Defect of index in the theory of non-local problems and the $\eta$-invariant

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.