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Surgery and the relative index in elliptic theoryNazaikinskii, Vladimir E., Sternin, Boris Yu. January 1999 (has links)
We prove a general theorem on the local property of the relative index for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions) this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities as well as for elliptic boundary value problems with a symmetry condition for the conormal symbol.
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On the homotopy classification of elliptic operators on manifolds with singularitiesSchulze, Bert-Wolfgang, Nazaikinskii, Vladimir E., Sternin, Boris Yu. January 1999 (has links)
We study the homotopy classification of elliptic operators on manifolds with singularities and establish necessary and sufficient conditions under which the classification splits into terms corresponding to the principal symbol and the conormal symbol.
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On surgery in elliptic theoryNazaikinskii, Vladimir, Sternin, Boris January 2000 (has links)
We prove a general theorem on the behavior of the relative index under surgery for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions), this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities.
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Localization problem in index theory of elliptic operatorsNazaikinskii, Vladimir, Schulze, Bert-Wolfgang, Sternin, Boris January 2001 (has links)
This is a survey of recent results concerning the general index locality principle, associated surgery, and their applications to elliptic operators on smooth manifolds and manifolds with singularities as well as boundary value problems. The full version of the paper is submitted for publication in Russian Mathematical Surveys.
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