series, Princeton Univ. Press, 1989.

    MATH  Google Scholar 

  87. M. Llarull, Scalar curvature estimates for (n+4k)-dimensional manifolds, preprint.

    Google Scholar 

  88. M. Llarull, Sharp estimates and the Dirac operator, preprint.

    Google Scholar 

  89. J. Lohkamp, Curvature h-principles, to appear in Ann. Math.

    Google Scholar 

  90. J. Lohkamp, Global and local curvatures, preprint.

    Google Scholar 

  91. J. Lohkamp, On the geometry of negative Ricci and scalar curvature, Max-Planck-Institut für Math., Bonn.

    Google Scholar 

  92. J. Lott, Heat kernels on covering spaces and topological invariants, J. Diff. Geom., 35 (1992), 471–510.

    MathSciNet  MATH  Google Scholar 

  93. G. Lusztig, Novikov’s higher signature and families of elliptic operators, J. Diff. Geom. 7 (1971), 229–256.

    MathSciNet  Google Scholar 

  94. V. Mathai, Non negative scalar curvatures, preprint, Univ. of Adelaide.

    Google Scholar 

  95. W. Meyer, Die Signatur von lokalen Koeffizientensystemen und Faserbündlen, Bonn. Math. Schr. 53 (1972).

    Google Scholar 

  96. M. Micallef, J. Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two planes, Ann. Math. 127:1 (1988), 199–227.

    Article  MathSciNet  MATH  Google Scholar 

  97. A.S. Mishchenko and A.T. Fomenko, The index of elliptic operators over C*-algebras, Math. USSR-Izv. 15 (1980), 87–112.

    Article  MATH  Google Scholar 

  98. J. Milnor and D. Husemoller, Symmetric bilinear forms, Springer Verlag, Heidelberg, New York, 1973.

    MATH  Google Scholar 

  99. J. Milnor and J. Stasheff, Characteristic classes, Ann. Math. Stud. 76 (1974), Princeton.

    Google Scholar 

  100. M. Min-Oo, A partial extension of a theorem of Llarull to other compact symmetric spaces, preprint.

    Google Scholar 

  101. M. Min-Oo, Scalar curvature rigidity of asymptotically hyperbolic spin manifolds, Math. Ann. 285, 527–539 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  102. A.S. Mishchenko, Infinite dimensional representations of discrete groups and higher signatures, English transl., Math. USSR-Izv 8, 85–112 (1974).

    Article  Google Scholar 

  103. Novikov conjectures, index theorems and rigidity, Proc. of an Oberwollfach meeting held on Sept. 1993. To be published by Cambridge Univ. Press, in the London Math. Soc. Lecture Notes series.

    Google Scholar 

  104. S.P. Novikov, M.A. Shubin, Morse theory and von Neumann invariants on non-simply connected manifolds, Uspekhi Matem. Nauk, 41 (1986), 222–223 (in Russian.

    Google Scholar 

  105. K. Ono, The scalar curvature and the spectrum of the Laplacian on spin manifolds, Math. Ann., (1988), 163–168.

    Google Scholar 

  106. G. Perelman, Spaces with curvature bounded below, Proc. ICM-1994, Zurich, to appear.

    Google Scholar 

  107. E.K Pedersen, J. Roe and S. Weinberger, On the homotopy invariance of the boundedly controlled analytic signature of a manifold over an open cone, preprint, 1994.

    Google Scholar 

  108. A. Ranicki, Algebraic L-theory and topological manifolds, 102, Cambridge Tracts in Mathematics, Cambridge University Press, 1992.

    Google Scholar 

  109. A. Ranicki, The Hauptvermutung book, a collection of papers by Casson, Sullivan, Armstrong, Rourke, Cooke and Ranicki, K-theory Journal book series, to appear.

    Google Scholar 

  110. A. Ranicki, Lower K- and L-theory, London Math. Soc., Lecture Note Series 178, Cambridge University Press.

    Google Scholar 

  111. A. Ranicki, On the Novikov conjecture, preprint.

    Google Scholar 

  112. J. Roe, Coarse cohomomology and index theory on complete Riemannian manifolds, Memoirs of the Amer. Math. Soc., 497 (1993), Vol. 104.

    Google Scholar 

  113. J. Roe, Partitioning non-compact manifolds and the dual Toeplitz problem, preprint.

    Google Scholar 

  114. J. Rosenberg, Analytic Novikov for topologists, preprint.

    Google Scholar 

  115. J. Rosenberg, C*-algebras, positive scalar curvature, and the Novikov conjecture, Part I, IHES Publ. Math. 58, 197–212 (1983), Part III, Top. Vol.25 (1986), no.3, 319–336.

    Google Scholar 

  116. J. Rosenberg, K and KK: Topology and operator algebras, preprint.

    Google Scholar 

  117. J. Rosenberg and S. Stolz, Manifolds of positive scalar curvature, Algebraic Topology and its applications (G. Carlsson, R. Cohen, W.-C. Hsiang and J.D.S. Jones eds.), M.S.R.I. Publ., Vol. 27 (1994), Springer, New York, 241–267.

    Chapter  Google Scholar 

  118. R. Schoen, Minimal manifolds and positive scalar curvature, Proc. Internat. Congress of mathematicians, Warsaw, 1983, Polish Scientific Publishers and North Holland, Warsaw, 1984, 575–578.

    Google Scholar 

  119. R. Schoen and S.T. Yau, Existence of incompressible minimal surfaces and the topology of manifolds with nonnegative scalar curvature, Ann. of Math. 110 (1979), 127–142.

    Article  MathSciNet  MATH  Google Scholar 

  120. R. Schoen and S.T. Yau, Proof of the positive action conjecture in quantum relativity, Phys. Rev. Let. 42 (1979) (9), 547–548.

    Article  MathSciNet  Google Scholar 

  121. R. Schoen and S.T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), 45–76.

    Article  MathSciNet  MATH  Google Scholar 

  122. R. Schoen and S.T. Yau, On the structure of manifolds with positive scalar curvature, Man. Math. 28 (1979), 159–183.

    Article  MathSciNet  MATH  Google Scholar 

  123. S. Stolz, to appear in Proc. ICM-1994, Zurich.

    Google Scholar 

  124. D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Inv. Math. 36 (1976), 225–255.

    Article  MATH  Google Scholar 

  125. C. Vafa and E. Witten, Eigenalue inequalities for fermions in gauge theories, Comm. Math. Phys. 95, 257–276 (1984).

    Article  MathSciNet  Google Scholar 

  126. N. Varopoulos, L. Saloff-Coste and Th. Coulhon, Analysis and geometry on groups, Cambridge Univ. Press, 1993.

    Book  Google Scholar 

  127. L. van den Dries and A.J. Wilkie, Gromov’s Theorem on groups of polynomial growth and elementary logic, Journal of Algebra 89 (1984), 349–374.

    Article  MathSciNet  MATH  Google Scholar 

  128. S. Weinberger, Aspect of the Novikov conjecture, preprint.

    Google Scholar 

  129. E. Witten, A New proof of the positive energy theorem, Comm. Math. Phys., 80 (1981), 381–402.

    Article  MathSciNet  Google Scholar 

  130. S.-T. Yau, Minimal surfaces and their role in differential geometry, Global Riemannian Geometry, T.J. Willmore and N.J. Hitchin, eds., Ellis Horwood and Halsted Press, Chichester, England, and New York, 1984, 99–103.

    Google Scholar 

  131. G. Yu, Cyclic cohomology and higher indices for noncompact complete manifolds, preprint (1994)

    Google Scholar 

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  1. Institut des Hautes Etudes Scientifiques, 35 Route de Chartres, 91440, Bures-sur-Yvette, France

    M. Gromov

  2. University of Maryland, 20742, College Park, MD, USA

    M. Gromov

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  1. Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA

    Simon Gindikin , James Lepowsky  & Robert L. Wilson ,  & 

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Gromov, M. (1996). Positive Curvature, Macroscopic Dimension, Spectral Gaps and Higher Signatures. In: Gindikin, S., Lepowsky, J., Wilson, R.L. (eds) Functional Analysis on the Eve of the 21st Century Volume II. Progress in Mathematics, vol 132. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4098-3_1

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