Geodesic
Defect of index in the theory of non-local problems and the $\eta$-invariant
Sbornik. Mathematics, Tome 195 (2004) no. 9, pp. 1321-1358

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This paper is concerned with elliptic theory on manifolds the boundary of which is a cover. Non-local boundary value problems are considered and their indices are calculated. The Atiyah–Patodi–Singer problem is studied on such manifolds. For non-trivial covers the defect of the index is calculated. The Poincaré duality is constructed in $K$-theory on the corresponding singular manifolds. 
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     author = {A. Yu. Savin and B. Yu. Sternin},
     title = {Defect of index in the theory of non-local problems and the $\eta$-invariant},
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A. Yu. Savin; B. Yu. Sternin. Defect of index in the theory of non-local problems and the $\eta$-invariant. Sbornik. Mathematics, Tome 195 (2004) no. 9, pp. 1321-1358. http://geodesic.mathdoc.fr/item/SM_2004_195_9_a5/