Geodesic
The Index Locality Principle in Elliptic Theory
Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 2, pp. 37-52

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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 some additional conditions (like symmetry conditions), this theorem permits computing 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. 
@article{FAA_2001_35_2_a3,
     author = {V. E. Nazaikinskii and B. Yu. Sternin},
     title = {The {Index} {Locality} {Principle} in {Elliptic} {Theory}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {37--52},
     publisher = {mathdoc},
     volume = {35},
     number = {2},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2001_35_2_a3/}
}
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V. E. Nazaikinskii; B. Yu. Sternin. The Index Locality Principle in Elliptic Theory. Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 2, pp. 37-52. http://geodesic.mathdoc.fr/item/FAA_2001_35_2_a3/