  We study elliptic theory on manifolds with boundary represented as a covering
space. Firstly, we consider boundary value problems, where the boundary
conditions are allowed to mix the values of functions in the fibers of the
covering. We show that elliptic elements define Fredholm operators and prove an
index formula. For the identity covering, our formula reduces to the
Atiyah-Bott index formula for classical boundary value problems. Secondly, we
consider Atiyah-Patodi-Singer boundary value problems for operators adapted to
the covering. We show that the corresponding symbols have a natural homotopy
invariant. This invariant is equal to the index of the corresponding spectral
problem plus the relative eta-invariant. The computation of this invariant in
topological terms is one of the main results of the paper. For a trivial
covering, we recover the mod n-index theorem of Freed-Melrose. Finally, we
prove the Poincare duality and isomorphisms in K-theory of singular spaces
corresponding to our manifolds. The isomorphisms are defined in terms of the
two classes of elliptic operators from the above. Thus, the two elliptic
theories are dual.
