Abstract
We suggest a general version of the relative index formula related to surgery and apply it to obtain index formulas for elliptic pseudodifferential operators and Fourier integral operators on manifolds with conical singularities.
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References
Agranovich, M., Elliptic singular integro-differential operators, Uspekhi Matem. Nauk (5) 20 (1965), 3–20. [Russian].
Agranovich, M., Elliptic boundary problems, In Partial Differential Equations IX. Elliptic Boundary Value Problems, number 79 in Encyclopaedia of Mathematical Sciences, M. Agranovich, Yu.V. Egorov, and M.A. Shubin (eds.), Berlin–Heidelberg, Springer–Verlag 1997, pp. 1 – 144.
Anghel, N., An abstract index theorem on non-compact Riemannian manifolds, Houston J. of Math. 19 (1993), 223 – 237.
Atiyah, M.F., Global theory of elliptic operators, In Proc. of the Int. Symposium on Functional Analysis, Tokyo, University of Tokyo Press 1969, pp. 21 – 30.
Booß–Bavnbek, B. and Wojciechowski, K., Elliptic boundary problems forDirac operators, Birkhäuser, Boston–Basel–Berlin 1993.
Bunke, U., Relative index theory, J. Funct. Anal. 105 (1992), 63 – 76.
Dezin, A.A., Invariant differential operators and boundary value problems,vol. 68 of Trudy MIAN. AN SSSR, Moscow 1962. [Russian].
Donnelly, H, Essential spectrum and the heat kernel, J. Funct. Anal. 75 (1987), 362 – 381.
Dynin, A.S., Multidimensional elliptic boundary problems with one unknown function, DAN USSR 141 (1961), 285–287. [Russian].
Epstein, C. and Melrose, R., Contact degree and the index of Fourier integral operators, Math. Res. Lett. (3) 5 (1998), 363 – 381.
Fedosov, B.V., Schulze, B.-W., and Tarkhanov, N., The index of higher order operators on singular surfaces, Pacific J. of Math. (1) 191 (1999), 25– 48.
Fedosov, B.V., Schulze, B.-W., and Tarkhanov, N., A remark on the index of symmetric operators, Univ. Potsdam, Institut für Mathematik, Potsdam, February 1998, Preprint N 98/4.
Gilkey, P.B., Invariance theory, the heat equation and the Atiyah–Singer index theorem, Publish of Perish. Inc., Wilmington Delawaere 1984.
Gilkey, P.B. and Smith, L., The η invariant for a class of elliptic boundary value problems, Comm. Pure Appl. Math. 36 (1983), 85 – 132.
The twisted index problem for manifolds with boundary, J. Diff. Geometry (3) 18 (1983), 393–444.
Gromov, M. and Lawson Jr, H.B., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. IHES 58 (1983), 295 – 408.
Hsiung, Ch.-Ch., The signature and G-signarute of manifolds with boundary, J. Diff. Geometry 6 (1972), 595 – 598.
]Hsiung, Ch.-Ch., A remark on cobordism of manifolds with boundary, Arch. Math. XXVII (1976), 551 – 555.
Leichtnam, E., Nest, R., and Tsygan, B., Local formula for the index of a Fourier integral operator, preprint math.DG/0004022, 2000.
Melrose, R., Transformation of boundary problems, Acta Math. 147 (1981), 149 – 236.
Nazaikinskii, V., Schulze, B.-W., and Sternin, B., The index of quantized contact transformations on manifolds with conical singularities, Univ. Potsdam, Institut für Mathematik, Potsdam, August 1998, Preprint N 98/16.
Nazaikinskii, V. and Sternin, B., Localization and surgery in index theory of elliptic operators. In Conference: Operator Algebras and Asymptotics on Manifolds with Singularities, Warsaw, Stefan Banach International Mathematical Center, Universität Potsdam, Institut für Mathematik 1999, pp. 27 – 28.
Nazaikinskii, V. and Sternin, B., Localization and surgery in the index theory to elliptic operators, Russian Math. Dokl. (1) 370 (2000), 19 – 23.
Nazaikinskii, V. and Sternin, B., Surgery of manifolds and the relative index to elliptic operators, Univ. Potsdam, Institut für Mathematik, Potsdam, Juli 1999, Preprint N 99/17.
Nazaikinskii, V. and Sternin, B., The index locality principle in elliptic theory, Funktsionalayi Ancliz i Prilozhen. (2) 35 (2001), 37–52. [Russian].
Schulze, B.-W., Sternin, B., and Shatalov, V., Differential equations on singular manifolds. Semiclassical theory and operator algebras, vol. 15 of Mathematics Topics, Wiley–VCH Verlag, Berlin–New York 1998.
Schulze, B.-W., Sternin, B., and Shatalov, V., On the index of differential operators on manifolds with conical singularities, Annals of Global Analysis and Geometry (2) 16 (1998), 141 – 172.
Stong, R.E., Manifolds with reflecting boundary, J. Diff. Geometry 9 (1974), 465 – 474.
Teleman, N., The index of signature operators on Lipschitz manifolds, Publ. Math. IHES 58 (1984), 39 – 78.
Volpert, A.I., On index and normal solvability to boundary value problems for elliptic systems of differential equations on the plane, Trudy Mosk. Matem. Ob-va 10 (1961),41–87. [Russian].
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Nazaikinskii, V.E., Sternin, B.Y. (2003). A Remark on Surgery in Index Theory of Elliptic Operators. In: de Gosson, M. (eds) Jean Leray ’99 Conference Proceedings. Mathematical Physics Studies, vol 24. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2008-3_24
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DOI: https://doi.org/10.1007/978-94-017-2008-3_24
Publisher Name: Springer, Dordrecht
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