Abstract and Applied Analysis

Surgery and the relative index in elliptic theory

V. E. Nazaikinskii and B. Yu. Sternin

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This is a survey article featuring the general index locality principle introduced by the authors, which can be used to obtain index formulas for elliptic operators and Fourier integral operators in various situations, including operators on stratified manifolds and manifolds with singularities.

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Abstr. Appl. Anal., Volume 2006, Special Issue (2006), Article ID 98081, 16 pages.

First available in Project Euclid: 19 March 2007

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Nazaikinskii, V. E.; Sternin, B. Yu. Surgery and the relative index in elliptic theory. Abstr. Appl. Anal. 2006, Special Issue (2006), Article ID 98081, 16 pages. doi:10.1155/AAA/2006/98081. https://projecteuclid.org/euclid.aaa/1174321759

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  • M. S. Agranovich, Elliptic boundary problems, Partial Differential Equations, IX, Encyclopaedia Math. Sci., vol. 79, Springer, Berlin, 1997, pp. 1–144, 275–281.
  • M. S. Agranovich and A. S. Dynin, General boundary-value problems for elliptic systems in higher-dimensional regions, Doklady Akademii Nauk SSSR 146 (1962), 511–514.
  • N. Anghel, An abstract index theorem on noncompact Riemannian manifolds, Houston Journal of Mathematics 19 (1993), no. 2, 223–237.
  • M. F. Atiyah, Global theory of elliptic operators, Proceedings of the International Symposium on Functional Analysis, University of Tokyo Press, Tokyo, 1969, pp. 21–30.
  • M. F. Atiyah and R. Bott, The index problem for manifolds with boundary, Differential Analysis, Bombay Colloquium, Oxford University Press, London, 1964, pp. 175–186.
  • Yu. V. Egorov and B.-W. Schulze, Pseudo-differential operators, singularities, applications, Operator Theory: Advances and Applications, vol. 93, Birkhäuser, Basel, 1997.
  • C. Epstein and R. Melrose, Contact degree and the index of Fourier integral operators, Mathematical Research Letters 5 (1998), no. 3, 363–381.
  • M. Gromov and H. B. Lawson Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Institut des Hautes Études Scientifiques. Publications Mathématiques 58 (1983), 83–196 (1984).
  • E. Leichtnam, R. Nest, and B. Tsygan, Local formula for the index of a Fourier integral operator, Journal of Differential Geometry 59 (2001), no. 2, 269–300.
  • V. P. Maslov, Théorie des Perturbations et Méthod Asymptotiques, Dunod, Paris, 1972, French translation from the Russian 1965 edition.
  • R. B. Melrose, Transformation of boundary problems, Acta Mathematica 147 (1981), no. 3-4, 149–236.
  • A. S. Mishchenko, V. E. Shatalov, and B. Yu. Sternin, Lagrangian Manifolds and the Maslov Operator, Springer Series in Soviet Mathematics, Springer, Berlin, 1990.
  • V. E. Nazaikinskii, A. Savin, B.-W. Schulze, and B. Yu. Sternin, Elliptic Theory on Singular Manifolds, CRC-Press, Florida, 2005.
  • V. E. Nazaikinskii and B. Yu. Sternin, Localization and surgery in index theory of elliptic operators, Conference: Operator Algebras and Asymptotics on Manifolds with Singularities, Stefan Banach International Mathematical Center, Universität Potsdam, Institut für Mathematik, Warsaw, 1999, pp. 27–28.
  • ––––, A remark on elliptic theory on manifolds with isolated singularities, Rossiĭskaya Akademiya Nauk. Doklady Akademii Nauk 374 (2000), no. 5, 606–610.
  • ––––, Localization and surgery in the index theory to elliptic operators, Russian Mathematics. Doklady 370 (2000), no. 1, 19–23.
  • ––––, On the local index principle in elliptic theory, Functional Analysis and Its Applications 35 (2001), no. 2, 111–123.
  • ––––, Surgery and the relative index in elliptic theory, University of Potsdam, Institut für Mathematik, preprint N 99/17, Juli 1999.
  • B.-W. Schulze, B. Yu. Sternin, and V. E. Shatalov, On the index of differential operators on manifolds with conical singularities, Annals of Global Analysis and Geometry 16 (1998), no. 2, 141–172.
  • A. Weinstein, Fourier integral operators, quantization, and the spectra of Riemannian manifolds, Géométrie symplectique et physique mathématique (Colloq. Internat. CNRS, No. 237, Aix-en-Provence, 1974), Éditions Centre Nat. Recherche Sci., Paris, 1975, pp. 289–298.
  • ––––, Some questions about the index of quantized contact transformations, Sūrikaisekikenkyūsho Kōkyūroku (1997), no. 1014, 1–14.