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. 
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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/

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